⚛️ nuclear theory

Factorization and virtuality evolution of jet functions in heavy-ion collisions

This paper introduces a new perturbative framework that defines an in-medium jet function to extend the BDMPS-Z formalism, enabling a virtuality-resolved description of jet propagation in heavy-ion collisions that elucidates the interplay between vacuum-like evolution and medium-induced dynamics.

Original authors: Carlos Lamas, Carlos A. Salgado, Bin Wu

Published 2026-09-29

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CC BY 4.0

Original authors: Carlos Lamas, Carlos A. Salgado, Bin Wu

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 heart of the universe, matter is not a solid, unchanging block but a seething soup of fundamental particles. When scientists smash heavy atomic nuclei together at nearly the speed of light, they recreate conditions similar to those just moments after the Big Bang. In these collisions, a fleeting, super-hot fluid known as the quark-gluon plasma is born. This fluid is so dense and energetic that it behaves differently than the empty space we experience every day. To understand how this exotic matter works, researchers fire high-energy particles, called jets, through it. These jets are like tiny, high-speed probes that carry information about the medium they traverse. However, understanding exactly how these jets change as they move through the plasma has been a persistent challenge. The difficulty lies in tracking the internal energy and "virtuality" of the particles as they evolve, a process that standard theories often simplify by averaging out the details.

A team of physicists has now developed a new way to look at this problem, creating a more precise map of how these jets change as they travel through the hot fluid. Instead of treating the jet as a single, static object, their new framework tracks the jet's internal energy state, known as virtuality, at every step of its journey. In this view, a jet is not just a stream of particles but a dynamic entity that is constantly forming and evolving. The researchers found that whether a jet interacts with the hot fluid or simply flies through it depends heavily on the timing of its formation relative to the creation of the fluid itself. If the fluid forms quickly after the collision, the jet can enter it while still highly energetic and change its path. But if the fluid takes longer to form, the jet often sheds its excess energy in the empty space before it even arrives, effectively becoming a different kind of particle before it ever touches the plasma.

The team focused on a specific scenario: a high-energy particle colliding with a nucleus to produce a jet and a photon. By using a mathematical approach that separates the initial collision from the subsequent journey through the medium, they could isolate the effects of the fluid. They discovered that the jet's behavior is controlled by a race between two clocks: the time it takes for the jet to form and the time it takes for the hot fluid to appear. When the fluid appears after the jet has already formed, the jet behaves mostly as it would in a vacuum, radiating energy in a predictable way. However, when the fluid appears while the jet is still forming, the interaction becomes complex, and the fluid significantly alters the jet's path and energy.

Crucially, the researchers showed that their new framework connects seamlessly with previous models. They demonstrated that if one integrates over the complete range of virtualities, the standard BDMPS-Z results are exactly recovered. However, by keeping the virtuality resolution, they revealed that for jets with high internal energy, the dominant process is actually the emission of particles in the empty space before the fluid even exists. This means that many of the jets observed in experiments are not interacting with the fluid in the way previously thought; instead, they have already transformed into lower-energy particles by the time they reach the plasma. The study confirms that the standard way of calculating energy loss is recovered when the full virtuality range is considered, but it highlights that specific initial virtualities contribute differently depending on the timing of the medium's formation.

The team also explored how the temperature of the fluid affects this process. They found that while hotter fluids cause more energy loss overall, the fundamental rule about the timing of the jet's formation remains the same. The key factor is not just how hot the fluid is, but how quickly it appears relative to the jet's own development. This insight helps explain why different experiments have sometimes produced conflicting results about how much energy jets lose. By accounting for the specific timing of the jet's formation, scientists can now better interpret experimental data and refine their understanding of the quark-gluon plasma.

This work provides a clearer picture of the early moments of a heavy-ion collision. It suggests that the universe's most extreme states of matter are best understood not by looking at the final result alone, but by tracing the history of the particles that probe them. The researchers' new framework allows them to distinguish between the vacuum-like evolution of a jet and the modifications caused by the medium. This distinction is vital for accurately measuring the properties of the quark-gluon plasma, such as its density and how it resists the passage of high-energy particles. By resolving the details of how jets evolve, the study offers a more reliable tool for exploring the fundamental forces that govern the behavior of matter at its most extreme.

Technical Summary: Factorization and Virtuality Evolution of Jet Functions in Heavy-Ion Collisions

Problem Statement
While QCD factorization is a cornerstone of perturbative calculations in hadron collider physics (e.g., deep inelastic scattering and Drell-Yan processes), its rigorous justification in heavy-ion collisions remains incomplete. Existing theoretical formalisms for jet quenching (such as BDMPS-Z, GLV, ASW, and AMY) often yield disparate extractions of medium properties, such as the transport coefficient $\hat{q}$ . A primary source of this discrepancy is the lack of a systematic framework that properly treats the interplay between vacuum-like evolution and medium-induced radiation. Specifically, current approaches often integrate out the initial virtuality of the parton, obscuring the dynamical evolution of the jet as it traverses the QCD medium. This paper addresses the need for a factorized description of jet cross sections in heavy-ion collisions that explicitly retains dependence on the initial jet virtuality, $m_I^2$ .

