Disentangling baryon stopping and neutron skin effects in heavy-ion collisions

This paper proposes a statistical model analysis of heavy-ion collisions that quantitatively extracts baryon-stopping parameters from isobar data and introduces a new oxygen-baseline observable to systematically probe neutron-skin thickness across various collision systems and energies.

The Gist

Imagine two heavy atomic nuclei smashing into each other at nearly the speed of light. Inside these tiny, super-dense balls of matter, there are two main types of "passengers": protons (which carry a positive electric charge) and neutrons (which are neutral). When the crash happens, these passengers get slowed down and scattered. Physicists want to know exactly how they stop and where they end up.

This paper is like a detective story trying to solve a mystery: Is the slowing down of these particles caused by a new, strange force, or is it just because the "shape" of the nuclei is slightly uneven?

Here is a breakdown of their investigation using simple analogies:

1. The Mystery: Two Suspects

When protons and neutrons crash, they lose energy (they "stop"). The scientists are looking at two possible reasons for this:

• Suspect A (The "Baryon Junction"): A theoretical, exotic mechanism where protons and neutrons might get separated or slowed down differently than their electric charges. Think of this like a traffic jam where the trucks (protons) get stuck, but the cars (charged particles) keep moving freely.
• Suspect B (The "Neutron Skin"): Atomic nuclei aren't perfect spheres. They often have a "skin" of extra neutrons on the outside, like a fuzzy coat. If the nuclei are fuzzy, the collision geometry changes. It's like trying to stop a smooth billiard ball versus a fuzzy tennis ball; the fuzzy one behaves differently just because of its shape.

The problem is that in a crash, both suspects act at the same time. It's hard to tell if the weird behavior is due to the exotic traffic jam (Suspect A) or just the fuzzy coat (Suspect B).

2. The First Clue: The "Twin" Crashes

The scientists first looked at a specific experiment involving two "twin" nuclei: Ruthenium (Ru) and Zirconium (Zr).

• These twins are almost identical in weight and size, but they have slightly different numbers of protons and neutrons.
• Because they are so similar, any difference in how they stop must be due to their tiny structural differences (the "fuzzy coat" or neutron skin).
• By comparing these twins, the authors created a mathematical tool (a ratio) to measure the "excess stopping." They found that the exotic traffic jam (Suspect A) is indeed real, but you have to be very careful to subtract the effect of the fuzzy coat first. They calculated that the "excess stopping" is about 60% stronger than what you'd expect if protons and neutrons were just simple passengers.

3. The New Tool: The "Oxygen Baseline"

To solve the mystery for other heavy nuclei (like Gold, Lead, or Uranium), the authors needed a better ruler. They realized that if they compare a heavy, fuzzy nucleus against a very small, perfectly smooth nucleus, they could isolate the "fuzziness."

• The Smooth Ruler: They chose Oxygen-16. In their model, Oxygen is treated as a perfect, smooth sphere with no "fuzzy coat" (no neutron skin).
• The Test: They imagined crashing Oxygen into various heavy nuclei (like Copper, Gold, or Lead).
• The Result: Because Oxygen is smooth and predictable, any weirdness in the crash results comes entirely from the heavy nucleus's "fuzzy coat."

They created a new measurement called $R_{OX}$. Think of this as a "Fuzziness Score."

• If the heavy nucleus has a thick neutron skin, the score changes significantly depending on whether the crash was a direct hit (central) or a glancing blow (peripheral).
• If the nucleus is smooth, the score stays the same.

4. The Conclusion

The paper claims that by using this "Oxygen Baseline" method, scientists can now:

1. Measure the "Fuzzy Coat": They can determine exactly how thick the neutron skin is for heavy nuclei like Lead or Gold, just by looking at how the crash particles stop.
2. Separate the Suspects: They have built a framework that allows them to calculate the "excess stopping" (the exotic physics) without it being confused by the shape of the nucleus.

