The size of the quark-gluon plasma in ultracentral collisions: impact of initial density fluctuations on the average transverse momentum

This paper analytically demonstrates that the variation of the quark-gluon plasma volume in ultracentral collisions depends on initial density fluctuations and is negligible when total entropy scales with the mass number, thereby establishing that measurements of average transverse momentum can probe detailed nuclear structure and pre-equilibrium dynamics.

The Gist

Imagine two giant, fluffy clouds of particles (atomic nuclei) smashing into each other at nearly the speed of light. When they collide, they create a tiny, super-hot drop of liquid called Quark-Gluon Plasma (QGP). This is the hottest, densest matter in the universe, similar to what existed just microseconds after the Big Bang.

For a long time, physicists thought they knew exactly how this drop behaved. But a new paper by Fabian Zhou, Giuliano Giacalone, and Jean-Yves Ollitrault suggests that the story is a bit more complicated—and that looking at how "fast" the particles fly out of this drop can tell us secrets about the very structure of the atoms themselves.

Here is the breakdown of their discovery using simple analogies.

1. The Setup: The "Perfect" Collision

Usually, when we smash two nuclei together, we can hit them slightly off-center (like two cars grazing each other). But in ultracentral collisions, the scientists aim for a perfect bullseye. The two nuclei hit dead center.

Because the hit is so perfect, the only thing that makes one collision different from another is quantum randomness. Think of it like two identical snowflakes hitting each other. Even though they look the same from a distance, the tiny ice crystals inside are arranged slightly differently. In the nucleus, these "crystals" are the protons and neutrons.

2. The Old Theory: The "Fixed Balloon"

For years, physicists believed that if you increased the number of particles coming out of the collision (the "multiplicity"), it was like pumping more air into a fixed-size balloon.

• More particles = Higher pressure and temperature inside the same space.
• Result: The particles flying out would move faster (higher average momentum, or $\langle p_T \rangle$).

This worked perfectly for a long time. It was like saying, "If you squeeze a balloon harder, the air shoots out faster, but the balloon doesn't get bigger."

3. The New Twist: The "Stretchy Balloon"

The authors of this paper asked a simple question: What if the balloon doesn't stay the same size?

They ran simulations using a popular computer model called TRENTo. They realized that depending on how the "ingredients" (the protons and neutrons) are mixed inside the nucleus, the resulting plasma might actually shrink or swell when you have more particles, rather than just getting hotter.

• The Analogy: Imagine you are making a soup.
  • Old View: If you add more ingredients (multiplicity), the pot stays the same size, so the soup gets thicker and hotter.
  • New View: Depending on how you chop the ingredients, adding more might actually make the pot expand (swell) or contract (shrink).

4. The Secret Ingredient: The "Recipe" (The Exponent $\nu$)

The paper introduces a variable called $\nu$ (nu), which acts like a recipe parameter. It determines how the density of the plasma is calculated based on the thickness of the colliding nuclei.

• The "Standard Recipe" ($\nu = 0.5$): If you use the standard recipe, the size of the plasma stays constant no matter how many particles you have. The "fixed balloon" theory holds true.
• The "Weird Recipes" ($\nu \neq 0.5$): If you tweak the recipe, the plasma changes size.
  • If $\nu$ is low, the plasma swells as you add more particles (like a balloon expanding).
  • If $\nu$ is high, the plasma shrinks (like a balloon being squeezed tighter).

5. The Detective Work: Listening to the "Pop"

How do we know which recipe nature is using? We can't see the plasma directly because it evaporates too fast. Instead, we listen to the "pop" of the particles flying out.

The authors show that the speed of the outgoing particles ($\langle p_T \rangle$) is a direct clue.

• If the plasma stays the same size, the speed increases at a specific, predictable rate.
• If the plasma changes size, that rate changes.

By measuring the speed of particles in the most perfect collisions at the Large Hadron Collider (LHC), we can figure out if the plasma is swelling, shrinking, or staying put.

