⚛️ phenomenology

Geometric Scaling and the Odderon

This paper derives scattering amplitudes based on geometric scaling to accurately reproduce most $\rho$ parameter data, demonstrating that a slight modification of the $p\bar{p}$ total cross-section to include the Odderon is necessary to accommodate the 13 TeV TOTEM and ATLAS measurements.

Original authors: Michal Praszalowicz

Published 2026-07-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Michal Praszalowicz

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

Imagine the universe as a giant, invisible dance floor where tiny particles called protons are the dancers. Sometimes, they zoom past each other without touching, but other times, they crash into one another. Physicists are obsessed with understanding these crashes because the way particles bounce off each other reveals the hidden rules of the universe. To make sense of this, scientists use a special map called "scattering amplitudes." Think of this map not just as a list of numbers, but as a complex recipe that tells us two things: how likely a crash is to happen (the "imaginary" part, like the size of the dance floor), and how the particles wiggle or shift during the crash (the "real" part, like the specific dance move).

For decades, physicists have noticed a strange pattern in these crashes called "geometric scaling." It's like noticing that no matter how fast the dancers spin, the ratio of their "bumps" to their "dips" stays exactly the same. This rule works beautifully for most collisions, but recently, a new mystery appeared at the world's biggest particle collider, the Large Hadron Collider (LHC). The data showed a tiny but stubborn mismatch between what the old recipes predicted and what the machines actually saw. This mismatch has led scientists to wonder if there is a ghostly, invisible partner in the dance that we haven't fully accounted for yet, a mysterious force known as the "Odderon."

This paper dives into that mystery by asking a simple question: Can we fix the mismatch in our particle crash recipes by adding this ghostly partner, the Odderon? The authors, Michał Praszałowicz and colleagues, start by using a clever trick called "geometric scaling." They treat the interaction between protons like a balloon that grows as the energy increases. They define two "interaction radii"—think of them as the size of the invisible bubbles the particles carry with them. One bubble, $R(s)$ , handles the standard, predictable collisions (the "even" part), while a second, newer bubble, $Q(s)$ , is introduced to handle the tricky, "odd" part of the collision where things behave differently for protons hitting protons versus protons hitting anti-protons.

When the authors tested their new recipe using the standard data available up to 13 TeV (the highest energy the LHC has reached), they found something interesting. Their formula worked perfectly for almost all the data, matching the "bouncing" behavior of particles with high accuracy. However, they hit a wall with the very latest measurements from the TOTEM and ATLAS experiments at 13 TeV. Specifically, the "real part" of the collision (the wiggle factor, known as the $\rho$ parameter) was lower than their standard recipe predicted. The paper suggests that the standard model, which assumes the "odd" forces fade away at high energies, is missing something crucial at these extreme speeds.

To solve this, the authors propose a slight tweak to the recipe. They suggest that while the standard "odd" forces (like the $\rho$ and $\omega$ particles) do fade away, a new, persistent force—the Odderon—might be stepping in at high energies. The Odderon is described as a "C-odd" amplitude, which is a fancy way of saying it behaves differently when you swap a particle for its anti-particle. The authors didn't just guess; they mathematically showed that if you add a specific, small contribution from the Odderon to the anti-proton collision data, the recipe suddenly fits the 13 TeV measurements perfectly.

Crucially, the paper argues that this fix doesn't break the rest of the physics. By adjusting the "odd" bubble just a tiny bit, they can match the mysterious 13 TeV data points without changing the total size of the collisions (the total cross-section) by much—only about 2 to 3 millibarns, which is a tiny fraction of the total. The authors admit that this is a "modest" approach; they aren't claiming to have proven the Odderon exists beyond a shadow of a doubt, but rather showing that the data allows for it and that it is a highly probable explanation for the discrepancy. They suggest that while the standard model works great at lower energies, the LHC is revealing a new layer of the universe where this ghostly Odderon partner is necessary to make the dance make sense. The study concludes that this modification provides a strong hint, if not yet a final proof, that the Odderon has been discovered at the LHC.

