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Precision Studies of Boosted Polarized Vector Bosons with Energy Correlators

This paper initiates precision studies of hadronic decays from boosted polarized vector bosons using energy correlators, demonstrating that polarization induces significant spectral shifts and distinct angular signatures that enable new applications in electroweak polarimetry, standard-candle measurements, and resonance searches.

Original authors: Anjie Gao, Kyle Lee, Xiaoyuan Zhang

Published 2026-10-30

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CC BY 4.0

Original authors: Anjie Gao, Kyle Lee, Xiaoyuan Zhang

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

In the high-energy collisions that take place inside particle accelerators, protons smash together with such force that they create a shower of new particles. Among the most common heavy particles born from these collisions are the W and Z bosons, the carriers of the weak nuclear force. When these particles are created with tremendous speed, they are said to be "boosted," meaning they travel so fast that their decay products are squeezed into a single, narrow stream of particles, looking much like a single jet of ordinary matter. For decades, physicists have studied these jets to understand the fundamental laws of nature. A key tool in this study is the measurement of how energy flows within the jet. By looking at the angles between the particles inside the jet, scientists can map out the internal structure of the decay. This mapping reveals a distinct peak in the distribution of energy, a signature that tells researchers the mass of the original particle. However, a crucial detail has often been overlooked: the W and Z bosons are not just heavy objects; they are spinning particles with a specific orientation, or polarization, at the moment they are created. Just as a spinning top has a specific axis of rotation, these bosons have a preferred direction of spin that changes how they break apart.

A new study by researchers Anjie Gao, Kyle Lee, and Xiaoyuan Zhang brings this missing piece of the puzzle into focus. They have performed a precise calculation of how the spin of a boosted W or Z boson affects the pattern of energy inside its resulting jet. The team found that the way these particles decay depends heavily on whether they are spinning along their direction of travel or perpendicular to it. When the researchers combined their detailed calculations of the decay patterns with the known rates at which these particles are produced in collisions, they discovered that ignoring the spin leads to a significant error. Specifically, for high-speed W and Z bosons produced in proton collisions, treating them as if they had no preferred spin direction causes the predicted energy pattern to be off by roughly ten percent. This is a large discrepancy in the world of precision physics, where measurements are often expected to be accurate to within a fraction of a percent. The study demonstrates that the spin of the boson shifts the entire energy distribution, making the peak of the signal appear slightly different depending on the particle's orientation.

The researchers developed a mathematical framework to separate the effects of production from the effects of decay. They showed that the energy pattern observed in the laboratory is a combination of how the particle was created and how it fell apart. By calculating the "boost kernels"—the rules that translate the decay of a stationary particle into the decay of a fast-moving one—they found that the spin information is preserved and amplified in the final measurement. For a particle moving at four times the speed of light relative to its rest mass, the height of the energy peak differs by about seventy percent between a particle spinning along its path and one spinning sideways. This difference is so pronounced that the energy pattern itself acts as a polarimeter, a device that measures spin, using only the angles of the particles inside the jet. This is a powerful result because it allows physicists to determine the spin of the boson without needing to detect the elusive neutrinos or other particles that often escape the detector, which is the traditional method.

The study also explored whether these effects could be seen in real experimental data without needing to perfectly reconstruct the original particle. The researchers simulated a scenario where they looked at a jet of particles recoiling against a known signal, without applying any filters to identify the jet as coming from a W or Z boson. Even in this messy environment, buried within a sea of ordinary particle jets, the specific energy pattern of the W and Z bosons emerged as a small, localized bump. This bump represented a seven percent excess of events at a specific angle, standing out clearly against the smooth background of random particle collisions. Although the actual W and Z bosons made up only about one percent of the total jets in the simulation, their unique spin-dependent signature was strong enough to be detected. This suggests that scientists can identify these particles and measure their properties simply by looking at the shape of the energy flow, even when the particles are hidden inside a complex spray of debris.

The implications of this work extend beyond just measuring known particles. The ability to precisely predict how polarized bosons decay opens the door to more sensitive searches for new physics. If a new, unknown particle were discovered, its spin would leave a distinct fingerprint on the energy pattern, different from that of the known W and Z bosons. By using these refined calculations, physicists can now distinguish between different types of new particles with much greater confidence. Furthermore, the study provides a new way to measure the properties of the W and Z bosons themselves, offering a cross-check against existing measurements that rely on leptonic decays. The researchers calculated these effects for both the Large Hadron Collider, where protons collide, and for future electron-positron colliders, showing that the spin effects are significant in both environments. The findings confirm that to truly understand the behavior of these fundamental particles, one must account for their spin, turning what was once a subtle correction into a central feature of precision measurements.

Technical Summary: Precision Studies of Boosted Polarized Vector Bosons with Energy Correlators

Problem Statement
Boosted, hadronically decaying $W$ and $Z$ bosons are abundant heavy objects at colliders. Unlike the Higgs boson or unpolarized vector bosons, these particles carry spin, and their production mechanisms generate specific polarization states (longitudinal vs. transverse). Previous precision studies of boosted decays using energy correlators assumed an unpolarized source, treating the decay spectrum as a simple convolution of a rest-frame correlator with a universal boost kernel. However, realistic production mechanisms do not yield unpolarized vector bosons; the spin density matrix $\rho_{\lambda\lambda'}$ is non-trivial. Ignoring polarization leads to inaccuracies in precision predictions for the substructure of these jets. Furthermore, while polarization measurements have historically relied on leptonic decays, there is a need to understand how polarization imprints itself on hadronic decays via energy correlators, particularly for precision measurements and searches for new resonances where reconstruction may be difficult.

