⚛️ phenomenology

Impact of coherent scattering on relic neutrinos boosted by cosmic rays

This paper calculates the diffuse flux of relic neutrinos boosted by ultra-high-energy cosmic rays, accounting for coherent scattering enhancements to constrain neutrino overdensity using current observatory data and exploring the potential explanation for a recent KM3NeT event.

Original authors: Jiajie Zhang, Alexander Sandrock, Jiajun Liao, Baobiao Yue

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiajie Zhang, Alexander Sandrock, Jiajun Liao, Baobiao Yue

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Cosmic "Ping-Pong" Game

Imagine the universe is filled with a ghostly, invisible fog made of neutrinos. These are tiny particles left over from the Big Bang, like the dust settling after a massive explosion. They are everywhere, but they are so shy and light that they almost never hit anything. Scientists call this the Cosmic Relic Neutrino Background (CνB). Detecting them directly is like trying to catch a single grain of sand in a hurricane using a butterfly net.

Now, imagine Ultra-High-Energy Cosmic Rays (UHECRs). These are atomic nuclei (like the cores of atoms) shot through space at speeds almost as fast as light. They are the "bullets" of the universe, carrying massive amounts of energy.

The Paper's Discovery:
The authors of this paper realized that when these super-fast "bullets" (Cosmic Rays) crash into the invisible "fog" (Relic Neutrinos), something special happens. It's not just a simple bump; it's a coherent ping-pong game.

The "Coherent" Magic: The Synchronized Swimmer Effect

Usually, when a fast particle hits an atom, it hits the individual protons and neutrons inside, like a billiard ball hitting a single marble.

But here is the twist: Because the relic neutrinos are so light and the cosmic rays are so fast, the neutrino sees the entire atomic nucleus (like a heavy Iron atom) as a single, solid target. Instead of hitting one tiny part, the neutrino interacts with all the parts of the nucleus at the same time.

The Analogy:
Think of a crowd of people (the protons and neutrons) standing in a tight circle.

Incoherent Scattering (Normal): If you throw a ball at them, it might hit one person, who bumps into the next. It's messy and weak.
Coherent Scattering (This Paper): If you throw a giant, soft net (the neutrino) at the whole circle, and everyone in the circle moves in perfect synchronization, the impact is massive. The force isn't just $N$ times stronger; it's $N^2$ times stronger.

The paper shows that because the cosmic rays are mostly made of heavy nuclei (like Iron, not just Hydrogen), this "synchronized" effect makes the collision much more powerful than previously thought. It's like upgrading from a pebble to a cannonball because the target is a heavy, synchronized team.

The Result: A "Boosted" Signal

When these heavy cosmic rays hit the relic neutrinos, they give the neutrinos a massive energy boost.

Before: The neutrino is a sleepy, slow-moving ghost.
After: The neutrino gets kicked by the cosmic ray and becomes a high-speed "super-neutrino."

The authors calculated that these "boosted" neutrinos should arrive at Earth with a specific energy signature, peaking around 200 PeV (that's 200 quadrillion electron-volts—an energy level that is hard to imagine).

The Detective Work: Checking the Evidence

The team used data from two giant detectors:

IceCube: A massive telescope buried in the Antarctic ice.
Pierre Auger Observatory: A giant array in Argentina that watches for cosmic rays hitting the atmosphere.

They asked: "If our theory is right, and there are a lot of these relic neutrinos, we should see a specific number of these super-high-energy neutrinos hitting our detectors."

The Findings:

The Limit: They didn't see enough of these events to prove the neutrinos are super dense, but they did set a new, stricter limit. They proved that the "fog" of relic neutrinos cannot be more than about 100 million times denser than the standard prediction. This is a huge improvement over previous guesses.
The Mystery Event: Recently, a detector called KM3NeT spotted a single, incredibly high-energy neutrino event (named KM3-230213A). The energy of this event matches the "peak" predicted by the authors' theory.
- The Analogy: It's like looking for a specific type of rare bird. You haven't seen a flock, but you saw one bird that looks exactly like the one you were hunting. The authors say, "Hey, maybe this one bird is a boosted relic neutrino!" However, they also warn that this is a long shot and might just be a coincidence or a different type of cosmic particle.

