Momentum fraction and hard scale dependence of double parton scattering in heavy-ion collisions

This paper extends the study of double parton scattering in proton-proton collisions to heavy-ion systems by incorporating nuclear effects and a model for transverse parton distributions, demonstrating that the effective cross section's dependence on final-state observables can probe the transverse structures of both bound nucleons and nuclei.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: When Two Cars Crash, Do They Hit Twice?

Imagine two massive trucks speeding toward each other on a highway. Usually, when they crash, we think of it as one big impact: the front bumper of Truck A hits the front bumper of Truck B. In physics, this is called Single Parton Scattering (SPS). It's the standard, expected crash.

But, these trucks aren't solid steel; they are actually made of thousands of tiny, invisible marbles (called partons) flying around inside them. Sometimes, during the crash, it's not just one marble hitting another. It's possible that two pairs of marbles collide at the exact same time. This is called Double Parton Scattering (DPS).

Think of it like this: Two people are throwing handfuls of confetti at each other. Usually, one piece of confetti from Person A hits one from Person B. But occasionally, two pieces from A hit two pieces from B simultaneously. That's a "double hit."

The Problem: The "Pocket Formula" is Too Simple

For a long time, physicists used a simple rule (a "pocket formula") to guess how often these double hits happen. They assumed that the marbles inside the trucks were spread out evenly, like sprinkles on a donut. If you know how big the donut is, you can guess the odds of two sprinkles hitting.

However, recent experiments showed this simple rule is wrong. The "donut" isn't uniform. The marbles clump together or spread out depending on how fast they are moving and how hard the crash is. The "effective cross-section" (a fancy way of saying "how big the target looks") changes depending on what kind of particles are created in the crash.

The New Study: Crashing Heavy Trucks (Heavy-Ion Collisions)

The authors of this paper took their improved model (which works well for regular proton crashes) and applied it to Heavy-Ion Collisions.

• Proton-Proton (pp): Two small trucks crashing.
• Proton-Lead (pA): A small truck crashing into a massive, dense freight train.
• Lead-Lead (AA): Two massive freight trains crashing into each other.

In these heavy collisions, things get weird because the "freight train" (the nucleus) is huge and crowded. The marbles inside the train cars (nucleons) behave differently than marbles in a single truck.

The Three Big Ideas (The "Secret Sauce")

The authors proposed three main changes to their model to make it work for these heavy crashes:

1. The "Crowded Room" Effect (Shadowing)

Imagine a crowded elevator. If you try to walk through it, you have to squeeze past people. In a heavy nucleus, the "marbles" (partons) are so crowded that they interfere with each other.

• Shadowing: At low speeds (low energy), the marbles hide behind each other, making the nucleus look "thinner" or less dense in the middle.
• Antishadowing: At medium speeds, they actually push together, making the nucleus look "denser" or "fatter."
  The authors created a model where the shape of the nucleus changes based on how fast the marbles are moving. It's like a crowd that shrinks when it's too packed and expands when it's just right.

2. The "Stretched Rubber Band" (Bound Nucleons)

The authors hypothesized that a marble inside a truck that is part of a massive train (a bound nucleon) is more spread out than a marble in a solo truck (a free proton).

• Analogy: Imagine a rubber band. In a free truck, the rubber band is tight. In a heavy train, the rubber band is stretched out because the train is pulling on it.
• Why it matters: If the marbles are more spread out, it's harder for two of them to hit two others at the same time. This changes the math significantly.

3. The "Two-Track" Crash

In a crash between a small truck and a train (pA), there are two ways the double hit can happen:

• 1x1: Both marbles come from the same car in the train.
• 1x2: One marble comes from one car, and the other comes from a different car in the train.
  The authors found that the "same car" (1x1) scenario is actually the most sensitive to how the marbles are arranged inside that single car.

What They Found

The team ran their new math on data from the Large Hadron Collider (LHC), specifically looking at crashes involving Lead (Pb) nuclei.

