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Weak (quasi-)PDFs from the lattice

This paper utilizes lattice methods to investigate Higgs and W/Z parton distribution functions (PDFs) within a reduced Standard Model, successfully determining quasi-PDFs and resulting PDFs that exhibit gauge-invariant structures despite remaining uncertainties.

Original authors: Patrick Jenny, Axel Maas, Sofie Martins, Georg Wieland

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Jenny, Axel Maas, Sofie Martins, Georg Wieland

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

To understand how the universe works at its most fundamental level, physicists often look at how particles smash into one another in massive machines called colliders. When these collisions happen at incredibly high speeds, the particles involved are not just solid, indivisible points; they are complex clouds of smaller constituents. In the familiar world of the strong nuclear force, these clouds are described by maps known as parton distribution functions. These maps tell scientists exactly how the momentum, or the "oomph" of motion, is shared among the tiny pieces inside a larger particle. For decades, these maps have been essential for predicting what happens in experiments involving protons. However, as we look toward the future of particle physics, with plans for even larger colliders, a new question arises. At these extreme energies, the particles involved in the collisions are not just protons, but also the heavy carriers of the weak nuclear force and the Higgs boson, the particle responsible for giving others mass. The question is whether these particles also have internal clouds of smaller parts that carry momentum, and if so, what those clouds look like.

For a long time, the standard way to answer this was through mathematical calculations that treat these particles as simple, elementary points. This approach suggested that the Higgs boson and the weak force carriers are just single, indivisible units, carrying all their own momentum without sharing it with anything else. However, a deeper look at the laws of physics suggests this picture might be incomplete. The mathematical symmetries that govern these particles are not truly broken but are merely hidden, meaning the physical particles we observe are actually composite objects made of more fundamental fields. If this is true, then the Higgs and the weak force carriers should have their own internal structure, with momentum shared among their constituents in a way that differs from the simple point-particle view. Until now, no one had been able to calculate this internal structure from first principles without relying on the simplifying assumptions of the standard mathematical approach.

A team of researchers has now taken the first step to solve this puzzle by using a powerful computational technique called lattice simulation. Instead of trying to solve the equations on paper, they built a digital grid that mimics the fabric of space-time and simulated the behavior of the Higgs and weak force fields directly. They created a simplified version of the universe that contains only the Higgs boson and the weak force carriers, stripping away other complexities to focus on the core interaction. By running millions of calculations on this grid, they were able to measure how the momentum is distributed inside these particles. The results reveal a striking difference between the Higgs boson and the weak force carriers. While the weak force carriers behave somewhat like the familiar particles in the strong nuclear force, sharing their momentum across a wide range of internal parts, the Higgs boson is very different. Its momentum distribution is substantially modified from the simple "single component" expectation, yet it is characterized by a distinct lack of the small-momentum contributions seen in other particles. Specifically, the Higgs PDF is strongly suppressed at low momentum fractions, meaning it does not develop the rich, sea-like internal structure that the weak force carriers do.

This finding challenges the long-held assumption that the Higgs boson is a simple, indivisible particle in the context of high-energy collisions. The simulations show that the Higgs does not behave like the trivial, single-object picture predicted by basic tree-level calculations; instead, its internal structure is significantly altered by the composite nature of the theory. However, unlike the gauge bosons which show a broad distribution, the Higgs remains unique in that it does not exhibit the enhanced low-momentum components typical of complex parton clouds. The researchers also tested whether adding other heavy particles, which act as a stand-in for the full complexity of the real world, would change this picture. They found that these extra particles had almost no effect on the results, confirming that the behavior they observed is a robust feature of the Higgs and weak force interaction itself.

The implications of this work are significant for the next generation of particle physics experiments. If future colliders are to accurately predict what happens when these particles collide, they cannot rely on the old, simple maps that assume the Higgs is just a single point, nor can they assume it behaves like the other force carriers. The new maps generated by this study show that the Higgs boson has a unique internal character that must be accounted for: it is neither a trivial point nor a complex sea of parts, but a distinct entity with a suppressed low-momentum profile. While the study is a simulation and not a direct measurement from a real collider, it provides the first non-perturbative, or non-approximated, view of how these particles are built. The researchers are careful to note that their results are a first look and that more work is needed to refine the precision and connect these findings to the full complexity of the real world. Nevertheless, the study offers a clear and concrete picture: the Higgs boson is not just a passive participant in the high-energy dance of the universe; it has a distinct, modified structure that sets it apart from its neighbors, a fact that will likely shape how physicists interpret the data from the next great machines of discovery.

Technical Summary: Weak (quasi-)PDFs from the Lattice

Problem Statement
Future high-energy colliders, such as the FCC, will probe the electroweak substructure of particles, necessitating an accurate description of electroweak Parton Distribution Functions (PDFs). While perturbative calculations suggest these contributions arise solely from the sea and are suppressed by the weak coupling, this framework relies on the assumption that asymptotic states are fully understood. However, due to Elitzur's theorem, gauge symmetry is not broken but hidden, implying that physical asymptotic states must be weakly gauge-invariant. The Fröhlich-Morchio-Strocchi (FMS) mechanism suggests that these gauge-invariant states possess a different composition than perturbative gauge-variant states, potentially introducing non-trivial valence contributions (e.g., an electroweak valence contribution of the Higgs) that perturbative approaches might miss. Data-driven extractions of PDFs could obscure such effects. Consequently, a non-perturbative determination of these PDFs is required to test the structure of composite gauge-invariant asymptotic states.

