⚛️ phenomenology

Doubly-polarized WZ hadronic cross sections at NLO QCD+EW accuracy

This paper presents next-to-leading order electroweak corrections for doubly-polarized $WZ$ production at the LHC, revealing that these corrections significantly reduce the transverse momentum distributions of doubly longitudinal polarization by approximately 10% at high energies.

Original authors: Duc Ninh Le, Julien Baglio

Published 2026-11-04

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CC BY 4.0

Original authors: Duc Ninh Le, Julien Baglio

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

Deep within the heart of the European Organization for Nuclear Research, known as CERN, scientists operate the Large Hadron Collider, a massive machine that smashes protons together at nearly the speed of light. When these particles collide, they briefly create a shower of new, heavy particles, including the W and Z bosons. These bosons are the carriers of the weak nuclear force, one of the four fundamental forces that govern how matter behaves. For decades, physicists have used these collisions to test the Standard Model, the best theory we have for how the universe works at the smallest scales. A key part of this testing involves looking at how these bosons spin, or their polarization. Just as a spinning top can rotate in different ways, these particles can be oriented in specific directions as they are created. Understanding these orientations helps scientists confirm if the Standard Model holds true or if there are subtle cracks in the theory that might point to new, undiscovered physics.

A team of researchers has now taken a closer look at a specific type of collision where a W boson and a Z boson are produced together and immediately decay into a quartet of particles: a positron, a neutrino, and two muons. While previous studies had calculated how these collisions happen using the rules of quantum chromodynamics, which describes the strong force, this new work adds a crucial layer of precision by including the effects of the electroweak force. The team focused on separating the different ways the W and Z bosons could be spinning. Since each boson can spin in three different ways, the pair can exist in nine different combined spin states. The researchers wanted to know how the laws of physics change when they look at these specific spin combinations rather than just the average of all of them.

To do this, the scientists used powerful computer simulations to calculate the probability of these collisions happening at the energy levels currently available at the LHC. They specifically looked at the "doubly longitudinal" state, where both bosons are spinning in a way that aligns with their direction of travel, as well as other combinations where one or both are spinning sideways. The study revealed that for the total number of collisions, the new calculations made very little difference; the corrections were small, shifting the predicted numbers by only a few percent. This suggests that for general counts of these events, the previous calculations were already quite accurate.

However, the story changes when the researchers looked at the energy of the particles flying out of the collision. When the positron produced in the decay carries a very high amount of energy, specifically around 200 GeV, the new calculations showed a significant change for the doubly longitudinal case. In this specific high-energy scenario, the probability of the event occurring drops by about ten percent compared to what was predicted by the strong-force calculations alone. This drop is not a small fluctuation; it is a distinct effect caused by the electroweak force. The researchers found that this reduction grows even larger as the energy increases further, a behavior that can be described by specific mathematical patterns known as Sudakov logarithms, which account for how forces behave at very high energies.

For the other spin combinations, the new corrections remained small and did not significantly alter the predictions. The study also examined the angles at which the particles fly apart and the distance between them, finding that the new calculations did not change the shape of these distributions in any major way. The most important takeaway is that while the overall number of events is well understood, the specific behavior of the doubly longitudinal spin state at high energies requires this new, more precise level of calculation to match what experiments might see. This work provides a clearer map for experimentalists at the LHC, ensuring that when they measure these high-energy collisions, they are comparing their data against the most accurate theoretical predictions possible, ready to spot any deviations that might hint at physics beyond our current understanding.

Technical Summary: Doubly-polarized WZ hadronic cross sections at NLO QCD+EW accuracy

Problem and Motivation
The study of gauge boson polarization in the three-lepton channel ( $WZ \to \ell\nu\ell\ell$ ) at the LHC is a critical avenue for probing the Standard Model (SM) and searching for new physics. While higher-order QCD and electroweak (EW) corrections for unpolarized $WZ$ production have reached high precision (including NNLO QCD and NLO EW), the theoretical description of doubly-polarized signals has been incomplete. Previous studies of polarization effects in the three-lepton channel either lacked NLO EW corrections or did not separate polarization states at the amplitude level. Specifically, while NLO QCD corrections for doubly-polarized $WZ$ production in the double-pole approximation (DPA) were established in Ref. [32], the corresponding NLO EW corrections were missing. This gap is significant because experimental collaborations (e.g., ATLAS) are increasingly measuring angular observables and polarization fractions, requiring precise theoretical predictions that include both QCD and EW effects to interpret data correctly.

