Probing the pion gluon distribution at small-$x$ in photon-induced interactions at LHC

This paper proposes analyzing heavy quark photoproduction associated with a leading neutron in proton-proton and proton-lead collisions at the LHC as a feasible method to probe the pion gluon distribution at small Bjorken-$x$ values, a kinematical range not previously covered by experiments.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks. For decades, physicists have been trying to figure out exactly how these bricks are arranged inside the most famous building block of all: the proton. We have a pretty good map of the proton's interior.

But there's another, smaller, and much more elusive building block called the pion. Think of the pion as the "ghost" of the particle world. It's light, it's short-lived, and it's incredibly hard to catch in a net to study. Because we can't easily hold a pion still, our map of its internal structure is full of blank spots, especially regarding a specific type of glue inside it called the gluon.

This paper is a proposal for a clever new way to map those blank spots using the world's biggest particle accelerator, the Large Hadron Collider (LHC).

Here is the story of their plan, broken down into simple concepts:

1. The Problem: The "Ghost" Particle

To see what's inside a pion, you usually have to smash it into something else. But pions are so unstable they fall apart almost instantly. You can't just put one in a jar and look at it.

Previously, scientists tried to study pions by smashing them into protons or electrons. But these methods only let us see the pion under specific conditions (like looking at it through a narrow keyhole). We need a way to see the pion in a different light, specifically when it's moving very fast and interacting with high-energy light (photons).

2. The Solution: The "Sullivan Process" (The Shadow Puppet Trick)

The authors propose using a trick called the Sullivan process. Imagine you are standing in front of a wall with a bright light behind you. You hold up a puppet, and a shadow appears on the wall.

In this experiment:

• The Light: The LHC accelerates protons (or lead nuclei) to near the speed of light. As they zoom past each other, they emit a massive burst of "light" (photons), even though they are charged particles.
• The Puppet: One of the protons is actually surrounded by a cloud of virtual pions (like a fuzzy halo).
• The Shadow: When the "light" from one proton hits the "pion cloud" of the other proton, it creates a shadow. By studying what gets created in that collision (specifically heavy particles called charm and bottom quarks), we can reverse-engineer the shape of the shadow.

This tells us about the gluons inside the pion. Gluons are the "glue" that holds the pion together. The paper suggests that by looking at how these heavy particles are produced, we can finally count how many gluons are in the pion and where they are hiding.

3. The "Leading Neutron" Clue

How do we know the collision happened with a pion and not just a regular proton? The paper suggests looking for a specific "witness" in the aftermath: a leading neutron.

Think of it like a game of billiards. If you hit a ball, and a specific piece of chalk flies off in a straight line, you know exactly what happened.

• In this collision, the proton breaks apart.
• Most of it turns into a mess of debris.
• But one piece, a neutron, flies straight ahead, carrying most of the original energy.
• Detecting this "leading neutron" is like finding a receipt that proves the collision happened with a pion. It acts as a tag, telling the scientists, "Hey, this event involved a pion!"

4. The Experiment: Two Types of Collisions

The team calculated what would happen in two scenarios at the LHC:

1. Proton-Proton (pp): Two protons colliding.
2. Proton-Lead (pPb): A proton colliding with a heavy lead nucleus.

They found that the Proton-Lead collisions are like using a super-bright flashlight. Because lead has many protons, it emits a much stronger "light" (more photons), making the signal much louder and easier to detect. They predict that the number of these events in lead collisions will be 1,000 times higher than in proton-only collisions.

5. The "Recipe" Test

The paper also proposes a clever way to double-check their results. They suggest comparing the production of Charm particles (lighter) vs. Bottom particles (heavier).

Imagine you are trying to figure out the strength of a specific type of flour by baking two cakes: one light sponge cake and one heavy fruitcake. If you change the flour, the heavy fruitcake will change its texture much more dramatically than the light sponge cake.

By looking at the ratio of Charm to Bottom particles, the scientists can cancel out many of the messy, confusing variables (like how the particles absorb energy). This ratio acts as a pure, clean signal that points directly to the structure of the pion's gluons.

Why Does This Matter?

Currently, our understanding of the pion's "glue" (gluons) at very small scales is a guess. This paper proposes a concrete, feasible experiment to turn that guess into a fact.

• If successful: We will finally have a complete map of the pion, filling in the last major gaps in our understanding of how matter is built.
• The Future: The authors say this isn't just theory; the LHC has the tools to do this now. They are essentially handing the experimentalists a blueprint and saying, "Look here, tag the neutron, count the heavy particles, and you'll see the pion's secret structure."

In short, this paper is a proposal to use the world's most powerful microscope to catch a ghost, using a specific "receipt" (the neutron) to prove we found it, and a special "recipe test" (Charm vs. Bottom) to understand exactly what it's made of.

Technical Summary

1. Problem Statement

Despite decades of research, the partonic structure of the pion, particularly the behavior of sea quarks and gluon distributions at small values of the Bjorken-$x$ variable, remains poorly understood. Previous constraints have come from pion-nucleus scattering (CERN, Fermilab) and leading neutron electroproduction at HERA. However, these experiments cover specific kinematic ranges. The authors aim to explore a complementary kinematical range accessible at the Large Hadron Collider (LHC) to probe the pion structure, specifically the pion gluon distribution ($g_\pi$), in a regime of high center-of-mass energy and small $x$ not previously accessible.