• The Match: Their new model, which includes the "stretched rubber band" idea, matched the real-world data much better than the old, simple models.
• The Prediction: They predicted what would happen in future crashes. They found that if you look at different types of particles being created (like heavy J/ψ particles vs. lighter W particles), the "effective size" of the collision changes.
  • In pA collisions: The results tell us about the shape of the marbles inside a single bound nucleon.
  • In AA collisions: The results tell us about the shape of the entire nucleus and how the marbles are distributed across the whole train.

Why Should You Care?

This isn't just about crashing trucks. It's about understanding the architecture of matter.

By studying these "double hits," physicists can map out the invisible 3D shape of protons and nuclei. It's like using X-rays to see inside a black box. The authors show that by looking at how the collision changes based on what comes out of it, we can learn:

1. How tightly packed the "marbles" are inside a single proton.
2. How the "crowd" of marbles behaves when they are squeezed into a giant nucleus.

In short: The paper proves that heavy-ion collisions are a powerful microscope. By refining how we calculate these double crashes, we can finally see the hidden, shifting shapes of the building blocks of our universe.

Technical Summary

Here is a detailed technical summary of the paper "Momentum fraction and hard scale dependence of double parton scattering in heavy-ion collisions" by Lovato, Huayra, and de Oliveira.

1. Problem Statement

Double Parton Scattering (DPS) occurs when two hard parton-parton interactions happen within a single inelastic hadron-hadron collision. While extensively studied in proton-proton ($pp$) collisions, the understanding of DPS in heavy-ion collisions (proton-nucleus $pA$ and nucleus-nucleus $AA$) remains limited.

The core challenges addressed in this work are:

• Universality Assumption: Standard theoretical models often assume the effective cross section ($\sigma_{\text{eff}}$) is a universal constant independent of the final state observables. However, experimental data in $pp$ collisions shows $\sigma_{\text{eff}}$ depends on the kinematics (momentum fractions $x$ and hard scales $\mu$).
• Nuclear Effects: Existing studies of DPS in $pA$ and $AA$ collisions typically employ fixed transverse profiles for nucleons and nuclei, neglecting how nuclear effects (shadowing and antishadowing) modify the transverse distribution of partons.
• Bound Nucleon Structure: It is unclear how the transverse separation of partons inside a nucleon bound within a nucleus differs from that in a free proton.

2. Methodology

The authors extend a phenomenological model previously developed for $pp$ collisions to the heavy-ion sector. The approach relies on a factorized formalism where the DPS cross section is expressed as the product of two Single Parton Scattering (SPS) cross sections and a scale factor $\Theta$ that encodes transverse geometry.

Key Theoretical Components:

• Factorized Formalism: The effective cross section is defined as:
  $$\frac{1}{\sigma_{\text{eff}}(CD)} = \int \dots \Theta(\dots) \sigma_C \sigma_D$$
  Here, $\Theta$ encapsulates the transverse distribution of partons, their correlations, and dependence on longitudinal momentum fractions ($x$) and hard scales ($\mu$).
• Transverse Profiles:
  • Free Proton: Uses a Gaussian profile with a width $B(x, \mu)$ fitted to $pp$ data, dependent on $x$ and $\mu$.
  • **Bound Nucleon ($1\times1$mechanism):** The authors hypothesize that partons inside a bound nucleon are more widely separated than in a free proton, particularly at small$x$. This is implemented by adding a term$\gamma_A$to the Gaussian variance in the scale factor$\Theta_{1\times1}$.
  • **Nuclear Profile ($1\times2$and$2\times2$mechanisms):** For partons originating from different nucleons, the authors propose a new transverse profile$\rho(x, r)$that depends on the nuclear modification factor$R_g(x)$.
    • Shadowing ($R_g < 1$): Leads to a broadening of the transverse profile (partons are more spread out).
    • Antishadowing ($R_g > 1$): Leads to a narrowing of the profile (partons are more compact).
    • This profile is constructed using the Woods-Saxon distribution modified by an exponential term involving $R_g$, ensuring normalization without free parameters.
• Collision Decomposition:
  • $pA$ Collisions: Decomposed into $1\times1$(both partons from one nucleon) and$1\times2$ (partons from two different nucleons) contributions.
  • $AA$ Collisions: Decomposed into four terms: $1\times1$,$1\times2$,$2\times1$, and$2\times2$. The$2\times2$ term (partons from two nucleons in each nucleus) is found to be dominant.
• Inputs: The model utilizes PYTHIA 8.3 for SPS cross sections, NNPDF2.3 for protons, and nNNPDF3.0 (NLO) for nuclear PDFs (specifically for Lead-208).