Methodology
The authors calculate the next-to-leading order (NLO) EW corrections to the doubly-polarized cross sections for the process $pp \to W^+Z \to e^+\nu_e\mu^+\mu^- + X$ at $\sqrt{s} = 13$ TeV. The calculation is performed within the Double-Pole Approximation (DPA), where the production and decay amplitudes are calculated using on-shell momenta for the intermediate $W$ and $Z$ bosons, while retaining full phase-space integration in the propagators.

Polarization Separation: The $WZ$ system possesses nine polarization states (combinations of two transverse and one longitudinal state for each boson). The authors classify these into four groups: doubly longitudinal ( $W_L Z_L$ ), doubly transverse ( $W_T Z_T$ ), and two mixed states ( $W_L Z_T$ and $W_T Z_L$ ). Interference terms between these states are also computed.
Framework: The calculation utilizes the DPA to separate polarization states at the amplitude level. The NLO QCD corrections were re-calculated to match the setup of Ref. [32] for consistency. The NLO EW corrections are more complex than in $ZZ$ production due to the possibility of photon radiation off the on-shell $W$ boson.
Kinematics and Cuts: Results are presented in the $WZ$ center-of-mass system (c.m.s.), consistent with recent ATLAS measurements. The analysis applies ATLAS fiducial cuts (transverse momentum and pseudorapidity cuts on leptons, invariant mass window for the $Z$ , and transverse mass for the $W$ ). Lepton-photon recombination is implemented for $\Delta R < 0.1$ .
Parameters: The calculation uses standard input parameters ( $G_\mu$ scheme), neglects light quark and lepton masses, and employs the LUXqed17 PDF set. Scale uncertainties are estimated via a three-point variation.

Key Contributions and Results

Integrated Cross Sections:
- The authors provide the first NLO QCD+EW predictions for integrated doubly-polarized cross sections.
- For the unpolarized cross section, NLO QCD corrections are large (+80%), while NLO EW corrections are negative (-4.2%). The relative EW correction with respect to the NLO QCD result ( $\bar{\delta}_{EW}$ ) is -2.3%.
- For polarized components, the doubly transverse ( $W_T Z_T$ ) dominates (~~63% at NLO QCDEW), followed by mixed states (~~15% each) and the doubly longitudinal ( $W_L Z_L$ ) at ~5.6%.
- The impact of NLO EW corrections on integrated polarized cross sections is generally small, with $\bar{\delta}_{EW}$ ranging from -0.2% to -3.3% across different polarization states.
- Scale uncertainties are significantly reduced at NLO QCDEW compared to LO, particularly for the $W_L Z_L$ component (reducing from ~6% to ~0.6%).
Kinematic Distributions:
- Angular Distributions: For angular observables ( $\cos \theta_{e^+}^{WZ}$ , $\cos \theta_{\mu^-}^{WZ}$ , and azimuthal angles $\Delta\phi$ ), EW corrections are uniformly small (between -4.5% and +0.5%) across all polarization states and kinematic ranges. The shapes of the distributions are largely insensitive to EW corrections.
- Transverse Momentum ( $p_T$ ) Distributions: A significant finding is that EW corrections become sizable specifically for the doubly longitudinal ( $W_L Z_L$ ) component in the high- $p_T$ $p_{T}$ tail.
  - At $p_{T,e} \approx 200$ GeV (accessible with current LHC data), the EW correction for $W_L Z_L$ is approximately -10%.
  - At $p_{T,e} \approx 450$ GeV, the correction grows to about -20%.
  - This behavior is attributed to negative Sudakov corrections in the virtual contribution. The authors provide a fit using single and double Sudakov logarithms that accurately describes this behavior.
  - For other polarization states ( $W_T Z_T$ , mixed), EW corrections remain below 5% and are negligible.

Significance and Claims
The paper claims to close a theoretical gap by providing the first NLO EW corrections for doubly-polarized $WZ$ production in the three-lepton channel. The primary significance lies in the quantification of EW effects on polarization signals:

The authors assert that for integrated cross sections and angular distributions, EW corrections are negligible (typically < 3-5%) relative to NLO QCD predictions.
However, for transverse momentum distributions of the doubly longitudinal state, EW corrections are non-negligible and grow rapidly with energy, reaching -10% in the currently accessible kinematic range.
The study emphasizes that while the unpolarized cross section is dominated by transverse polarizations, the specific high- $p_T$ behavior of the longitudinal component is sensitive to EW physics, necessitating the inclusion of these corrections for precision phenomenology in the high-energy tail of the spectrum.

The work is presented as a necessary step for interpreting current and future LHC data on gauge boson polarization, ensuring that theoretical predictions match the precision of experimental measurements in the $WZ$ channel.

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