Why This Matters

New Physics: It shows that we need to stop treating cosmic rays as just simple protons. The heavy elements (Iron, Silicon) play a huge role because of this "coherent" effect.
A New Way to Hunt: Instead of trying to catch the slow, shy relic neutrinos directly (which is incredibly hard), we can look for the "footprints" they leave when they get hit by cosmic rays.
Future Hope: With new, bigger telescopes coming online (like IceCube-Gen2), we might finally be able to see these boosted neutrinos clearly. If we do, it would be the first direct proof that the "ghost fog" of the Big Bang actually exists right here, right now, surrounding us.

Summary in One Sentence

This paper suggests that heavy cosmic rays act like giant slingshots, flinging the invisible ghosts of the Big Bang (relic neutrinos) toward Earth at super speeds, and by looking for these flung ghosts, we can finally prove they are there.

1. Problem Statement

The Cosmic Relic Neutrino Background (CνB) is a fundamental prediction of the standard $\Lambda$ CDM cosmological model, yet it has never been directly detected due to its extremely low energy ( $\sim 10^{-4}$ eV). While direct detection experiments like PTOLEMY face significant technical hurdles, an indirect method involves Ultra-High-Energy Cosmic Rays (UHECRs) scattering off CνB neutrinos, "boosting" them to observable high energies.

Previous studies investigating this mechanism suffered from two major limitations:

Composition Assumption: They predominantly assumed UHECRs consist of protons, neglecting the growing observational evidence (from Pierre Auger Observatory and Telescope Array) that UHECRs at energies $>10$ EeV are dominated by heavy nuclei (e.g., Iron, Silicon).
Cross-Section Simplification: They often used oversimplified scattering cross-sections (e.g., approximating differential cross-sections as $\sigma/E_{max}$ ) that ignored the suppression of form factors at high momentum transfers and failed to account for coherent elastic neutrino-nucleus scattering (CE $\nu$ NS).

The paper addresses the need to recalculate the boosted CνB flux by incorporating the heavy-nuclei composition of UHECRs and the significant enhancement of the scattering cross-section due to coherent interactions.

2. Methodology

A. Scattering Cross-Sections

The authors derive the differential cross-sections for UHECR nuclei ( $N_i$ ) scattering off non-relativistic relic neutrinos. They distinguish between two channels:

Coherent Scattering (CE $\nu$ NS): Occurs when the momentum transfer $q$ $q$ is small ( $qR \lesssim 1$ $q R ≲ 1$ , where $R$ $R$ is the nuclear radius). The neutrino interacts with the entire nucleus collectively. The cross-section is enhanced by a factor of $N^2$ $N^{2}$ (where $N$ $N$ is the number of neutrons) and depends on the nuclear weak charge $Q_W$ $Q_{W}$ .
- Formula: $\frac{d\sigma_{coh}}{dE_\nu} \propto G_F^2 m_\nu Q_W^2 F^2(q^2)$ .
Incoherent Scattering: Occurs at high momentum transfers where the neutrino resolves individual nucleons. The cross-section scales linearly with the number of nucleons ( $A$ $A$ ).
- Formula: $\frac{d\sigma_{incoh}}{dE_\nu} \propto \sum (Z \sigma_{\nu p} + N \sigma_{\nu n}) (1 - F^2(q^2))$ .

Key Theoretical Nuance: The authors explicitly calculate these for both Dirac and Majorana neutrinos. They demonstrate that for the non-relativistic CνB, the difference between Dirac and Majorana cross-sections is negligible (suppressed by $O(m_\nu/E_\nu)^2$ ), unlike in capture scenarios where a factor of 2 difference exists. They also correct previous literature by noting that a simple Lorentz boost of standard CE $\nu$ NS formulas overestimates the cross-section by a factor of $\sim 2$ because CνB neutrinos are non-relativistic helicity eigenstates, not ultra-relativistic chiral states.

B. UHECR Flux Modeling

To calculate the boosted flux, the authors employ two distinct approaches for the UHECR source distribution and composition:

PriNCe Simulation: A numerical propagation code simulating UHECRs from sources to Earth, accounting for cosmological expansion, energy losses (pair production, photohadronic interactions), and magnetic deflection. They use the Star Formation Rate (SFR) as the primary source evolution model.
Hillas Parametrization: A phenomenological model fitting the current energy spectrum, normalized to Pierre Auger Observatory (PAO) data. This model assumes a higher fraction of heavy nuclei compared to the PriNCe simulation.

