Addressing uncertainties of model predictions for extensive air showers initiated by high energy cosmic rays

This paper utilizes a new hadronic collision Monte Carlo generator, QGSb, to investigate how specific model modifications affect predictions for extensive air shower characteristics—namely shower maximum depth and ground-level muon number—while analyzing the underlying physics and consistency with accelerator data to address uncertainties in cosmic ray composition studies.

The Gist

Imagine the universe is constantly bombarding Earth with invisible, ultra-fast bullets called cosmic rays. These aren't normal bullets; they are subatomic particles traveling at nearly the speed of light. When one of these high-energy bullets hits the Earth's atmosphere, it doesn't just stop. Instead, it smashes into an air molecule and triggers a massive, cascading explosion of other particles. Scientists call this a "shower" (specifically, an Extensive Air Shower).

Think of it like throwing a single bowling ball into a stack of dominoes. The first hit knocks over a few, which knock over more, creating a massive wave of falling dominoes that spreads out across the floor.

The paper you provided is about a team of scientists trying to build a better simulation (a computer model) of how these cosmic ray showers behave. They are using a new tool called QGSb, which is like a sophisticated video game engine for particle physics. Their goal is to figure out how much "wiggle room" or uncertainty exists in their predictions.

Here is a breakdown of their two main experiments, explained simply:

1. The "Depth" Problem: How deep does the shower go?

When a cosmic ray hits the atmosphere, the shower grows bigger and bigger until it reaches a peak (the most particles at once), and then it starts to die out. Scientists measure the depth of this peak, called $X_{max}$.

• The Mystery: Real-world experiments (like the Pierre Auger Observatory) are seeing showers that peak deeper in the atmosphere than the current computer models predict. It's like the dominoes are falling further down the hallway than the physics teacher expected.
• The Fix Attempt: The scientists tried to tweak the rules of their simulation to make the showers go deeper.
  • Idea A: They tried making the initial collision "softer" (less energetic), hoping the shower would take longer to build up.
  • The Result: They found that to make the shower go significantly deeper, they had to change the fundamental rules of how particles share energy. However, when they checked these new rules against data from the Large Hadron Collider (LHC) (the world's biggest particle accelerator), the rules failed. The LHC data said, "No, particles behave this way, not that way."
  • The Twist: They also tried a theory called "diquark breaking" (imagine a tightly held pair of particles suddenly letting go of each other). They thought this would make the shower develop faster (shallower depth), but the simulation showed it barely changed anything.
• The Conclusion: The models are likely already as "deep" as they can get without breaking the laws of physics as we know them. If the real showers are deeper, it might mean the cosmic rays are made of heavier, stranger particles than we thought, not that our physics models are just slightly off.

2. The "Muon" Puzzle: Where are all the muons?

Muons are a specific type of particle produced in these showers. When scientists count the muons hitting the ground, they find more of them than the computer models predict. This is known as the "Muon Puzzle."

• The Mystery: The simulation is underestimating the number of muons. It's like the dominoes are producing more "special tokens" (muons) than the math says they should.
• The Fix Attempt: The scientists tried to tweak the simulation to produce more muons.
  • Idea A: They tried changing how particles decay (break apart). They hoped that by making certain particles live longer or decay differently, more energy would stay in the "particle chain" and create more muons.
  • Idea B: They tried to increase the production of heavy particles (like protons and kaons) at the front of the shower.
  • The Result: They managed to increase the predicted number of muons by a small amount (up to about 5%). However, to do this, they had to make the simulation predict particle behaviors that contradicted other experimental data. For example, changing the rules to get more muons made the simulation predict the wrong number of other particles (like pions) that we can actually measure in the lab.
• The Conclusion: You can't just "turn up the volume" on muons without breaking the rest of the physics. The uncertainty in the model is limited by what we know from accelerator experiments. The "Muon Puzzle" remains a puzzle because the current models are already doing the best they can within the known rules of physics.

The Big Picture

The authors are essentially saying: "We tried to break our own model to see how wrong it could be."

They tested extreme scenarios to see if they could force the model to match the strange data from the sky (deep showers, too many muons). Every time they tried to force a match, the model broke the rules established by the Large Hadron Collider.

The Takeaway:
The uncertainty in our predictions isn't as huge as we might hope. The models are tightly constrained by real-world lab data. If the cosmic ray data still doesn't match the models, it suggests that either:

1. We are missing a fundamental piece of physics (a new rule of the universe).
2. The cosmic rays hitting us are made of something much heavier and stranger than we currently believe.

They didn't find a simple "tweak" to fix the models; instead, they proved that the models are robust, and the mystery lies in the nature of the cosmic rays themselves.

Technical Summary

Problem Statement
Experimental studies of ultra-high energy cosmic rays (UHECRs) rely on reconstructing primary particle properties from extensive air shower (EAS) characteristics, specifically the average shower maximum depth ($X_{max}$) and the muon number at ground level ($N_\mu$). These reconstructions depend heavily on Monte Carlo (MC) generators modeling hadronic interactions in the atmosphere. A significant challenge lies in quantifying the uncertainties of these generators. While comparing different existing models offers some insight, current models may share common deficiencies or miss critical physics, potentially leading to underestimated uncertainties or systematic deviations from reality. The Pierre Auger Observatory, for instance, indicates that current models may require substantially larger $X_{max}$ values to be consistent with observed mass compositions, while a "muon puzzle" suggests models underpredict the number of muons in UHECR-induced showers.