Methodology
The authors develop a framework to compute the two-point energy-energy correlator (EEC) for boosted, polarized vector bosons ( $V \to q\bar{q}'$ ) by combining precision calculations of polarized production rates with polarized decay correlators.

Rest-Frame Decomposition: In the boson rest frame, the correlation matrix between helicity states is decomposed into three real structure functions: $W^{(0,0)}$ , $W^{(2,0)}$ , and $W^{(2,2)}$ . These correspond to different angular momentum ranks and projections relative to the bisector of the detector directions.
- $W^{(0,0)}$ represents the unpolarized EEC.
- $W^{(2,0)}$ and $W^{(2,2)}$ encode the polarization dependence.
- The authors compute these functions with high precision: $W^{(0,0)}$ is known at NNLO fixed order with NNLL collinear and N4LL back-to-back resummation. The polarized structure functions are derived using fixed-order ratios $a^{(j,m)}(z)$ computed at NNLO, combined with resummation techniques adapted for the specific spin projections.
Boost Kernels: The transition from the rest frame to the laboratory frame is handled via polarization-weighted boost kernels. Unlike the scalar case where a single kernel applies, the boosted spectrum for a polarized boson is a sum of convolutions of the three structure functions with distinct kinematic kernels ( $K^{[1]}_\gamma$ , $K^{[3\cos^2\vartheta-1]}_\gamma$ , and $K^{[\sin^2\vartheta \cos 2\phi]}_\gamma$ ). These kernels are derived in closed form using complete elliptic integrals.
Production Rates: The authors compute the polarized production density matrices ( $\rho_{\lambda\lambda'}$ ) for:
- $pp \to V + X$ at NNLO in QCD using Nnlojet.
- $e^+e^- \to W^+W^-$ at leading order (tree-level electroweak).
  These rates are differential in the boost factor $\gamma$ .
Synthesis: The final laboratory-frame spectrum is constructed by convolving the polarized production rates with the boosted decay spectra, weighted by the longitudinal fraction $\rho_{00}$ .

Key Contributions

Formalism: The paper establishes the first precision framework for calculating the EEC of boosted polarized vector bosons. It generalizes the scalar boost kernel formalism to include spin-dependent boost kernels that act on polarized decay structure functions.
Precision Calculation: The authors provide the first NNLO fixed-order + NNLL collinear + N4LL back-to-back resummed predictions for the polarized decay correlators. They demonstrate that while the $m=2$ channel requires only leading-order inputs for the collinear region, the $m=0$ channel (longitudinal) achieves NNLL accuracy, sufficient for the leading-power boosted spectrum.
Boost Kernels: Closed-form expressions for the polarization-weighted boost kernels are derived, allowing for the exact mapping of rest-frame angular distributions to laboratory-frame opening angles.

Results

Polarization Shifts: For high- $p_T$ $W/Z$ production in $pp$ collisions, the polarization of the boson shifts the EEC spectrum by approximately $O(10\%)$ relative to an unpolarized treatment. This effect is most pronounced at the decay peak ( $z' \simeq 1/\gamma^2$ ), where the longitudinal spectrum is enhanced and the transverse spectrum is depleted. At a boost of $\gamma=4$ , the peak height difference between purely longitudinal and purely transverse samples is roughly 70%.
Production Dependence: In $pp \to V+X$ , high- $p_T$ production is dominantly transverse, leading to a suppression of the spectrum compared to the unpolarized assumption. In $e^+e^- \to W^+W^-$ , the polarization content varies significantly with energy: near threshold ( $\sqrt{s} \approx 162$ GeV), the longitudinal fraction is substantial ( $\sim 33\%$ ), while at high energies ( $\sqrt{s} = 500$ GeV), the $W$ bosons become predominantly transversely polarized ( $h=-1$ ), with the longitudinal fraction dropping to $\sim 3.6\%$ .
Untagged Detection: Monte Carlo studies of untagged jet samples recoiling against a leptonic $Z$ show that hadronic $W/Z$ decays manifest as a localized angular bump in the EEC distribution at $\Delta R \simeq 2m_V/p_T$ . Despite making up only $\sim 1\%$ of the sample, these decays create a $\sim 7\%$ excess over the smooth QCD background, visible without mass reconstruction or jet tagging.

Significance
The paper claims to open the door to using energy correlators for the polarimetry of electroweak bosons through their hadronic decays. By accounting for polarization, the authors enable:

Standard Candle Measurements: Precise measurements of electroweak boson rates and polarization fractions inside jets, treating the decay peak as a standard candle for the boson mass.
New Physics Searches: Sharper searches for new resonances, where the polarization-dependent shape of the decay peak can provide spin discrimination for hypothetical new particles, distinguishing them from QCD backgrounds.
Theoretical Precision: A rigorous treatment of spin effects in boosted heavy-particle decays, correcting the unpolarized approximation which is insufficient for precision phenomenology at the LHC and future lepton colliders.

The work emphasizes that polarization effects are not degenerate with resummation or matching uncertainties and must be treated correctly in any precision program for boosted vector bosons.

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