⚛️ phenomenology

Odderon exchange in high-energy $K^0_S$ regeneration at the LHC

This paper evaluates the feasibility of detecting Odderon exchange via high-energy neutral-kaon regeneration at the LHC by analyzing both coherent-forward and non-forward diffractive processes, ultimately identifying significant experimental challenges such as primary $K^0_S$ contamination and competing electromagnetic or Regge backgrounds that necessitate specific mitigation strategies.

Original authors: P. Filip, M. Taševský, V. A. Khoze, R. Pasechnik, M. G. Ryskin, B. G. Zakharov

Published 2026-08-04

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CC BY 4.0

Original authors: P. Filip, M. Taševský, V. A. Khoze, R. Pasechnik, M. G. Ryskin, B. G. Zakharov

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 Invisible Ghost and the Quantum Mirror

Imagine the universe as a giant, chaotic dance floor where subatomic particles are the dancers. For decades, physicists have been trying to understand the rules of this dance, specifically how particles bounce off each other. Most of the time, the dance is governed by a force called the "Pomeron," which acts like a polite, invisible hand that gently pushes particles apart without changing their fundamental nature. But there's a rumor of a ghostly partner in this dance, a mysterious force called the Odderon. Unlike its polite partner, the Odderon is a "crossing-odd" ghost; it behaves differently when you swap a particle with its anti-particle (like swapping a dancer for their mirror image).

Finding this ghost is incredibly hard because it's shy and rarely shows up on its own. It usually hides behind the much louder, more common Pomeron. To catch a glimpse of it, scientists need a special trick: a "quantum mirror." In the world of particle physics, this mirror is a block of material (a regenerator) that can turn one type of particle, a long-lived neutral kaon ( $K^0_L$ ), into its short-lived cousin ( $K^0_S$ ). If the Odderon is there, it will nudge this transformation just enough to change the timing and pattern of the particles' decay, like a subtle shift in a dancer's step that reveals the ghost's presence. The big question is: Can we build a mirror big enough and sensitive enough at the world's most powerful particle collider, the Large Hadron Collider (LHC), to finally see this ghost?

The Paper's Quest: Hunting the Odderon with Kaon Mirrors

This paper is a detailed feasibility study by a team of physicists who are asking, "Can we actually do this at the LHC?" They revisit an old idea from the 1980s—using high-energy neutral kaons to detect the Odderon—but they update it for the modern, high-energy environment of the LHC, where particles collide at energies up to 13.6 TeV. The authors, P. Filip and colleagues, break their investigation into two main strategies: looking at the "straight-on" bounce and the "glancing" bounce.

Strategy 1: The Straight-On Bounce (Coherent Forward Regeneration)
First, the team looked at what happens when a beam of high-energy kaons (around 2 TeV) smashes straight into a block of material (like copper, carbon, or lead) and converts into short-lived kaons without changing direction. They calculated that if the Odderon exists, it would create a specific "phase shift"—a kind of timing glitch—that distorts where the particles decay.

The Good News: Their simulations show that with a 2 TeV beam, this distortion would be visible, changing the decay pattern by about 20–40%.
The Bad News: There are two massive hurdles. First, the "primary" short-lived kaons created directly at the collision point are so long-lived at these high speeds that they survive all the way to the detector, drowning out the signal from the regenerated ones. To fix this, the detector would need to be moved hundreds of meters away (about 620–650 meters), requiring a major construction project in the CERN tunnels. Second, there's a "fake" signal caused by electromagnetic forces (photon exchange) that looks just like the Odderon signal. Unless they can perfectly calculate and subtract this electromagnetic noise, they can't be sure what they are seeing is the Odderon.
The Verdict: While theoretically possible, this method is currently too messy and requires too much infrastructure to be a clean measurement right now.

Strategy 2: The Glancing Bounce (Non-Forward Regeneration)
Next, the authors looked at a lower-energy beam (0.2–0.8 TeV) where the kaons bounce off the material at a slight angle. This is the "non-forward" mode.

