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Proposed measurement of longitudinally polarised vector bosons in $WH$ and $ZH$ production at Hadron colliders

This paper proposes and evaluates analysis strategies using kinematic observables and machine learning to measure longitudinally polarized $W$ and $Z$ bosons in associated Higgs production at the LHC, projecting that the High-Luminosity LHC will achieve 5 $\sigma$ discovery significance and percent-level cross-section precision for these states in the $H \to b\bar{b}$ and $H \to \gamma\gamma$ channels.

Original authors: Rosa Colyer, Dominik Duda

Published 2026-09-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rosa Colyer, Dominik Duda

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

The universe is held together by invisible forces that dictate how particles interact, and at the heart of this system lies a mechanism that gives matter its mass. Without this mechanism, the particles that make up our world would zip around at the speed of light, unable to clump together to form atoms, stars, or people. This mechanism involves a field that fills all of space, and when particles move through it, they acquire mass. A crucial part of this story involves the W and Z particles, which carry the weak nuclear force. When they gain mass from this field, they also acquire a specific way of spinning, known as longitudinal polarization. This spin state is a direct fingerprint of the mass-giving process itself. Understanding how these particles behave in this specific state is like reading the fine print of the universe's rulebook; it tells scientists whether the current theory of particle physics is complete or if there are hidden rules waiting to be discovered.

A team of researchers at the University of Edinburgh has proposed a new way to look for these specific spinning particles, focusing on a rare event where a Higgs boson is created alongside a W or Z particle. The Higgs boson is the particle associated with the field that gives mass, so studying it alongside these force-carrying particles offers a unique window into how the mass-giving mechanism works. The researchers simulated billions of collisions that could occur at the Large Hadron Collider, the massive machine in Switzerland that smashes protons together at nearly the speed of light. They focused on two specific ways the Higgs boson can decay: into two photons, which are particles of light, or into two bottom quarks, which are heavy types of matter particles. By analyzing the patterns of the debris left behind in these collisions, the team developed a method to tell the difference between the long-sought longitudinal spin states and the more common transverse spin states.

To find these rare events, the researchers had to build a sophisticated filter, much like a sieve designed to catch only the most elusive grains of sand. They used advanced computer programs to simulate what the collisions would look like if the Standard Model of physics were correct, generating millions of potential signal events and an even larger number of background events that could mimic them. The background noise comes from other common particle interactions that produce similar-looking debris. The team then trained a type of artificial intelligence, known as a boosted decision tree, to recognize the subtle differences between the signal and the noise. This computer program learned to weigh various clues, such as the angles at which particles fly apart and their speeds, to make a judgment call on whether an event contained the specific longitudinal spin state they were hunting for.

The results of these simulations are promising. The researchers found that with enough data, they could distinguish the longitudinal spin states from the transverse ones with high confidence. They calculated that if the Large Hadron Collider runs at its highest planned intensity, known as the High-Luminosity LHC, it would be possible to confirm the existence of these longitudinal Z particles with a statistical certainty that physicists call a five-sigma discovery. This level of certainty is the gold standard in the field, meaning the chance of the result being a fluke is less than one in a million. For the longitudinal W particles, the team projected that a similar level of discovery could be reached with slightly more data. The study suggests that while the Higgs boson decaying into bottom quarks provides a huge number of events, the background noise is so heavy that it is harder to see the signal. Conversely, the decay into two photons is much cleaner, with less background noise, making it easier to spot the rare events even though they happen far less often.

Looking ahead, the researchers estimate that once the collider has collected its full dataset, they will be able to measure the production rates of these longitudinal particles with remarkable precision. For the longitudinal Z particles produced with a Higgs boson that decays into bottom quarks, they expect to measure the rate with an uncertainty of just eight percent. For the longitudinal W particles in the same channel, the precision would be around thirteen percent. Even in the cleaner but rarer photon channel, they project precisions of ten percent for the W particles and thirty-five percent for the Z particles. These numbers represent a significant step forward, turning what was once a theoretical possibility into a measurable reality. The study also looked at the transverse spin states, finding that while they are harder to isolate in some channels, they can still be measured with reasonable precision, particularly in the photon channel for the W particles.

This work does not claim to have found new physics yet, but it lays out a clear roadmap for how to find it. By establishing a method to measure these specific spin states directly, the researchers provide a new tool for testing the fundamental laws of nature. If the measurements in the future differ from the predictions made by the current theory, it would be a clear sign that new, unknown particles or forces are at play. The team's approach is designed to be model-independent, meaning it does not assume what the new physics might look like, but simply looks for deviations from the expected behavior. This makes the proposed measurements a powerful, neutral test of the universe's underlying structure. The study concludes that isolating these longitudinal vector bosons is well within the reach of future experiments, offering a direct path to probing the deepest secrets of how the universe acquired its mass.

Technical Summary: Proposed Measurement of Longitudinally Polarised Vector Bosons in WH and ZH Production at Hadron Colliders

Problem and Motivation
The longitudinal polarisation states of the $W$ and $Z$ bosons are intrinsically linked to the mechanism of electroweak (EW) symmetry breaking, arising as these bosons absorb the massless Goldstone degrees of freedom of the Higgs field. Measuring the production of these longitudinally polarised bosons in association with a Higgs boson ( $VH$ , where $V=W^\pm, Z$ ) offers a unique probe of the EW symmetry breaking mechanism and potential sensitivity to physics beyond the Standard Model (BSM). While previous studies have constrained EW symmetry breaking through vector-boson scattering or Effective Field Theory (EFT) fits, this work proposes a direct measurement of the production cross-sections for longitudinally ( $V_LH$ ) and transversely ( $V_TH$ ) polarised vector bosons. Such measurements serve as a model-independent benchmark to test EW unitarity and restoration at the TeV scale, potentially revealing deviations indicative of composite Higgs states or new heavy gauge bosons.