Methodology
The authors employ first-principles lattice methods to determine weak PDFs within a reduced Standard Model setup. This theory includes an $SU(2)_W$ Yang-Mills theory coupled to a fundamental scalar field $X$ (Higgs) and two generations of degenerate vectorial fermions $\psi$ . The study utilizes three distinct lattice parameter sets:

Set H: A coarse lattice with a heavy Higgs mass ( $1.85 m_W$ ) and strong couplings.
Set L: A finer lattice with a Standard Model-like mass hierarchy and weaker couplings.
Set F: An exploratory set including dynamical fermions (Set L parameters plus fermions), though with reduced statistics due to computational cost.

The methodology follows the quasi-PDF approach:

Operator Construction: Matrix elements are computed for gauge-invariant composite operators representing the physical Higgs ( $0^+$ ) and the physical $W/Z$ bosons ( $1^-$ , custodial triplet). The currents used include scalar singlet currents (for the Higgs) and field-strength tensor-based currents (for the vector bosons), connected by Wilson lines.
Quasi-PDF Extraction: The normalized three-point matrix elements are calculated as a function of the hadron momentum $P_3$ and the separation distance $z$ . The quasi-PDFs are obtained via a Fourier transform of the matrix elements in the limit of large Euclidean time.
Bjorken Limit Extrapolation: To approximate the physical PDFs, the quasi-PDFs are extrapolated to the infinite momentum frame ( $P_3 \to \infty$ ). This involves fitting the $x$ -dependence of the quasi-PDFs at various $P_3$ values and then extrapolating the fit parameters to $P_3 \to \infty$ using various ansätze.
Normalization: Since the factorization scale dependence is lost in the $P_3 \to \infty$ approximation (effectively a tree-level result), the results are normalized using sum rules. For the Higgs, a constituent counting rule ( $\int dx H_H(x) = 1$ ) is adopted based on the FMS mechanism. For the vector bosons, a sum rule involving the Higgs contribution to the custodial charge is used ( $\int dx W_H(x) = 2$ ), acknowledging that the elementary gauge field approximation is insufficient for longitudinal modes in certain gauges.

Key Contributions

First Non-Perturbative Calculation: This work presents the first lattice determination of weak PDFs for a theory containing the Higgs and $W/Z$ bosons, moving beyond perturbative assumptions.
Gauge-Invariant Asymptotic States: The study explicitly treats the physical states as gauge-invariant composites, investigating how this alters the PDF structure compared to perturbative expectations.
Systematic Control: The authors perform a detailed analysis of systematic uncertainties, including finite volume effects (demonstrating they are negligible for large volumes) and the extrapolation to the Bjorken limit. They also compare results across different lattice spacings and the inclusion of dynamical fermions.

Results

Higgs vs. Gauge Boson Behavior: The Higgs PDF ( $H_H$ ) is found to be qualitatively different from the gauge boson PDFs. The Higgs distribution is never enhanced at small $x$ , in strong contrast to quark PDFs, and is substantially suppressed at low $x$ compared to the gauge boson PDF. In contrast, the gauge boson PDFs ( $W_W$ ) exhibit a $1/x$ behavior at low $x$ , resembling QCD-like PDFs.
Polarization Dependence: The difference between transversely and longitudinally polarized vector boson PDFs is found to be minor.
Impact of Fermions: The inclusion of heavy, dynamical vector-like fermions (Set F) yields results qualitatively and quantitatively similar to the pure gauge-scalar sets (H and L), indicating a negligible impact on the studied quasi-PDFs within the current statistical precision.
Deviation from Tree-Level: The results show substantial modifications from the low-energy tree-level expectations (where Higgs PDFs are trivial $\delta(1-x)$ and $W$ PDFs are zero). The gauge bosons dominate massively at low $x$ , suggesting significant non-trivial structure at high energies.
Current Dependence: Different choices of currents for the vector bosons yield overlapping results within error bands, supporting the anticipated universality of the quasi-PDF approach.
Coupling Dependence: The Higgs PDF obtained from the stronger coupled set H is more heavily suppressed at larger $x$ compared to the sets L and F with weaker couplings.

Significance and Claims
The paper claims to provide the first non-perturbative insight into the structure of electroweak PDFs, confirming that the physical asymptotic states possess a complex internal structure distinct from simple perturbative gauge-variant states. The authors emphasize that their results are leading-order accurate due to the neglect of factorization scale evolution and the $P_3 \to \infty$ approximation. They state that while the results are not yet at the precision of QCD lattice PDFs, they serve as a crucial first input for phenomenological calculations involving physical asymptotic states. The work suggests that future colliders may be sensitive to these non-perturbative effects, particularly in processes like vector boson scattering at multi-TeV scales. The authors explicitly avoid claiming that these results explain current data or predict specific new physics signals beyond the scope of the reduced model studied, noting that further work is needed to incorporate factorization and improve statistical precision.

Technical Summary: Weak (quasi-)PDFs from the Lattice

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