In short: The authors built a mathematical "filter" that separates the signal of new physics (how particles stop) from the noise of nuclear structure (how fuzzy the nuclei are). They proved that by using Oxygen as a smooth reference point, we can measure the "fuzziness" of heavy atoms with high precision, which helps us understand both the structure of atoms and the fundamental forces inside them.

Technical Summary

1. Problem Statement

In ultra-relativistic heavy-ion collisions, the yields of particles at midrapidity encode information about the transport of conserved charges (net baryon number $B$ and net electric charge $Q$). Recent proposals by the STAR Collaboration suggest using isobar collisions (Ru+Ru and Zr+Zr) to probe baryon-junction transport, a mechanism where baryon number is transported independently of valence quarks.

The primary observable for this is the ratio $r$:
$$r = \frac{B}{\Delta Q} \frac{\Delta Z}{A}$$
where $\Delta Q$ and $\Delta Z$ are differences in net charge and proton number between the two systems, and $B$ is the average net baryon number. A value $r > 1$ suggests excess baryon stopping relative to charge stopping.

The Core Challenge: The interpretation of $r$ is ambiguous. The observable is a convolution of:

1. Stopping Physics: The dynamical mechanism of charge transport (e.g., baryon junctions).
2. Nuclear Structure: The initial spatial distributions of protons and neutrons, specifically the neutron-skin thickness ($\Delta R_{np}$), which affects the fraction of participant protons in collisions.
3. Medium Evolution: The expansion of the quark-gluon plasma.

Current interpretations struggle to separate the "excess baryon stopping" parameter from the "neutron-skin" effects, particularly because the two nuclei in isobar collisions have different neutron skins.

2. Methodology

The authors develop a theoretical framework using the Thermal-FIST package to model the statistical hadronization of the system at chemical freeze-out.

A. Statistical Model Setup

• Assumptions: Midrapidity matter is a Hadron Resonance Gas (HRG) in chemical equilibrium.
• Inputs: Temperature ($T$) and chemical potentials ($\mu_B, \mu_Q, \mu_S$).
• Constraints:
  • Strangeness neutrality ($n_S = 0$).
  • A generalized charge-to-baryon ratio constraint:
    $$\frac{n_Q}{n_B} = \frac{p_{frac}(c)}{\gamma_B}$$
    Here, $p_{frac}(c)$ is the participant proton fraction (centrality-dependent), and $\gamma_B$ is a phenomenological parameter quantifying excess baryon stopping.
  • If $\gamma_B = 1$, baryon and charge stopping are identical (valence quark transport). If $\gamma_B > 1$, there is additional baryon transport (e.g., baryon junctions).

B. Nuclear Geometry and Neutron Skin

• 3D Glauber Model: Used to calculate participant nucleons ($N_{part}$) based on Woods-Saxon density profiles for protons and neutrons.
• Neutron Skin Modeling: The neutron skin thickness $\Delta R_{np}$ is modeled primarily by a difference in diffuseness parameters ($\Delta a = a_n - a_p$) between neutron and proton distributions, while keeping the half-density radii nearly identical.
• Baseline Nucleus: Oxygen-16 ($^{16}$O) is proposed as a reference nucleus with negligible neutron skin ($\Delta R_{np} \approx 0$) and spherical symmetry, serving as a "clean" baseline.

C. Observable Construction

The authors generalize the isobar ratio $r$ to arbitrary pairs of nuclei $(\Gamma, X)$:
$$r_{\Gamma X} = \frac{\Delta Z}{A} \frac{\tilde{B}_X}{\Delta \tilde{Q}} + \dots$$

• Experimental Proxy: Since experiments cannot measure neutrons directly, they reconstruct net baryon number ($\tilde{B}$) and net charge ($\Delta \tilde{Q}$) using identified hadron yields ($\pi, K, p$) and deuteron yields (to infer neutrons), following the STAR strategy.
• Scaling: To compare systems of vastly different sizes (e.g., O vs. Pb), they employ multiplicity scaling ($\sigma_X = \langle N_{ch} \rangle_{ev}$) to normalize yields.

