⚛️ phenomenology

Femtoscopic correlation functions between the lightest charmed hadrons

This paper calculates femtoscopic momentum correlation functions for lightest charmed hadron pairs ( $D^{+}D^{+}$ , $D^{0}D^{0}$ , $D^{+}D^{0}$ , and $\Lambda_c^+\Lambda_c^+$ ) in Pb--Pb collisions at 5.02 TeV using calibrated strong interaction models and PHSD sources to provide ALICE-ready benchmarks that isolate weak repulsive interactions and quantum effects.

Original authors: Qi-Wei Yuan, Ming-Zhu Liu

Published 2026-10-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Qi-Wei Yuan, Ming-Zhu Liu

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 subatomic world, matter is not a static collection of solid blocks but a dynamic interplay of forces that bind particles together. To understand how these particles form the universe we see, physicists must map the invisible forces that act between them when they come close. For decades, scientists have been able to measure these forces for long-lived particles, like protons, by firing beams at them and watching how they scatter. However, a vast and mysterious class of particles known as charmed hadrons behaves differently. These particles are heavy, unstable, and vanish in a fraction of a second, making it impossible to create a beam of them or use them as a target. Because they disappear before they can be studied in a traditional collision, their mutual attraction or repulsion has remained a blind spot in our understanding of the strong force, the fundamental glue that holds atomic nuclei together.

To solve this puzzle, researchers have turned to a technique called femtoscopy, which acts like a high-speed camera for the subatomic realm. Instead of trying to catch a single particle, scientists look at pairs of particles created together in high-energy collisions, such as those between heavy lead nuclei. By measuring how often these pairs emerge with very similar speeds, they can infer the distance between them at the moment they were born and the forces that acted on them as they flew apart. This method allows physicists to probe the interactions of short-lived particles without ever needing to hold them in a beam. The question driving this new work is simple yet profound: how do the lightest particles carrying a "charm" quark interact with one another? Do they attract, repel, or form exotic new structures?

A team of physicists from Lanzhou University in China has taken a significant step toward answering this by simulating the collisions of lead nuclei at the energy levels produced by the Large Hadron Collider. They focused on the lightest possible pairs of charmed particles: combinations of two D mesons and two Lambda-c baryons. These specific pairs were chosen because, unlike other systems that might form complex, short-lived molecules, these channels are relatively clean. There is no known exotic state dominating their behavior, which makes them ideal reference points for understanding the fundamental forces at play. The researchers used powerful computer simulations to model the chaotic environment of a lead-lead collision, tracking exactly where and when these charmed particles were born and how they moved.

The team discovered that the space from which these particles emerge is not a uniform, simple shape as often assumed in older models. Instead, the distribution of birth points depends on the specific type of particle and the violence of the collision. In the most violent, central collisions, the particles are born from a broad, diffuse cloud. In more peripheral, glancing collisions, this cloud shrinks into a much tighter, more compact source. This finding is crucial because the size of the source directly influences the strength of the signal scientists can detect. The researchers found that the tighter sources in peripheral collisions significantly sharpen the signal, making it easier to spot the subtle effects of the strong force.

To understand the forces themselves, the team compared two different theoretical descriptions of how these particles interact. One approach treats the interaction as the exchange of other, lighter particles, while the other uses a simplified mathematical contact point. Both models were tuned to match the most reliable data available, which comes from lattice quantum chromodynamics—a method that calculates particle interactions from first principles using supercomputers. The results showed that for the D meson pairs, the interaction is repulsive, meaning the particles push each other away. This repulsion is weak but consistent. The team identified a specific pair, the positively charged D-plus and the neutral D-zero, as the "cleanest" way to observe this force. Because this pair does not suffer from electrical repulsion or quantum interference that complicates other measurements, it offers a direct window into the repulsive strong force between charmed particles.