The Good News: At these lower energies, the annoying "primary" kaons die out before reaching the detector, leaving a much cleaner signal. The Odderon signal here is predicted to be strong, potentially 3 to 10 times louder than the background noise from other known forces (like the $\omega$ -Reggeon).
The Bad News: A new enemy appears: neutrons. The beam is always accompanied by a swarm of neutrons. When these neutrons hit the regenerator, they can create fake short-lived kaons that look exactly like the signal the scientists are hunting. The authors estimate that this "neutron-induced background" is about 20 times stronger than the signal they want to find.
The Proposed Solution: To beat the neutrons, the authors suggest a clever "double-regenerator" trick. By using two different blocks of material (like one made of Carbon and one of Lead) and comparing the results, they could mathematically subtract the neutron noise, leaving only the Odderon signal. However, this requires a very specific setup and a deep understanding of how neutrons interact with matter, which needs more study.

What They Rule Out and What They Suggest
The paper explicitly rules out the idea that the current LHC setup (with detectors placed 140 meters away) can successfully measure the Odderon using high-energy kaons. The "primary" kaon contamination is simply too high (about 27% at 2 TeV), making the measurement impossible without moving the detector. They also caution that the electromagnetic "photon" signal is a major confounder that must be solved before the straight-on method can work.

However, they do not claim to have found the Odderon. Instead, they suggest that the non-forward (glancing) method at lower energies (0.2–0.8 TeV) is the most promising path forward. They propose that with a specialized "active" detector setup that can veto neutrons and a double-regenerator subtraction strategy, it might be possible to see the Odderon. They calculate that the signal they are looking for is a cross-section of about 0.05–0.1 microbarns per nucleon, which is small but detectable if the background noise can be suppressed by a factor of 20 to 200.

In short, this paper doesn't announce a discovery. Instead, it draws a map. It tells us that the "straight-on" route is blocked by traffic (primary kaons) and fog (electromagnetic noise), but the "glancing" route is open, provided we can build a better shield against the neutron swarm. It's a call to action for engineers and physicists to build the right tools to finally catch the ghost of the Odderon.

Technical Summary: Odderon Exchange in High-Energy $K^0_S$ Regeneration at the LHC

Problem Statement
The existence of the Odderon, a $C$ -odd singularity in QCD formed by three $t$ -channel reggeized gluons, remains a subject of experimental verification. While the TOTEM collaboration at the LHC claimed evidence for the Odderon through elastic $pp$ and $p\bar{p}$ scattering (specifically via the real-to-imaginary amplitude ratio $\rho$ and the diffractive dip structure), these results face challenges regarding normalization uncertainties and the dominance of the $C$ -even Pomeron amplitude. Consequently, there is a need for independent observables where the Odderon enters a different amplitude combination. This paper revisits the proposal of detecting the Odderon via high-energy neutral-kaon regeneration ( $K^0_L \to K^0_S$ conversion in matter), a method suggested in the 1980s but not yet realized at TeV energies. The study addresses the feasibility of this measurement within the specific constraints of the LHC infrastructure, focusing on the interplay between the Odderon signal, competing $C=-1$ amplitudes (photon exchange), and various experimental backgrounds.

Methodology
The analysis is divided into two complementary regimes: coherent forward regeneration and non-forward (diffractive) regeneration.

Coherent Forward Regeneration (TeV Energies):
- The authors model the conversion of $\sim 0.2\text{--}2$ TeV $K^0_L$ mesons (produced in $\sqrt{s}=13.6$ TeV $pp$ collisions) into $K^0_S$ mesons in a regenerator (Liquid Hydrogen, Carbon, Copper, or Lead).
- They reproduce previous estimates of the regeneration phase shift ( $\Delta \phi_o$ ) induced by the Odderon, extending them to realistic regenerator geometries and including neutral-kaon attenuation.
- The observable is the distortion of the $K^0 \to \pi^0\pi^0$ decay-vertex distribution behind the regenerator.
- The analysis explicitly accounts for the competing electromagnetic $C=-1$ amplitude (photon exchange) and the survival probability of primary $K^0_S$ mesons originating from the interaction point (IP).
Non-Forward Regeneration (0.2–0.8 TeV Energies):
- This mode examines $K^0_S$ production at finite momentum transfer ( $|t| > 0$ ), where the scattering angle is measurable.
- The regeneration amplitude is calculated as a sum of the Odderon exchange ( $T_{Odd}$ ), the Pomeron–Odderon cut ( $T_{P\otimes Odd}$ ), the $\omega$ -Reggeon exchange ( $T_R$ ), and photon exchange.
- Calculations utilize a one-loop running $\alpha_s$ frozen in the infrared, effective gluon masses ( $m_g = 0.17$ and $0.7$ GeV), and Gaussian or pole form factors for the proton and kaon.
- The authors estimate background contributions from neutron-induced strangeness production and inelastic triple-Reggeon exchanges.