3. Key Contributions

1. Extension to Heavy Ions: Successfully generalized the $pp$ DPS model to $pA$ and $AA$ collisions, accounting for nuclear modifications.
2. Dynamic Transverse Profiles: Introduced a novel framework where the transverse parton distribution is not static but dynamically modified by shadowing and antishadowing effects derived from nuclear PDFs.
3. Bound Nucleon Hypothesis: Proposed and tested the hypothesis that partons in bound nucleons are more spatially separated than in free protons at small $x$, introducing a specific parameter ($\gamma_A$) to model this.
4. Final-State Dependence: Demonstrated that $\sigma_{\text{eff}}$ is not universal in heavy-ion collisions but varies significantly based on the final state (particle type) and rapidity.

4. Results

The study compares theoretical predictions with available experimental data and provides new predictions for the LHC.

• Comparison with $pPb$ Data:
  • The model was tested against $pPb$ data at $\sqrt{s} = 8.16$ TeV (CMS and LHCb measurements of $J/\psi$, $D^0$, and combinations).
  • A model assuming bound nucleons are identical to free protons ($\gamma_A = 0$) failed to reproduce the rapidity dependence of the data (specifically the trend between forward and backward rapidities).
  • Introducing the hypothesis of wider separation in bound nucleons ($\gamma_A = 1$ mb) brought the theoretical predictions into reasonable agreement with experimental lower limits.
• Predictions for $pPb$:
  • Predictions for various final states ($W^+W^+$, $\Upsilon\Upsilon$, $J/\psi J/\psi$, $D^0 D^0$, etc.) show a hierarchy: $\sigma_{\text{eff}}^{\text{central}} \ll \sigma_{\text{eff}}^{\text{backward}} \lesssim \sigma_{\text{eff}}^{\text{forward}}$.
  • This variation is driven primarily by the $1\times1$ contribution and the sensitivity of different processes to the shadowing region.
• Predictions for $PbPb$:
  • For $PbPb$ collisions at $\sqrt{s} = 5.5$ TeV, the $2\times2$ contribution dominates (by two orders of magnitude).
  • Here, $\sigma_{\text{eff}}$ is driven by the nuclear profile modifications (shadowing/antishadowing) rather than nucleon-nucleon correlations.
  • The effective cross section varies by approximately 10% across different final states. Lighter observables (probing smaller $x$) generally yield larger $\sigma_{\text{eff}}$ due to shadowing-induced broadening.

5. Significance

• Probing Nuclear Structure: The work establishes that DPS in heavy-ion collisions is a powerful tool to probe the transverse structure of both free protons and bound nucleons.
• Differentiating Mechanisms: It highlights a clear distinction between $pA$ and $AA$ collisions:
  • In $pA$, DPS is sensitive to the internal transverse correlations of a single bound nucleon.
  • In $AA$, DPS is sensitive to the global transverse profile of the nucleus modified by multiple scattering effects (shadowing/antishadowing).
• Future Experimental Guidance: The paper provides specific predictions for unmeasured channels at the LHC. It suggests that measuring the dependence of $\sigma_{\text{eff}}$ on final states and rapidity can constrain the value of $\gamma_A$ (separation in bound nucleons) and refine our understanding of nuclear parton distributions.
• Theoretical Advancement: By moving beyond the "pocket formula" (constant $\sigma_{\text{eff}}$), the authors provide a more realistic, kinematic-dependent description of multi-parton interactions in nuclear environments.