Methodology
The study focuses on $pp$ collisions at $\sqrt{s} = 14$ TeV, targeting four final states: $\ell^\pm\nu\gamma\gamma$ , $\ell^+\ell^-\gamma\gamma$ , $\ell^\pm\nu b\bar{b}$ , and $\ell^+\ell^- b\bar{b}$ (where $\ell = e, \mu$ ).

Simulation and Generation: Signal samples for $W^\pm H$ and $ZH$ were generated using MadGraph5_aMC@NLO at Leading Order (LO) with up to two additional parton emissions, interfaced with Pythia8 for parton showering and hadronisation. Separate samples were generated for longitudinally ( $V_L$ ) and transversely ( $V_T$ ) polarised bosons. Backgrounds (triboson, $t\bar{t}$ , single-top, diboson, and $V+b\bar{b}$ ) were also simulated at LO. Event samples were normalised to state-of-the-art NNLO QCD and NLO EW cross-sections and branching ratios. Detector effects were simulated using DELPHES with the default ATLAS card.
Event Selection: Selection criteria mimic experimental strategies, requiring specific lepton multiplicities, photon/jet kinematics, and $b$ -tagging. For $H \to b\bar{b}$ channels, events are further divided into four mutually exclusive regions based on the transverse momentum of the vector boson candidate ( $p_{T,V}$ ).
Discrimination Strategy: To separate $V_LH$ from $V_TH$ , the analysis employs kinematic and angular observables sensitive to polarisation, including the production angle $\theta^*$ , decay angles $\theta_1$ and $\Phi_1$ , and the azimuthal separation $\Delta\Phi(L, \bar{L})$ .
Machine Learning: Boosted Decision Trees (BDTs) are trained to maximise separation:
- BDT $_{Pol}$ : Trained exclusively on signal samples to distinguish $V_LH$ from $V_TH$ using kinematic and angular features.
- BDT $_{VH}$ : Trained to separate the combined signal ( $V_LH + V_TH$ ) from background, specifically for the $H \to b\bar{b}$ channels where background suppression is critical.
Statistical Analysis: A binned maximum-likelihood fit is performed on the BDT response score distributions. The fit extracts signal strength modifiers ( $\mu_{L}$ and $\mu_{T}$ ) for longitudinal and transverse modes, incorporating systematic uncertainties (cross-section, branching ratios, background normalisation) and statistical uncertainties via nuisance parameters. Asimov datasets are used to project sensitivities for integrated luminosities of 300, 1000, and 3000 fb $^{-1}$ .

Key Contributions

Direct Cross-Section Measurement: The paper establishes a roadmap for measuring $W^\pm_L H$ , $W^\pm_T H$ , $Z_L H$ , and $Z_T H$ cross-sections as fundamental observables, rather than relying solely on indirect EFT constraints.
Optimised Analysis Strategy: It proposes a tailored analysis using specific kinematic regions ( $p_{T,V}$ bins) and machine learning classifiers to handle the distinct challenges of the clean $H \to \gamma\gamma$ channel versus the high-statistics but high-background $H \to b\bar{b}$ channel.
Feasibility Demonstration: The study demonstrates that isolating longitudinally polarised vector bosons in $VH$ production is feasible with future LHC datasets, providing a complementary approach to existing di-boson and Vector Boson Fusion (VBF) studies.

Results
The statistical projections indicate the following capabilities at the High-Luminosity LHC (HL-LHC):

Discovery Significance: A $5.0\sigma$ discovery significance for $Z_L H$ in the $H \to b\bar{b}$ channel is achievable with 210 fb $^{-1}$ . For $W^\pm_L H$ , $5.0\sigma$ significance is projected with 340 fb $^{-1}$ ( $H \to b\bar{b}$ ) and 425 fb $^{-1}$ ( $H \to \gamma\gamma$ ).
Precision Measurements: With the full HL-LHC dataset (3000 fb $^{-1}$ $^{- 1}$ ), the inclusive cross-section measurement precisions are expected to reach:
- $W^\pm_L H$ : 13% ( $H \to b\bar{b}$ ) and 10% ( $H \to \gamma\gamma$ ).
- $Z_L H$ : 8% ( $H \to b\bar{b}$ ) and 35% ( $H \to \gamma\gamma$ ).
- Transverse Modes: Constraints on $Z_T H$ ( $H \to b\bar{b}$ ) and $W^\pm_T H$ ( $H \to \gamma\gamma$ ) are projected with precisions of 28% and 41%, respectively.
Channel Comparison: The $H \to \gamma\gamma$ channel, despite lower branching ratios, proves highly competitive for $W^\pm_L H$ due to exceptionally low background contamination, enabling high signal purity.

Significance and Claims
The authors claim that this work provides a "direct experimental roadmap" to measure polarised $VH$ cross-sections, establishing a "clean, model-independent benchmark" for testing EW symmetry breaking and unitarity. The study asserts that these measurements are "well within the experimental reach of future LHC analyses." By avoiding assumptions about specific BSM coupling structures, the proposed strategy offers a robust method to probe the underlying structure of the Higgs mechanism. The paper notes that while interference effects between $V_LH$ and $V_TH$ were neglected in the fit, dedicated studies suggest these effects are negligible at the expected precision. The authors conclude that these projections are realistic and that the ultimate precision may be even superior as systematic uncertainties improve over the lifespan of the HL-LHC.

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