They consider three source evolution models: SFR, Quasistellar Objects (QSO), and Gamma-Ray Bursts (GRB).

C. Flux Calculation

The differential flux of boosted neutrinos at Earth is calculated by integrating the scattering rate over redshift ( $z=0$ to $z=6$ ), weighting by the UHECR flux at each redshift, the CνB number density (scaled by an overdensity factor $\eta$ ), and the derived differential cross-sections.

3. Key Contributions

Inclusion of Coherent Enhancement: The paper establishes that for heavy nuclei (like Iron) scattering off CνB neutrinos, the coherent channel dominates at lower boosted neutrino energies ( $E_\nu \lesssim 10^7$ GeV). This results in a cross-section enhancement of $\sim 2$ orders of magnitude compared to proton-only scattering in certain energy regimes.
Correction of Cross-Section Approximations: The authors demonstrate that previous works (e.g., Ref. [15]) overestimated the boosted flux by neglecting the nucleon form factor suppression at high energies. Their rigorous treatment of $q$ -dependence yields a more accurate, albeit lower, flux prediction at the highest energies.
Heavy Nuclei Dominance: By incorporating the heavy-nuclei composition of UHECRs, the study shows that the total boosted flux is significantly higher than proton-only predictions, particularly in the coherent regime.
Dirac vs. Majorana Equivalence: They provide a rigorous derivation showing that for UHECR-boosted CνB scattering, the distinction between Dirac and Majorana neutrinos is negligible, simplifying future constraints.

4. Results

Boosted Flux Spectrum: The boosted CνB flux peaks at approximately 200 PeV.
- At lower energies ( $E_\nu \lesssim 10^7$ GeV), the flux is dominated by coherent scattering of heavy nuclei (Fe, Si).
- At higher energies ( $E_\nu \gtrsim 10^7$ GeV), incoherent scattering and proton elastic scattering become dominant.
- The Hillas model (higher heavy-nuclei fraction) predicts a flux $\sim 1$ order of magnitude higher than the PriNCe simulation.
Constraints on Overdensity ( $\eta$ ): Using non-observation of excess events by IceCube and Pierre Auger Observatory (PAO), the authors place upper limits on the local CνB overdensity $\eta$ $η$ :
- For a lightest neutrino mass $m_1 = 0.1$ $m_{1} = 0.1$ eV:
  - IceCube: $\eta < 7.8 \times 10^7$
  - PAO: $\eta < 7.0 \times 10^8$
- These limits are significantly stronger (by 2-3 orders of magnitude) than the current KATRIN limit ( $\eta < 9.7 \times 10^{10}$ ).
- The constraints are strongest for the GRB source distribution model and for the inverted neutrino mass ordering.
Explanation of KM3NeT Event (KM3-230213A):
- The paper investigates the recently observed 220 PeV neutrino event by KM3NeT.
- The energy of this event aligns with the peak of the boosted CνB flux.
- The authors find that explaining this event via boosted CνB requires an overdensity $\eta \in [5.9 \times 10^9, 2.3 \times 10^{11}]$ .
- Tension: This required $\eta$ is compatible with KATRIN but in tension with IceCube's diffuse flux limits. The authors suggest this tension could be resolved by new physics (e.g., sterile-active neutrino transitions or self-interactions) or if the event is a statistical fluctuation.

5. Significance

New Probe for CνB: This work revitalizes the UHECR-CνB scattering mechanism as a viable probe for the CνB, showing that heavy nuclei and coherent scattering make the signal much more detectable than previously thought.
Improved Constraints: It provides the most stringent constraints to date on the local CνB overdensity, surpassing direct capture experiments like KATRIN.
Interpretation of High-Energy Events: It offers a novel astrophysical explanation for the highest-energy neutrino events (like KM3-230213A), linking them to the fundamental relic background of the universe.
Future Outlook: The paper highlights that upcoming observatories (IceCube-Gen2, GRAND, POEMMA) with larger effective areas and better energy resolution could potentially detect the boosted CνB signal directly, distinguishing it from astrophysical backgrounds via spectral features and multimessenger correlations (absence of gamma-ray counterparts).

In summary, the paper corrects previous theoretical oversimplifications regarding UHECR composition and scattering kinematics, revealing that coherent scattering by heavy nuclei significantly enhances the potential to detect or constrain the Cosmic Neutrino Background.