3. Key Contributions

1. Quantitative Extraction of $\gamma_B$:
   The authors demonstrate that by comparing their thermal model predictions with hydrodynamic simulations (used as a proxy for experimental data), the excess baryon stopping parameter $\gamma_B$ can be quantitatively extracted from the centrality dependence of the isobar ratio $r_{ZrRu}$.

2. Disentanglement of Effects:
   They show that $\gamma_B$ acts primarily as an overall normalization factor (weak centrality dependence), whereas the neutron skin affects the centrality dependence of the ratio via the participant proton fraction $p_{frac}(c)$. This allows the two effects to be separated.

3. The Oxygen-Baseline Observable ($r_{OX}$):
   A novel contribution is the proposal to use Oxygen ($^{16}$O) as a reference nucleus. By forming ratios $r_{OX}$ (Oxygen vs. Target $X$), the neutron-skin effects of the reference are eliminated, isolating the neutron-skin physics of the target nucleus $X$.

4. The Central-to-Peripheral Ratio ($R_{OX}$):
   They define a robust observable:
   $$R_{OX} \equiv \frac{r_{OX}(0-5\%)}{r_{OX}(60-80\%)}$$
   This ratio is shown to be highly sensitive to the neutron-skin thickness $\Delta R_{np}$ of the target nucleus while being largely insensitive to the absolute value of $\gamma_B$.

4. Key Results

• Isobar Analysis (Ru+Ru vs. Zr+Zr):

  • Using hydrodynamic proxy data, the authors extract $\gamma_B = 1.61 \pm 0.06$.
  • They found that ignoring neutron-skin effects leads to a significant overestimation of $\gamma_B$ (e.g., using only central collision data yields $\gamma_B \approx 1.75$, an error $>10\%$).
  • The framework validates the STAR experimental strategy of using double ratios and deuteron yields to reconstruct conserved charges with high precision (few-percent level).

• Neutron Skin Sensitivity:

  • The centrality dependence of $r_{OX}$ closely tracks the participant proton fraction $p_{frac}(c)$.
  • $R_{OX}$ vs. $\Delta R_{np}$: A strong, approximately linear dependence is observed between $R_{OX}$ and the neutron-skin thickness $\Delta R_{np}$ for various nuclei (Cu, Ru, Zr, Au, U, Pb).
  • For Lead ($^{208}$Pb), the observable distinguishes between different theoretical predictions for neutron skin (e.g., PREX-II $\Delta R_{np} \approx 0.28$ fm vs. ab initio $\approx 0.17$ fm).

• Systematic Robustness:

  • The results are robust against different normalization schemes (density-matched vs. multiplicity-scaled).
  • Corrections arising from charge asymmetry and system mismatch in the double-ratio construction are found to be small (negligible for pions/kaons, manageable for protons).

5. Significance and Implications

• New Probe for QCD Mechanisms: The framework provides a controlled method to isolate and quantify baryon-junction transport (excess baryon stopping) from nuclear structure effects, a crucial step in understanding non-perturbative QCD dynamics.
• Neutron Skin Constraints: The proposed $R_{OX}$ observable offers a new, independent method to constrain neutron-skin thicknesses in heavy nuclei using heavy-ion collision data. This complements existing methods like PREX-II (parity-violating electron scattering) and dipole polarizability measurements.
• Experimental Roadmap: The paper provides a concrete strategy for upcoming RHIC and LHC experiments:
  1. Measure centrality-dependent $r_{ZrRu}$ to fix $\gamma_B$.
  2. Measure $r_{OX}$ (Oxygen vs. heavy nuclei) to extract $\Delta R_{np}$.
  3. Use the central-to-peripheral ratio $R_{OX}$ to minimize systematic uncertainties related to the stopping mechanism.

In summary, this work bridges the gap between statistical hadronization models and experimental observables, offering a rigorous tool to simultaneously study the transport properties of the QCD medium and the structural properties of atomic nuclei.