Equation of state and cumulants of proton multiplicity in equilibrium near critical point from Pade estimates

This paper proposes a method to constrain proton multiplicity cumulants in heavy-ion collisions by leveraging Pade-resummed lattice QCD data on the Lee-Yang singularity structure, revealing four distinct scenarios for critical point signatures that depend on the critical point's location and the slope of the chiral crossover curve.

The Gist

Imagine the universe as a giant, cosmic soup. When you heat this soup up (like in the early universe or inside a particle collider), the ingredients—quarks and gluons—melt together into a chaotic, super-hot fluid called Quark-Gluon Plasma (QGP). As the soup cools down, these ingredients freeze back into solid particles (protons and neutrons), much like water turning into ice.

Physicists have long suspected that this "freezing" process isn't always smooth. They think there might be a specific spot in the recipe book where the transition changes from a gentle melt to a sudden, explosive snap. This spot is called the Critical Point. Finding it is the "Holy Grail" of modern particle physics.

Here is a simple breakdown of what this paper does to help us find that spot.

1. The Problem: The Recipe is Missing Pages

We know the rules of the game (Quantum Chromodynamics, or QCD), but we can't calculate the exact "recipe" for the soup at high pressures and temperatures because the math gets too messy. It's like trying to predict exactly how a cake will rise without being able to see the oven or measure the heat perfectly.

We also can't just run the experiment at every possible setting because the machines (like the Large Hadron Collider or RHIC) can only hit certain "beam energies." We need a way to guess what happens in the spots we haven't visited yet.

2. The Detective Work: Using "Lee-Yang" Clues

The authors act like detectives. They know that near a Critical Point, the physics behaves in a very specific, universal way (similar to how magnets behave near their Curie point).

Instead of guessing the whole recipe, they look for singularities. Think of these as "ghosts" in the math. In the complex world of numbers, there are invisible points (called Lee-Yang singularities) that dictate how the soup behaves.

• The Analogy: Imagine you are trying to find a hidden island (the Critical Point) on a foggy ocean. You can't see the island, but you can see the ripples in the water caused by the island's gravity. By studying the pattern of these ripples (the singularities) using a mathematical tool called Padé resummation (a fancy way of connecting the dots between known data points), they can estimate where the island is.

3. The Map: Four Possible Scenarios

Using these mathematical ripples, the authors created a map. They realized that depending on exactly where the Critical Point is and how the "freezing line" (where the soup turns to ice) crosses the "melting line," there are four distinct scenarios:

1. Hot & No Crossing: The Critical Point is "hot" (above the freezing line), and the two lines never touch.
2. Hot & Crossing: The Critical Point is hot, but the freezing line cuts right through the melting line.
3. Cool & Crossing: The Critical Point is "cool" (below the freezing line), and the lines cross.
4. Cool & No Crossing: The Critical Point is cool, and the lines stay apart.

Why does this matter?
Think of the "freezing line" as the path a car takes down a mountain. The "Critical Point" is a pothole on the road.

• If the car drives over the pothole (Crossing), the ride gets bumpy in a specific way.
• If the car drives under the pothole (No Crossing), the bumps happen differently.
• If the pothole is on a steep hill (Hot) vs. a flat road (Cool), the car reacts differently.

4. The Measurement: Counting Protons

How do we know which scenario is real? We look at the protons (the particles) that fly out of the collision.

• The Analogy: Imagine throwing a handful of marbles into a room. If the room is calm, the marbles scatter evenly. If there is a hidden whirlpool (the Critical Point), the marbles will clump together or spread out in weird patterns.
• The authors calculated how these "clumps" (called cumulants) would look for each of the four scenarios.
  • Scenario A: You might see a big peak (a huge clump) followed by a dip.
  • Scenario B: You might see a deep dip (a sudden lack of marbles) followed by a peak.

5. The Big Conclusion

The paper says: "We don't know exactly where the Critical Point is yet, but we know the shape of the signal it will leave behind."

• If the experiments see a Peak in the data, the Critical Point is likely "Hot."
• If they see a Dip, it's likely "Cool."
• If the signal changes suddenly, the freezing line likely crossed the melting line.

Why is this important?

This paper is like giving the experimenters a Wanted Poster. It doesn't tell them exactly where the criminal (the Critical Point) is hiding, but it tells them exactly what the criminal looks like (a peak or a dip in the data).

By comparing their experimental data to these four "Wanted Posters," scientists can finally narrow down the search. If the data matches the "Cool & Crossing" poster, they know they are on the right track. If it matches "Hot & No Crossing," they can rule out the other three.