6. Why This Matters: The "Nuclear X-Ray"

This isn't just about hot soup; it's about the structure of the atom.

The paper argues that if the plasma size stays constant (which the standard recipe predicts), it means the fluctuations in the nucleus are distributed in a very specific, simple way. It implies that the "randomness" we see in the collision comes purely from the arrangement of protons and neutrons in a single nucleus, without any weird new physics happening during the crash.

The Big Picture:
If we measure the speed of particles and find that the plasma size does change, it would tell us that our understanding of how protons and neutrons are arranged inside the nucleus is incomplete. It would be like discovering that a snowflake has a hidden internal structure we never knew about, just by watching how it melts.

Summary

• The Problem: We thought smashing atoms together just made a hotter, denser drop of plasma.
• The Discovery: The drop might actually change its size depending on the internal "recipe" of the atoms.
• The Method: By measuring how fast the particles fly out, we can tell if the drop is swelling or shrinking.
• The Goal: This acts as a high-tech X-ray, allowing us to see the detailed, microscopic arrangement of protons and neutrons inside the heaviest atoms in the universe.

In short, this paper turns the speed of flying particles into a ruler, allowing physicists to measure the invisible shape of the universe's smallest building blocks.

Technical Summary

1. Problem Statement

Recent experiments at the Large Hadron Collider (LHC) have observed that the mean transverse momentum ($\langle p_T \rangle$) of outgoing particles increases with particle multiplicity ($N_{ch}$) in ultracentral (0–0.2% centrality) Pb+Pb collisions.

• Standard Interpretation: This was originally interpreted via hydrodynamic simulations assuming a constant volume. Under this assumption, an increase in multiplicity implies an increase in entropy density and temperature, leading to a higher $\langle p_T \rangle$ governed by the speed of sound ($c_s^2 = d\ln\langle p_T \rangle / d\ln N_{ch}$).
• The Conflict: Recent state-of-the-art simulations (e.g., Trajectum) suggest that the volume of the Quark-Gluon Plasma (QGP) is not constant in ultracentral collisions. Depending on the initial condition model, the system may shrink or swell as multiplicity increases.
• The Gap: There is currently a lack of experimental constraints on the spatial distribution of initial-state fluctuations (specifically the two-point function of the entropy density) in the transverse plane. The only existing constraint is on the total entropy variance.

The paper aims to elucidate how the variation of the QGP volume with multiplicity depends on the specific model of initial conditions and to determine if $\langle p_T \rangle$ measurements can constrain the microscopic structure of these fluctuations.

2. Methodology

The authors employ a combination of analytical derivations and Monte Carlo simulations using the TRENTo model for initial conditions.

• Generalized Initial Conditions: They parametrize the initial entropy density $s(x)$ as a function of the nuclear thickness functions $t_A(x)$ and $t_B(x)$:
  $$s(x) \propto (t_A(x) t_B(x))^\nu$$
  • The standard model uses $\nu = 0.5$ (derived from pre-equilibrium dynamics where $s \propto \sqrt{t_A t_B}$).
  • The authors vary $\nu$ (e.g., 0.33, 0.67, 0.75, 1.00) to test different scaling behaviors.
• Fluctuation Tuning: To match experimental multiplicity distributions (specifically the tail of the distribution at $b=0$), they tune the parameter $k$ of the gamma distribution used to weight participant nucleons. This ensures that the variance of the total entropy matches experimental data for all values of $\nu$.
• System Size Definition: They define the root-mean-square (RMS) transverse size $R$ of the QGP based on the initial entropy density profile:
  $$R^2 \equiv \frac{1}{S} \int d^2x \, |x|^2 s(x) - \left| \frac{1}{S} \int d^2x \, x s(x) \right|^2$$
• Analytical Decomposition: They decompose the entropy density into a mean profile $\kappa_1(x)$ and fluctuations $\delta s(x)$. They derive a perturbative relation linking the change in system size to the correlation between local density fluctuations and total entropy fluctuations (the two-point function integrated over one coordinate, $\kappa_2(x)$).