2. Methodology

The study proposes an exploratory analysis of inclusive heavy quark (charm and bottom) photoproduction associated with a leading neutron (LN) in ultraperipheral collisions (UPCs) at the LHC.

• Physical Process: The analysis relies on the Sullivan process, where a photon emitted by one incoming hadron (proton or nucleus) scatters off the virtual pion cloud of the target hadron ($p \to \pi + n$). The photon interacts with a gluon inside the pion ($\gamma g \to Q\bar{Q}$), producing a heavy quark pair.
• Final State Signature: The process is characterized by:
  • A heavy quark pair ($c\bar{c}$ or $b\bar{b}$).
  • A leading neutron carrying a large fraction of the proton momentum.
  • A rapidity gap (due to photon exchange).
  • The leading neutron can be tagged using a Zero Degree Calorimeter (ZDC).
• Formalism:
  • Equivalent Photon Approximation (EPA): Used to calculate photon fluxes from protons and nuclei.
  • Factorization: The cross-section is factorized into the pion flux ($f_{\pi/p}$), the photon-pion cross-section ($\sigma_{\gamma\pi}$), and an absorption factor ($K$) accounting for soft rescatterings.
  • Leading Order (LO) Calculation: The heavy quark photoproduction cross-section is calculated at LO in perturbative QCD, assuming the dominant channel is $\gamma g \to Q\bar{Q}$.
  • Parameterizations: The study utilizes three distinct parameterizations for the pion gluon distribution available in the LHAPDF library: GRV, JAM21, and xFitter. These differ in their initial DGLAP evolution conditions and data sets used for constraints.
• Collisions Studied: Proton-Proton ($pp$) at $\sqrt{s} = 13$ TeV and Proton-Lead ($pPb$) at $\sqrt{s} = 8.1$ TeV.

3. Key Contributions

• Novel Probe: This is the first proposal to use inclusive heavy quark photoproduction with a leading neutron in UPCs to probe the pion structure at the LHC.
• Kinematical Reach: Demonstrates that LHC UPCs probe photon-proton center-of-mass energies ($W_{\gamma p}$) up to $\sim 8.2$ TeV, significantly exceeding the reach of HERA, thereby accessing much smaller values of $x$.
• Sensitivity Analysis: Quantifies the sensitivity of the cross-sections and rapidity distributions to different pion gluon PDF parameterizations.
• Robust Observable: Proposes the ratio of charm to bottom rapidity distributions ($d\sigma_c/dY / d\sigma_b/dY$) as a sensitive probe that minimizes dependence on theoretical uncertainties such as absorptive effects, the pion flux model, and the hard scale ($\mu$).

4. Results

• Pion Gluon Distributions: The three parameterizations (GRV, JAM21, xFitter) all predict an increase in the gluon distribution at small $x$, but the magnitude and shape differ significantly. The uncertainty bands for JAM21 and xFitter widen at very small $x$, indicating a lack of current experimental constraints in this region.
• Cross-Sections:
  • The total cross-sections for heavy quark production increase with collision energy ($W_{\gamma\pi}$) and are larger for charm than for bottom (due to mass thresholds).
  • $pPb$ vs. $pp$: Cross-sections in $pPb$ collisions are approximately 3 orders of magnitude larger than in $pp$ collisions due to the $Z^2$ enhancement in the nuclear photon flux.
  • Magnitude: For $pp$ at 13 TeV, charm cross-sections are in the range of $\sim 100$ nb (depending on the PDF), while bottom cross-sections are $\sim 10$ nb. For $pPb$ at 8.1 TeV, charm cross-sections reach $\sim 100$ $\mu$b.
• Rapidity Distributions:
  • The distributions are symmetric in $pp$ collisions but asymmetric in $pPb$ (dominated by photon-proton interactions).
  • The differences between predictions using GRV, JAM21, and xFitter become more pronounced at large rapidities ($|Y|$), where smaller $x$ values are probed.
  • At mid-rapidity ($Y \approx 0$), the central predictions for charm production differ by a factor of $\approx 2$ between parameterizations.
• The Ratio ($c/b$): The ratio of charm to bottom rapidity distributions is shown to be largely independent of the absorption factor ($K$) and the hard scale ($\mu$), making it a robust observable to distinguish between different pion gluon PDFs.

5. Significance

• Constraining Pion Structure: The study demonstrates that measuring these processes at the LHC can provide crucial new constraints on the pion gluon distribution at small $x$, a regime currently unconstrained by existing data.
• Feasibility: The predicted cross-sections are large enough to be experimentally feasible at the LHC, particularly in $pPb$ collisions.
• Future Impact: The results motivate the implementation of these processes in Monte Carlo event generators and future experimental analyses. The proposed ratio observable offers a way to reduce theoretical systematic uncertainties, potentially allowing for a more precise determination of the pion's internal structure before the advent of future facilities like the Electron-Ion Collider (EIC) or EicC.

In conclusion, the paper establishes that inclusive heavy quark photoproduction with a leading neutron in LHC UPCs is a powerful and viable tool for exploring the gluonic content of the pion at high energies and small Bjorken-$x$.