For the pairs of Lambda-c baryons, the situation is more complex due to the rules of quantum mechanics that govern their spin. The researchers found that the strong interaction between these baryons is also repulsive, but this effect is hidden beneath a layer of other forces. The particles carry an electric charge, causing them to repel each other electrically, and their quantum nature creates additional patterns. The team showed that the repulsive strong force acts as a small, systematic shift on top of these larger effects. This shift is most visible in peripheral collisions where the source is compact. However, the researchers noted a significant caveat: the data used to calibrate their models for the Lambda-c baryons was calculated using a version of the universe where the mass of a key particle, the pion, is different from its real-world value. While other theories suggest the repulsive nature of the force would remain the same in our real universe, the exact strength of the effect at the physical mass of the pion is not yet quantified.

The work provides a detailed roadmap for future experiments, particularly for the ALICE detector at the Large Hadron Collider. By predicting exactly what the correlation signals should look like for different types of particle pairs and different collision intensities, the study gives experimentalists a benchmark to compare against real data. The authors emphasize that while their results are robust within the limits of current lattice data and their simulation models, they are conditional estimates. The uncertainty in the final predictions comes primarily from the input data regarding the scattering lengths, which are the fundamental measures of how far apart particles get before interacting.

Ultimately, this research transforms the study of charmed hadrons from a theoretical guessing game into a precise, testable framework. It demonstrates that by carefully separating the effects of electric charge, quantum statistics, and the strong force, scientists can isolate the specific interaction between these heavy particles. The findings suggest that the lightest charmed particles do not form bound states but instead repel one another, a conclusion that helps refine our understanding of the strong force in a regime that was previously inaccessible. As the Large Hadron Collider continues to collect data with higher precision, these theoretical predictions will serve as the standard against which the true nature of charmed matter is measured, potentially revealing new insights into the structure of the universe at its smallest scales.

Technical Summary: Femtoscopic Correlation Functions between the Lightest Charmed Hadrons

Problem and Motivation
Understanding low-energy hadron–hadron interactions is essential for bridging Quantum Chromodynamics (QCD) with the spectrum and structure of hadronic matter. While lattice QCD offers a first-principles approach to these interactions, experimental constraints are often limited by the short lifetimes of unstable hadrons, which preclude conventional scattering experiments. Intensity-frontier femtoscopic measurements in high-energy heavy-ion collisions offer a complementary pathway to probe final-state interactions (FSI) involving short-lived particles.

This work focuses on the interactions between the lightest charmed hadrons: the $D$ meson pairs ( $D^+D^+$ , $D^0D^0$ , $D^+D^0$ ) and the $\Lambda_c^+\Lambda_c^+$ baryon pair. These systems are selected because their threshold regions are comparatively "clean," lacking established near-threshold molecular states that would complicate the dynamics. Furthermore, lattice QCD calculations for these specific channels are available, providing model-independent constraints on their scattering lengths. The study aims to calculate femtoscopic momentum correlation functions for these pairs in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV, providing benchmark estimates for future measurements by the ALICE experiment.

Methodology
The theoretical framework relies on the Koonin–Pratt (KP) formalism, where the correlation function $C(k)$ is the integral of the relative source function $S_R(r)$ weighted by the squared two-body scattering wave function $|\Psi^{(-)}_k(r)|^2$ .

Source Functions: Instead of assuming a universal Gaussian source, the authors construct species-dependent, non-Gaussian relative source functions from Parton–Hadron–String Dynamics (PHSD) simulations of Pb–Pb collisions. These sources are derived from the hadronization positions of the charmed hadron pairs, propagated to an equal-time frame in the pair rest frame. The distributions are fitted using a generalized-gamma function, capturing non-Gaussian shapes that vary with collision centrality (narrowing from central to peripheral collisions).
Strong Interactions:
- $DD$ System: Two independent descriptions are employed: a One-Boson-Exchange (OBE) potential (involving $\sigma, \rho, \omega$ exchange) and an Effective Field Theory (EFT) contact interaction.
- $\Lambda_c\Lambda_c$ System: Only the EFT contact interaction is retained for the final results.
- Calibration: Parameters for both models are calibrated to reproduce lattice QCD scattering lengths ( $a_0$ ). For $DD$ , the physical-pion-mass extrapolation is used; for $\Lambda_c\Lambda_c$ , the calibration uses lattice results at a non-physical pion mass ( $m_\pi \approx 303$ MeV). The uncertainty bands in the results propagate only the statistical and systematic uncertainties of these lattice scattering lengths.
Wave Functions: The calculation includes strong interactions, Coulomb repulsion, and quantum-statistical (QS) effects.
- For identical bosons ( $D^+D^+$ , $D^0D^0$ ), Bose–Einstein symmetrization is applied.
- For the mixed-charge $D^+D^0$ , the particles are distinguishable, eliminating QS interference, though the strong interaction is constrained by isospin symmetry to the $I=1$ channel.
- For $\Lambda_c^+\Lambda_c^+$ (fermions), the total wave function must be antisymmetric. The spin-singlet ( $S=0$ ) state is symmetric in space and carries the strong interaction, while the spin-triplet ( $S=1$ ) state is antisymmetric in space (P-wave) and treated as a background with no S-wave strong interaction in this approximation. The final observable is a spin-weighted average ( $1/4$ singlet, $3/4$ triplet).