In short: The authors used advanced math to predict the "fingerprint" of the QCD Critical Point. Now, experimentalists just need to look at their proton data to see which fingerprint matches, bringing us one step closer to solving one of the universe's biggest mysteries.

Technical Summary

1. Problem Statement

The existence and location of the QCD critical point (CP) in the phase diagram of strongly interacting matter remain one of the most significant open questions in high-energy nuclear physics. The search for this point relies heavily on analyzing fluctuations of conserved quantities (specifically proton multiplicity) in heavy-ion collisions.

However, connecting experimental observables to the theoretical QCD equation of state (EoS) is challenging because:

• Non-perturbative nature: The critical region is inaccessible to standard perturbation theory.
• Lattice limitations: Lattice QCD calculations suffer from the "sign problem" at finite baryon chemical potential ($\mu_B$), preventing direct simulation of the critical region.
• Mapping ambiguity: While the critical point belongs to the 3D-Ising universality class, the non-universal mapping parameters that relate Ising variables ($r, h$) to QCD variables ($T, \mu_B$) are unknown.
• Freeze-out complexity: Translating hydrodynamic fluctuations in the Quark-Gluon Plasma (QGP) to final-state hadron observables requires a robust framework (freeze-out) that accounts for the interplay between the critical EoS and the experimental freeze-out curve.

The paper aims to constrain the shape of proton multiplicity cumulants by utilizing the analytical properties of the QCD EoS near the critical point, specifically leveraging Lee-Yang edge singularities estimated via Padé resummation of lattice data.

2. Methodology

The authors employ a multi-step theoretical framework combining lattice QCD constraints, universality mapping, and statistical freeze-out models:

A. Lee-Yang Singularities and Padé Resummation

• Concept: The QCD EoS possesses singularities in the complex $\mu_B$ plane. The Lee-Yang (LY) edge singularities are complex conjugate pairs that merge on the real axis at the critical point ($T_c, \mu_c$).
• Technique: The authors use conformal Padé approximants to extrapolate lattice QCD Taylor expansion coefficients (calculated at $\mu_B=0$) to higher $\mu_B$ and complex values.
• Trajectory Construction: By identifying the location of these singularities ($\mu_{LY}(T)$) at different temperatures, they reconstruct the "Lee-Yang trajectory."
• Parameter Extraction: The trajectory near the critical point follows a scaling form:
  $$\mu_{LY}(T) \approx \mu_c - c_1(T - T_c) \pm i x_{LY} c_2 (T - T_c)^{\beta\delta}$$
  From this, they extract:
  • $T_c, \mu_c$: Critical temperature and chemical potential.
  • $c_1 = \tan \alpha_1$: Slope of the crossover/first-order transition line.
  • $c_2$: A non-universal parameter related to the mapping scale.

B. Mapping to the Ising Model

The singular part of the QCD pressure is mapped to the 3D-Ising EoS ($G_{Ising}$) via linear (and quadratic) transformations involving reduced temperature $r$ and magnetic field $h$. The mapping depends on non-universal parameters:

• $\alpha_1, \alpha_2$: Angles determining the orientation of the $h=0$ (crossover) and $r=0$ axes in the $T-\mu$ plane.
• $\rho, w$: Parameters defining the size and shape of the critical region.
• Key Insight: The authors define a rescaled parameter $\bar{\rho} = \rho w^{1-1/\beta\delta}$. They demonstrate that the shape of the factorial cumulants depends on $\bar{\rho}$, while $w$ only scales the overall magnitude. Crucially, the imaginary part of the LY trajectory (via $c_2$) constrains $\bar{\rho}$ as a function of $\alpha_2$.

C. Maximum Entropy Freeze-out

To connect the hydrodynamic fluctuations (derived from the EoS) to experimental proton multiplicities, the authors use the Maximum Entropy Freeze-out framework.

• This method assumes that at chemical freeze-out, the system maximizes entropy subject to constraints where the average values and correlators of conserved quantities (energy, baryon number) in the hadron gas match those of the hydrodynamic fluid.
• They calculate the irreducible relative correlators (IRCs) to isolate "genuine" critical correlations from the background hadron resonance gas (HRG) contributions.
• The final observables are the normalized factorial cumulants of proton multiplicity, denoted as $\hat{\Delta}\omega_{kp}$ (for $k=2, 3, 4$).

D. Scenario Classification

The authors identify four topologically distinct scenarios based on the relative position of the critical point and the crossover curve with respect to the experimental freeze-out curve ($T_F(\mu)$):

1. H (Hot, No Crossing): $T_c > T_F$ everywhere; freeze-out stays below the crossover.
2. HX (Hot, Crossing): $T_c > T_F$, but the freeze-out curve intersects the crossover line before reaching the critical point.
3. C (Cool, No Crossing): $T_c < T_F$ everywhere; freeze-out stays above the crossover.
4. CX (Cool, Crossing): $T_c < T_F$, but the freeze-out curve intersects the crossover line.