Key Contributions and Results

Coherent Mode Constraints:
- Signal: A benchmark Odderon-induced phase shift of $|\Delta \phi_o| = 20^\circ$ (conservative) to $45^\circ$ produces a measurable 20–40% distortion in the $K^0 \to \pi^0\pi^0$ decay intensity at distances of 20–150 m behind the regenerator for 2 TeV kaons.
- Primary $K^0_S$ Contamination: A critical limitation is the survival of primary $K^0_S$ mesons from the IP. At the current TAN absorber location ( $L_{IP} \approx 140$ m), the survival fraction at 2 TeV is $\sim 27\%$ , which overwhelms the regenerated signal. Even at a Forward Physics Facility (FPF) distance ( $\sim 620\text{--}650$ m), the surviving fraction ( $\sim 10^{-3}$ ) remains significant due to the large $K^0_S$ branching ratio and short lifetime, requiring precise subtraction.
- Electromagnetic Background: The photon-exchange amplitude ( $C=-1$ ) is comparable to or larger than the Odderon-induced Pomeron–Odderon cut contribution, depending on the effective gluon mass and target $Z/A$ ratio. This prevents a "clean" Odderon extraction without constraining the electromagnetic component (e.g., via different targets).
Non-Forward Mode Analysis:
- Signal vs. Background: In the kinematic window $0.4 \lesssim |t| \lesssim 1$ GeV $^2$ for kaon energies of 0.2–0.8 TeV, the Odderon-dominated cross section is estimated at $0.05\text{--}0.1$ $\mu$ b per nucleon (for $m_g=0.17$ GeV), exceeding the $\omega$ -Reggeon contribution by factors of 3–10.
- Dominant Backgrounds: The primary limitation is neutron-induced neutral-only strangeness production (reactions like $n+A \to \Sigma^0/\Lambda^0 + K^0 + A'$ or $n+A \to n' + \bar{K}^0 + K^0 + A'$ ). The fiducial background cross section is estimated at $\sim 1.5$ $\mu$ b per nucleon, roughly 20 times larger than the signal.
- Mitigation Strategy: The authors propose an "active double-regenerator" subtraction strategy. By using two regenerators with different materials (e.g., C and Pb) and lengths, one can construct a combination where the neutron-induced backgrounds cancel (if scaled correctly) while the regeneration signals differ, provided the ratio of regeneration amplitudes differs significantly from the ratio of background cross sections.
Event Yield and Statistics:
- For the coherent mode, achieving a $5\sigma$ observation would require $\mathcal{O}(10^4)$ reconstructed $K^0 \to \pi^0\pi^0$ decays, whereas current LHCf special-run projections suggest $\sim 10^3$ events, yielding only a $\sim 3\sigma$ effect.
- For the non-forward mode, the statistical limitation is severe due to the need to select a narrow energy range ( $0.2\text{--}0.5$ TeV) to suppress primary $K^0_S$ contamination, combined with the high background rates.

Significance and Claims
The paper does not claim that an Odderon discovery is imminent with current LHCf capabilities. Instead, it establishes the feasibility boundaries and systematic requirements for such a measurement.

It demonstrates that while the Odderon signal is theoretically distinct in the regeneration amplitude, the primary $K^0_S$ contamination and electromagnetic photon exchange pose severe, potentially prohibitive, challenges for the coherent forward mode at the current LHC geometry.
It identifies neutron-induced backgrounds as the dominant limitation for the non-forward mode, quantifying the need for a background suppression factor of 20–200.
The study concludes that a definitive measurement requires a dedicated infrastructure (such as a neutral-kaon beamline in an FPF-like cavern) and a sophisticated active double-regenerator setup to subtract backgrounds.
The work serves as a benchmark for future proposals, defining the necessary signal-to-background scales and the specific measurements (e.g., neutron-induced exclusive cross sections) required to validate the sensitivity of such an experiment.

The Invisible Ghost and the Quantum Mirror

The Paper's Quest: Hunting the Odderon with Kaon Mirrors

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