Key Results

Source Characteristics: The PHSD sources are non-Gaussian and species-dependent. The Gaussian-equivalent radii ( $R_{src}$ ) decrease from central (0–10%) to peripheral (60–80%) collisions. For $DD$ , $R_{src}$ ranges from $\approx 4.2$ fm to $2.1$ fm; for $\Lambda_c\Lambda_c$ , from $\approx 3.9$ fm to $2.0$ fm.
$DD$ Correlations:
- $D^+D^+$ : The correlation is shaped by the competition between Coulomb suppression at very low momentum and Bose–Einstein enhancement at finite momentum. The repulsive strong interaction acts as a subleading modification, slightly suppressing the peak.
- $D^0D^0$ : Lacking Coulomb interaction, the low-momentum peak (originally $C(0)=2$ due to QS) is depleted by the repulsive strong interaction. The suppression is more pronounced in peripheral collisions due to the compact source.
- $D^+D^0$ : This channel is free of both Coulomb distortion and identical-particle interference. Consequently, the correlation function is driven solely by the strong interaction, showing a suppression below unity that rises monotonically toward 1. This provides the "cleanest" access to the repulsive $DD$ strong interaction, though the effect is small (a few percent) due to the small scattering length.
$\Lambda_c^+\Lambda_c^+$ Correlations: The signal is dominated by the repulsive Coulomb interaction and the large statistical weight of the spin-triplet ( $S=1$ ) background, which suppresses the correlation at low momentum. The strong interaction (acting only in the $S=0$ channel) manifests as a small, systematic displacement of the spin-averaged correlation function relative to a reference without strong interactions. This effect is most pronounced in peripheral collisions where the compact source enhances the low-momentum signal.
Model Comparison: The OBE and EFT models for $DD$ yield similar results in the low-momentum region where the signal is largest, as both are anchored to the same scattering length. Differences emerge at higher momenta, reflecting sensitivity to the effective range and interaction shape.

Significance and Claims
The paper presents these results as benchmark estimates and an analysis framework for forthcoming femtoscopic measurements by the ALICE experiment. The authors explicitly state that the quoted uncertainty bands propagate only the lattice scattering-length input; they do not include uncertainties from source fitting, regulator choices, coupling variations, or the extrapolation of $\Lambda_c\Lambda_c$ results from non-physical to physical pion masses.

The study highlights that:

The $D^+D^0$ channel offers the most transparent probe of the $DD$ strong interaction due to the absence of competing Coulomb and QS backgrounds.
Peripheral collisions are optimal for $\Lambda_c\Lambda_c$ measurements because the compact source maximizes the low-momentum signal.
A joint analysis of all four channels, utilizing the constructed model-independent references (Coulomb and QS only), could disentangle strong, Coulomb, and statistical contributions without external input.

The authors conclude that while the current results are conditional benchmarks, they provide a necessary foundation for interpreting future high-statistics data. They note that resolving the small signals (particularly for $D^+D^0$ ) will require the statistical precision of the upgraded ALICE heavy-ion program. Future improvements would require lattice QCD calculations at the physical pion mass for $\Lambda_c\Lambda_c$ and a quantification of the pion-mass dependence of the input.

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