3. Key Contributions

1. Constraint via Lee-Yang Trajectory: The paper establishes a direct link between the imaginary part of the Lee-Yang singularity trajectory (constrained by Padé resummation) and the shape of the proton multiplicity cumulants. This reduces the freedom in choosing non-universal mapping parameters.
2. Topological Scenarios: It systematically categorizes the possible experimental signatures into four distinct scenarios (H, HX, C, CX) based on the geometry of the freeze-out curve relative to the critical region.
3. Role of $\alpha_2$: The study highlights the critical role of the angle $\alpha_2$ (orientation of the Ising $h$-axis). It determines whether the freeze-out curve traverses regions of positive or negative higher-order cumulants, thereby dictating whether the signature is a peak or a dip.
4. Quantitative Predictions: The authors provide specific predictions for the behavior of $\hat{\Delta}\omega_{2p}, \hat{\Delta}\omega_{3p},$ and $\hat{\Delta}\omega_{4p}$ as a function of collision energy ($\sqrt{s}$) for each scenario, constrained by the uncertainty bands of the Padé estimates.

4. Results

The results are presented as contour plots of baryon density correlators and the resulting factorial cumulants along the freeze-out curve:

• General Trends:

  • $\hat{\Delta}\omega_{2p}$ (Variance): Generally exhibits a peak near the critical point. The position shifts slightly with $\bar{\rho}$ and $\alpha_1$.
  • $\hat{\Delta}\omega_{3p}$ (Skewness): Shows the most dramatic variation between scenarios.
    • Hot Scenarios (H, HX): Typically dominated by a peak. In scenario HX (with crossing), a dip may precede the peak if the curve crosses the $\Delta H_{3n}=0$ contour.
    • Cool Scenarios (C, CX): Typically dominated by a dip. In scenario CX, a small peak may appear at higher energies if the crossing occurs close to the critical point.
  • $\hat{\Delta}\omega_{4p}$ (Kurtosis): Generally shows a dip followed by a peak as collision energy decreases.

• Scenario Specifics:

  • Scenario H (Hot, No Crossing): A single prominent peak in $\hat{\Delta}\omega_{3p}$.
  • Scenario HX (Hot, Crossing): A dip followed by a peak in $\hat{\Delta}\omega_{3p}$. The "crossing" creates a sharper transition in the signal.
  • Scenario C (Cool, No Crossing): A prominent dip in $\hat{\Delta}\omega_{3p}$.
  • Scenario CX (Cool, Crossing): A dip in $\hat{\Delta}\omega_{3p}$, potentially preceded by a peak at higher energies.

• Impact of Parameters:

  • Increasing $\bar{\rho}$ stretches the contours, displacing the peaks/dips to higher collision energies (lower $\mu$).
  • The uncertainty in $\bar{\rho}$ (derived from Padé errors) constrains the location of these signatures, though the qualitative shape (peak vs. dip) remains robustly determined by the scenario type.

5. Significance and Implications

• Discriminating Power: The paper demonstrates that the qualitative shape of the third cumulant ($\hat{\Delta}\omega_{3p}$) is a powerful discriminator between "hot" and "cool" critical points and between scenarios with and without curve crossings. This offers a concrete way to interpret Beam Energy Scan (BES) data.
• Theoretical Consistency: By using Padé resummation to constrain the non-universal parameters, the authors ensure that their predictions are consistent with current lattice QCD data, moving beyond purely phenomenological models.
• Discrepancy with Current Data: The authors note that the predicted locations of the peaks/dips (based on Padé estimates placing the CP at $\mu_c \sim 600$ MeV) are at higher chemical potentials (lower collision energies) than where current experimental data (RHIC BES) shows non-monotonic behavior.
  • Conclusion: This suggests either the critical point is at a lower $\mu_c$ than predicted by current extrapolations, or non-equilibrium effects (which are not included in this equilibrium study) play a dominant role in shifting the signatures.
• Foundation for Future Analysis: The work provides a rigorous set of "informed priors" for future Bayesian analyses of BES data, helping to constrain the likelihood distribution of the QCD equation of state parameters.

In summary, this paper bridges the gap between lattice QCD constraints on the EoS singularity structure and experimental observables, providing a robust, scenario-dependent prediction for proton multiplicity fluctuations that can guide the search for the QCD critical point.