How uncertain are model predictions for the muon content of extensive air showers

This paper investigates how uncertainties in model predictions for the muon content of extensive air showers relate to specific hadron-air interaction properties, utilizing the new QGSb Monte Carlo generator to analyze interaction mechanisms and their constraints from accelerator data.

The Gist

Imagine the Earth's atmosphere as a giant, invisible bowling alley. When a cosmic ray (a super-fast particle from deep space) hits the top of the atmosphere, it's like throwing a bowling ball down the lane. But instead of hitting pins, it smashes into air molecules, creating a massive, chaotic explosion of smaller particles. This explosion is called an Extensive Air Shower (EAS).

As this shower crashes down toward the ground, it splits into two main groups:

1. The Electromagnetic Crowd: Particles that turn into light and heat (like photons and electrons). They get absorbed quickly by the air.
2. The Muon Team: Heavy, ghost-like particles called muons. They are tough; they can punch through the atmosphere and reach the ground, where our detectors catch them.

The Great Mystery: The "Muon Puzzle"

For years, scientists have been trying to predict exactly how many muons should reach the ground. They use powerful computer simulations (like a video game engine for physics) to model these collisions.

Here's the problem: The real world has more muons than the computer predicts. It's like the bowling alley simulation says there should be 100 pins left standing, but when you look, there are actually 110. This is the "Muon Puzzle."

The Detective Work: Why are there more muons?

The author of this paper, Sergey Ostapchenko, acts like a detective trying to figure out why the computer is underestimating the muon count. He realizes that the number of muons depends on a specific rule in the "game": How much energy is passed down to the heavy particles (muons) versus the light particles (photons)?

Think of the energy in the shower as a pie.

• If the pie is sliced mostly into neutral pions (which turn into light/photons), the energy disappears into the "electromagnetic sink," and fewer muons are made.
• If the pie is sliced mostly into charged pions and other stable particles, that energy stays in the "hadronic cascade," keeps bouncing around, and eventually creates more muons.

The paper asks: Can we tweak the computer models to slice the pie more in favor of the muon-makers without breaking the laws of physics?

The Three Suspects (and the Constraints)

The author tests three different ways to tweak the model to get more muons. He uses a new, sophisticated simulation tool called QGSb. However, he has a strict judge: Accelerator Data.

Imagine the computer model is a recipe. You can't just add more sugar (muons) if the taste testers (accelerator experiments like NA61/SHINE and EHS-NA22) say the cake tastes wrong.

Here are the three "tweaks" he tried:

1. The "Pion Exchange" Trick (The ρ-meson)

• The Idea: In a collision, a pion might swap places with a virtual particle, turning into a heavier particle called a rho meson ($\rho$). When this rho meson decays, it produces more charged pions (muon-makers) and fewer neutral ones.
• The Result: The author tried to increase this swapping rate.
• The Verdict: It worked a little bit (adding ~1% more muons), but it clashed with data from the NA61/SHINE experiment. It's like trying to add extra chocolate chips, but the experiment says, "No, the texture is wrong."

2. The "Strange Quark" Trick (Kaons)

• The Idea: Maybe the collisions are creating more Kaons (particles containing "strange" quarks) than we thought. Kaons are heavy and stable enough to keep the energy flowing toward muons.
• The Result: The author tweaked the model to create more Kaons.
• The Verdict: This boosted the muon count by about 5%. However, it created a huge conflict with other experiments. The model started predicting too many Kaons in proton collisions, which the data says is impossible.

3. The "Nucleon" Trick (Protons/Antiprotons)

• The Idea: Maybe the collisions are creating more protons and antiprotons moving forward than we thought.
• The Result: The author tweaked the model to create more of these heavy particles.
• The Verdict: This boosted the muon count by about 6%. But again, it failed the "taste test." The model predicted twice as many protons as the LEBC-EHS experiment actually saw.

The Final Conclusion: How Uncertain are we?

After testing all these ideas, the author draws a sobering conclusion:

• The Limit: Even if we push the models to their absolute limit—ignoring some conflicting data to maximize the muon count—we can only explain about 10% of the missing muons.
• The Real Problem: To explain the entire missing muon count, the models would need to assume that the production of these heavy particles increases as the energy gets higher.
• The Catch: There is no known physics theory that says this happens. In fact, the Large Hadron Collider (LHC) and the FASER experiment have already checked this and said, "No, the production doesn't rise with energy."

The Takeaway

The paper is essentially saying: "We have tried every legal move in the rulebook to fix the computer models, but we can't get them to match the real world."

The uncertainty in predicting muon numbers is currently capped at about 10% by what we know from particle accelerators. If the "Muon Puzzle" is bigger than that, it suggests we are missing a fundamental piece of physics—perhaps something "exotic" that we haven't discovered yet. Until then, our cosmic ray simulations are like a map that is accurate for 90% of the journey, but gets lost in the final, most important stretch.

Technical Summary

1. Problem Statement

The paper addresses the "muon puzzle" in cosmic ray physics: experimental measurements of the muon number ($N_\mu$) in extensive air showers (EAS) at ground level consistently exceed the predictions made by current Monte Carlo (MC) simulation models.