Methodology
The authors utilize a new MC generator of hadronic collisions, QGSb, which is based on a transparent theoretical formalism (Quark-Gluon String model) but possesses sufficient parameter freedom to explore model uncertainties. The study involves systematically modifying specific model parameters to alter predictions for $X_{max}$ and $N_\mu$ while strictly adhering to constraints imposed by accelerator data (e.g., LHC, TOTEM, ALICE, NA61/SHINE, EHS-NA22).

The investigation focuses on two main areas:

1. Modifying $X_{max}$: The authors explore mechanisms to either increase or decrease the predicted shower maximum depth. This involves adjusting the energy dependence of the inelastic proton-air cross-section, the inelasticity of interactions ($K_{inel}$), and the momentum distributions of constituent partons. Specific modifications include:
   • Tuning the overcriticality of the "semihard Pomeron" ($\Delta_{sh}$).
   • Altering the momentum distribution exponent of sea (anti)quarks ($\alpha^{(sh)}_{sea}$).
   • Implementing a "diquark breaking" mechanism where the coherence of valence quarks in a nucleon is destroyed during multiple scattering, potentially increasing inelasticity.
2. Modifying $N_\mu$: The authors investigate ways to enhance the muon content by altering the energy balance between stable secondary hadrons (nucleons, kaons) and neutral pions ($\pi^0$) in pion-air interactions. Since $N_\mu$ is dominated by forward production of stable hadrons, the study focuses on:
   • Enhancing $t$-channel pion exchange to favor charged over neutral pions.
   • Increasing the production of forward (anti)baryons and kaons in the projectile fragmentation region by modifying string fragmentation parameters (e.g., probabilities for creating strange quark pairs or diquark-antidiquark pairs from the vacuum).

Key Contributions and Results

• Increasing $X_{max}$: The study finds that the most effective way to increase $X_{max}$ is to assume softer momentum distributions for the constituent partons attached to color field strings. By increasing the exponent $\alpha^{(sh)}_{sea}$ from the default 0.5 to 0.8 and 0.95 (Tunes 3 and 4), the inelasticity of proton-nitrogen collisions is reduced by up to 11%, resulting in an increase of $X_{max}$ by up to $\approx 40$ g/cm$^2$ at $10^{20}$ eV.

  • Constraint: These extreme modifications are disproved by recent ALICE data at $\sqrt{s}=13$ TeV. The ALICE experiment observes a strong anticorrelation between central charged hadron multiplicity and forward neutron yield. Tunes 3 and 4 predict a much weaker anticorrelation, contradicting the data. Furthermore, these modifications lead to a maximal muon production depth ($X^\mu_{max}$) that would require primary nuclei heavier than iron to match Pierre Auger data, which is physically unlikely.
  • Alternative: A moderate reduction in the Pomeron overcriticality ($\Delta_{sh} = 0.21$ vs. 0.22, Tune 2) yields a smaller increase in $X_{max}$ ($\approx 7$ g/cm$^2$) and remains consistent with ALICE correlations, though it slightly underestimates energy loss in leading neutrons.

• Decreasing $X_{max}$: The authors tested the "diquark breaking" mechanism (Tune 5), where multiple scattering destroys diquark coherence, theoretically increasing inelasticity. Despite a predicted 3% increase in inelasticity at high energies, this mechanism produced no noticeable change in the average $X_{max}$ (within statistical uncertainties of 1–2 g/cm$^2$). The authors conclude that the default model's predictions for $X_{max}$ are likely already close to the lower bound, as peripheral interactions (which dominate the average) remain unaffected by this mechanism.

• Enhancing $N_\mu$: The study confirms that $N_\mu$ is highly sensitive to the forward production of stable hadrons.

  • Enhancing forward kaon production (Tune 7) increased $N_\mu$ by up to $\approx 2\%$. However, this tune overestimates forward kaon yields in other collision systems (e.g., $K^+$ in $\pi p$ collisions) and fails to match certain experimental spectra.
  • Enhancing forward (anti)nucleon production (Tune 8) by modifying the hadronization of "remnant" quarks to include more baryonic content increased $N_\mu$ by up to $\approx 5\%$. While this tune improves agreement with some NA61/SHINE data, it leads to a factor-of-two excess in proton/antiproton production compared to LEBC-EHS data at 360 GeV/c.
  • The study highlights that the uncertainty in $N_\mu$ is currently limited by serious contradictions between different accelerator measurements of (anti)proton yields in the fragmentation region.

Significance and Claims
The paper claims to provide a detailed mapping of the "parameter space" available for tuning hadronic interaction models within the limits of current accelerator data. Its primary significance lies in:

1. Ruling out extreme scenarios: It demonstrates that the most drastic modifications to $X_{max}$ (via soft parton distributions) are incompatible with LHC forward physics data (ALICE) and Auger muon depth data.
2. Identifying limits on $X_{max}$ reduction: It challenges literature claims that diquark splitting significantly lowers $X_{max}$, showing that such mechanisms have negligible impact on the average shower maximum when constrained by fixed-target proton spectra.
3. Quantifying the "Muon Puzzle" uncertainty: It establishes that increasing $N_\mu$ by 5% is theoretically possible within the QGSb framework but requires assumptions about forward (anti)nucleon production that are currently in conflict with existing experimental data.

The authors conclude that while the QGSb generator allows for exploring these uncertainties, the current constraints from accelerator data (particularly regarding forward particle production and correlations) significantly limit the ability to resolve the discrepancies between EAS observations and model predictions without introducing new physics mechanisms not yet considered.