3. Key Contributions and Theoretical Framework

A. The Special Case of $\nu = 0.5$

The authors analytically prove (Appendix A) that for the default model ($\nu = 0.5$), the excess entropy density (fluctuations) is distributed in the exact same spatial profile as the mean entropy density.

• Result: Consequently, the system size $R$ remains constant regardless of the total entropy $S$ (or multiplicity).
• Implication: If the physical world follows $\nu = 0.5$, the standard hydrodynamic interpretation (constant volume) holds true.

B. Deviation for $\nu \neq 0.5$

When $\nu \neq 0.5$, the spatial distribution of fluctuations differs from the mean profile:

• $\nu < 0.5$: Excess density is concentrated at the edges of the fireball. As multiplicity increases, the system expands (radius increases).
• $\nu > 0.5$: Excess density is concentrated in the center. As multiplicity increases, the system shrinks (radius decreases).
• Physical Mechanism: This is driven by the conservation of the total number of nucleons. For $\nu > 0.5$, high central density implies fewer participants at the edges, leading to a negative correlation between local edge density and total entropy.

C. Connection to Observables ($\langle p_T \rangle$)

The authors derive a modified relation for the slope of $\langle p_T \rangle$ vs. $N_{ch}$:
$$\frac{d \ln \langle p_T \rangle}{d \ln N_{ch}} = c_s^2 \left[ 1 - \frac{3}{2} \left( \frac{R_2^2}{R_1^2} - 1 \right) \right]$$
Where $R_1$ is the mean radius and $R_2$ is a radius defined by the fluctuation correlation function $\kappa_2(x)$.

• If $R_2 = R_1$ (the $\nu=0.5$ case), the slope is simply $c_s^2$.
• If $R_2 \neq R_1$, the slope is modified by the volume variation term.

4. Results

1. Volume Variation: Monte Carlo simulations confirm that for $\nu = 0.5$, the average squared radius $\langle R^2 \rangle$ is constant for ultracentral events ($b=0$). For $\nu < 0.5$, the radius increases with entropy; for $\nu > 0.5$, it decreases.
2. Agreement with Trajectum: The authors' perturbative formula (Eq. 27) successfully reproduces the trends seen in full hydrodynamic simulations (Trajectum) where the parameter $q$ (related to $\nu$) is varied. The deviation in the $\langle p_T \rangle$ slope observed in Trajectum is quantitatively explained by the volume variation derived in this paper.
3. Sensitivity: The slope parameter is highly sensitive to the ratio $R_2/R_1$. Even small deviations from $\nu=0.5$ lead to measurable changes in the $\langle p_T \rangle$ slope.

5. Significance and Implications

• Probing Nuclear Structure: The paper argues that measuring the precise slope of $\langle p_T \rangle$ in ultracentral collisions provides a direct probe of the two-point function of initial entropy fluctuations.
• Constraint on Initial Conditions: If experimental data confirms the standard slope ($c_s^2$), it strongly supports the $\nu=0.5$ model, implying that initial fluctuations are purely those of a single nucleus (no new correlations generated by the collision).
• Nuclear Ground State Correlations: If the volume is constant ($\nu=0.5$), it validates the theoretical link between multi-particle correlations in heavy-ion collisions and many-body correlations in the ground state of the nucleus. This would allow colliders to probe nuclear structure (e.g., nucleon-nucleon correlations) with unprecedented detail.
• Pre-equilibrium Dynamics: Conversely, if the slope deviates significantly, it implies that pre-equilibrium dynamics or nuclear structure models generate non-trivial spatial correlations that alter the volume scaling, challenging current assumptions about the thermalization process.

In summary, this work provides a rigorous analytical and numerical framework connecting the macroscopic observable $\langle p_T \rangle$ to the microscopic spatial distribution of initial-state fluctuations, offering a new tool to test models of nuclear structure and the early stages of heavy-ion collisions.