Methodology
The authors develop a new framework based on perturbative QCD and the Glauber model, where colliding nuclei are treated as ensembles of uncorrelated nucleons. The core of the methodology involves:

Derivation of a Factorized Cross Section: The authors derive a factorized formula for the impact-parameter-dependent cross section of $\gamma$ -jet production ( $A_1 + A_2 \to \gamma + \text{Jet} + X$ ). By expanding in the large momentum scale $Q$ and the medium length, they separate the cross section into a hard scattering part (involving on-shell partons) and a jet function. Crucially, this jet function is defined differentially with respect to the initial virtuality $m_I^2$ of the parton emerging from the hard interaction.
Adaptation of BDMPS-Z Formalism: To evaluate these virtuality-dependent jet functions, the authors adapt the BDMPS-Z formalism. They utilize light-cone coordinates defined by the jet direction and employ a path-integral approach in the harmonic oscillator approximation to model the interaction with the background medium.
Virtuality-Resolved Calculation: Unlike standard BDMPS-Z calculations which assume the jet is created at a single point ( $x^+=0$ $x^{+} = 0$ ), the uncertainty principle ( $\Delta x^+ \Delta p^-_I \sim 1$ $Δ x^{+} Δ p_{I}^{-} \sim 1$ ) implies that for a fixed virtuality, the jet creation point is not perfectly localized. The authors calculate the jet function by considering three distinct kinematic regions based on the creation time of the jet relative to the medium formation time $\tau_0$ $τ_{0}$ :
- The jet is created before the medium in both the amplitude and conjugate amplitude ( $|x^+| < 2\tau_0$ ).
- The jet is created inside the medium in one of the amplitudes ( $2\tau_0 < |x^+| < 2\tau_f$ ).
- The jet is created after the medium in one of the amplitudes ( $|x^+| > 2\tau_f$ ).
Order-by-Order Analysis: The authors compute the jet functions at Leading Order (LO) for transverse momentum broadening and at Next-to-Leading Order (NLO) for the medium-induced soft gluon spectrum. They perform numerical integrations over the virtuality phase space up to a cutoff $\mu^2$ to study the cumulative behavior.

Key Contributions

Formal Definition of In-Medium Jet Functions: The paper provides a rigorous definition of an in-medium jet function that is differential in the initial virtuality $m_I^2$ . This allows for the systematic study of how virtuality evolution influences jet quenching observables.
Extension of BDMPS-Z: The work extends the BDMPS-Z formalism to include virtuality-differential transverse momentum broadening and soft gluon spectra, recovering standard BDMPS-Z results only after integrating over the full virtuality range.
Identification of Dominant Scales: The authors identify the ratio between the medium formation time ( $\tau_0$ ) and the jet formation time ( $t_J = p^+_J / |m_I^2|$ ) as the critical variable controlling which initial virtualities contribute to the jet function.

Results

LO Transverse Momentum Broadening: Numerical results show that the contribution to transverse momentum broadening is dominated by jets with formation times $t_J > \tau_0$ . Jets with $t_J < \tau_0$ (high virtuality) do not contribute to the LO spectrum because they must radiate in vacuum before entering the medium. The total cumulative jet function converges to the standard BDMPS-Z result when the dimensionless ratio $\xi_\mu = \tau_0 \mu^2 / p^+_J \gtrsim 1$ .
NLO Soft Gluon Spectrum: Similar behavior is observed at NLO. The region where $t_J < \tau_0$ is dominated by vacuum-like radiation and does not contribute to the medium-induced spectrum at $\mathcal{O}(\alpha_s)$ . The total spectrum converges to the BDMPS-Z result as the virtuality cutoff increases, with destructive interference between the different creation-time regions accelerating this convergence.
Dependence on Medium Parameters:
- Formation Time ( $\tau_0$ ): Shorter thermalization times allow higher virtuality jets to enter the medium before radiating, shifting the convergence point of the cumulative spectrum to higher virtualities.
- Temperature ( $T$ ): While the normalization of the spectrum increases with temperature (via $\hat{q}$ ), the virtuality scale at which the asymptotic BDMPS-Z result is reached remains largely independent of $T$ .
- Instantaneous Medium ( $\tau_0 = 0$ ): In the limit where the medium forms immediately, the "before-medium" contribution vanishes, and the spectrum receives contributions from a wider virtuality range, as high-virtuality jets enter the medium before they can radiate.

Significance and Claims
The authors claim that this work represents the "very first step" toward a factorization formalism that coherently describes the virtuality evolution of jets in a QCD medium. The primary significance lies in:

Clarifying the Role of Virtuality: The paper demonstrates that standard BDMPS-Z results are recovered only after integrating over virtuality, masking the fact that high-virtuality jets undergo vacuum-like evolution before interacting with the medium.
Systematic Framework: It provides a systematic framework to distinguish between vacuum-like and medium-induced radiation, potentially resolving discrepancies in phenomenological models.
Limitations of Current Models: The authors explicitly note a limitation of the BDMPS-Z formalism: its assumption of a medium correlator local in light-cone time ( $\delta(x^+ - y^+)$ ) prevents a dynamical description of virtuality evolution. The current work isolates the initial virtuality dependence but does not yet describe the continuous change of virtuality along the path.
Future Directions: The paper posits that combining the derived virtuality dependence with a medium model that resolves light-cone energy exchange (finite support in $\Delta x^+$ ) could lead to an evolution kernel in both light-cone time and virtuality, enabling the construction of medium-induced parton showers.

The authors remain modest, stating that these fixed-order results motivate the expectation that in an all-orders treatment, high-virtuality jets will first undergo vacuum-like evolution before the medium is formed, with medium effects becoming relevant only for lower-virtuality constituents.

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