• Context: High-energy cosmic ray studies rely on reconstructing primary particle properties from EAS characteristics. This requires accurate modeling of hadronic interactions in the atmosphere.
• Core Issue: Current models (e.g., QGSJET, EPOS) underpredict the muon content. Since $N_\mu$ is closely correlated with the energy distribution of stable secondary hadrons (pions, kaons, nucleons) in pion-air interactions, the discrepancy suggests a gap in understanding forward particle production in high-energy hadron collisions.
• Objective: To quantify the uncertainty in $N_\mu$ predictions by analyzing specific interaction mechanisms within a new MC generator, QGSb, and determining how much these predictions can be adjusted without violating existing accelerator data.

2. Methodology

The author utilizes the QGSb Monte Carlo generator to simulate cosmic ray interactions and investigates three specific mechanisms that influence the forward production of stable hadrons:

1. Theoretical Framework: The study focuses on the moment $\langle x_E^{\alpha_\mu} n_{\pi-air}^{stable} \rangle$, where $\alpha_\mu \approx 0.9$. This moment represents the average fraction of energy carried by stable secondary hadrons in pion-air collisions. Enhancing this moment increases the hadronic cascade energy, thereby increasing $N_\mu$.
2. Mechanism Analysis:
   • Pion Exchange ($\rho$ meson production): Investigating the conversion of incident pions into $\rho$ mesons via virtual pion emission. This process alters the charge ratio of secondary pions (changing the energy split from 2:1 to 3:1 between charged and neutral pions), keeping more energy in the hadronic channel.
   • Kaon Production: Analyzing the fragmentation of valence quarks via the creation of strange quark-antiquark pairs ($\bar{s}s$) from the vacuum, which affects forward kaon yields.
   • (Anti)nucleon Production: Investigating the creation of diquark-antidiquark pairs ($\bar{u}u, \bar{d}d$) from the vacuum to enhance forward proton and antiproton yields.
3. Validation: The modified models are tested against experimental data from fixed-target accelerator experiments, specifically NA61/SHINE (pion-carbon collisions) and EHS-NA22/LEBC-EHS (pion-proton collisions).

3. Key Contributions

• Quantification of Uncertainty: The paper provides a rigorous upper bound on how much the predicted muon content can be increased by tuning interaction parameters within the constraints of current accelerator data.
• Mechanism-Specific Constraints: It demonstrates that different mechanisms (pion exchange, kaon production, nucleon production) face distinct tensions with experimental data.
• Role of Forward Spectra: It reinforces that the muon content is dominated by the forward production of stable hadrons (high Feynman-$x_F$), rather than the total multiplicity in the central rapidity region.

4. Results

A. Pion Exchange and $\rho$ Mesons

• Effect: Increasing the rate of pion exchange (enhancing $\rho$ meson production) shifts the energy balance toward charged pions.
• Constraint: While this improves agreement with EHS-NA22 data (pion-proton), it creates a tension with NA61/SHINE data (pion-carbon).
• Outcome: Even with an enhanced pion exchange rate to fit EHS-NA22, the resulting increase in $N_\mu$ is minimal (~1%).

B. Kaon Production ($\bar{s}s$ pairs)

• Effect: Increasing the probability of $\bar{s}s$ pair creation enhances forward charged kaon yields.
• Constraint: This modification improves the fit for $K^-$ in pion-carbon collisions but leads to a significant overestimation of $K^0_S$ and creates serious tension with pion-proton data.
• Outcome: This yields a 0.4–1% increase in the energy moment, translating to a ~5% enhancement in $N_\mu$ (due to the multiplicative nature of cascade steps where kaons interact rather than decay).

C. (Anti)nucleon Production (Diquark Pairs)

• Effect: Including the creation of $\bar{u}u$ and $\bar{d}d$ diquark pairs (in addition to $\bar{u}d$) enhances forward proton and antiproton production.
• Constraint: While this brings the model into agreement with NA61/SHINE data for both protons and antiprotons, it results in a factor of two overestimation of proton/antiproton spectra in pion-proton collisions compared to LEBC-EHS data.
• Outcome: This modification yields a ~0.5% increase in the energy moment and up to a ~6% increase in $N_\mu$.

D. Overall Uncertainty Limit

• By maximizing forward yields of stable hadrons within the bounds of current experimental uncertainties (accepting tensions with some datasets), the author concludes that the predicted EAS muon content can be enhanced by at most ~10%.
• Theoretical Limit: To achieve a stronger enhancement (solving the muon puzzle completely), models would need to assume that forward yields of stable hadrons rise with collision energy. However, no viable theoretical approach supports this, and such a rise is already disfavored by LHC data (e.g., FASER experiment for forward kaons).

5. Significance

• Limits of Standard Physics: The paper establishes that standard hadronic interaction mechanisms, even when pushed to their experimental limits, cannot fully resolve the muon puzzle (which often requires >20% enhancement).
• Exotic Scenarios: It suggests that if the muon puzzle is real and not a systematic error, it likely points to exotic physics (e.g., new interaction mechanisms) that would affect both pion-nucleus and proton-proton collisions.
• Discrimination via LHC: The author notes that such exotic mechanisms can be discriminated by Large Hadron Collider (LHC) measurements, particularly in the forward region, as they would manifest as an energy-dependent rise in forward particle yields not seen in current data.

In summary, the paper argues that the uncertainty in model predictions for EAS muon content is constrained to roughly 10% by current accelerator data. Resolving the remaining discrepancy likely requires physics beyond the standard hadronic interaction models.