Technical Summary: Geometric Scaling and the Odderon

Problem Statement
The paper addresses the discrepancy between theoretical predictions and experimental measurements of the $\rho$ parameter ( $\rho_{pp} = \text{Re}T_{el}(s, t=0)/\text{Im}T_{el}(s, t=0)$ ) in proton-proton ($pp$) scattering at the LHC energy of 13 TeV. While the COMPETE parametrization of total cross-sections successfully predicts the total cross-section ( $\sigma_{tot}$ ), it fails to reproduce the unexpectedly low $\rho_{pp}$ value measured by the TOTEM Collaboration ( $0.098 \pm 0.01$ ) and confirmed by ATLAS. This discrepancy has sparked debate regarding the necessity of the Odderon—a $C$ -odd exchange of three reggeized gluons—to explain the data. The authors aim to determine, using a modest approach based on geometric scaling, whether forward $pp$ and $p\bar{p}$ data necessitate the introduction of the Odderon.

Methodology
The authors employ the framework of Geometric Scaling (GS), previously applied to $C$ -even amplitudes, and extend it to crossing-odd amplitudes. The methodology proceeds as follows:

Geometric Scaling Ansatz: The elastic scattering amplitude is expressed in terms of interaction radii and universal scaling functions.
- For the crossing-even amplitude ( $T^+$ ), the interaction radius is $R(s)$ .
- For the crossing-odd amplitude ( $T^-$ ), a new interaction radius $Q(s)$ is introduced.
- The amplitudes are constructed to satisfy crossing symmetry ( $T(s) = \pm T^*(-s)$ ) using a rapidity variable $y$ and analytic continuation techniques (specifically the trick from Ref. [31] involving $R^2(-is) \to R^2(y - i\pi/2)$ ).
Derivation of Observables: Using the optical theorem and the derived amplitudes, the authors establish parameter-free relations connecting the total cross-sections ( $\sigma_{tot}$ ) and the $\rho$ parameters to the interaction radii $R^2$ and $Q^2$ and their derivatives with respect to rapidity:
- $\sigma_{tot}^{pp/p\bar{p}} = 2 \left( R^2 \pm \frac{\pi}{2} \frac{dQ^2}{dy} \right)$
- $\rho^{pp/p\bar{p}} = \frac{\frac{\pi}{2} \frac{dR^2}{dy} \mp Q^2}{R^2 \pm \frac{\pi}{2} \frac{dQ^2}{dy}}$
Phenomenological Input: The authors utilize the COMPETE parametrization for total cross-sections to constrain $R^2$ and $Q^2$ .
- $R^2$ is derived from the average of $pp$ and $p\bar{p}$ cross-sections.
- $Q^2$ is derived from the difference between $pp$ and $p\bar{p}$ cross-sections, which vanishes at high energies in the standard COMPETE fit (implying no Odderon contribution in the standard fit).
Odderon Modification: To address the 13 TeV data, the authors introduce a "gauge" freedom in the formalism. They modify the $p\bar{p}$ total cross-section parametrization by adding a function $f(y)$ that leaves $\sigma_{tot}^{pp}$ unchanged but alters $\sigma_{tot}^{p\bar{p}}$ and the $\rho$ parameters. Inspired by the Maximal Odderon hypothesis, they propose $f(y) = A \ln^2(s/s_3)$ , representing a $C$ -odd amplitude growing as $\ln^2 s$ .

Key Results

Standard Fit: Using the standard COMPETE parametrization (where the Odderon contribution vanishes at high energy), the model reproduces $\rho_{pp}$ and $\rho_{p\bar{p}}$ with high accuracy up to $\sim 1$ TeV. However, it fails to describe the 13 TeV TOTEM and ATLAS data points, confirming that a non-zero crossing-odd amplitude is necessary at LHC energies.
Odderon Inclusion: By introducing the Maximal Odderon Ansatz ( $f(y) = A \ln^2(s/s_3)$ ) with specific parameters ( $A=0.08$ mb, $s_3=1$ TeV, $q=-0.2$ mb), the authors successfully reproduce the 13 TeV TOTEM $\rho_{pp}$ data point.
Consistency: This modification allows the model to touch the 13 TeV data without spoiling the fit at lower energies ( $< 1$ TeV). Crucially, the modification results in only a minimal change (2–3 mb) to the total $p\bar{p}$ cross-section at LHC energies, which remains consistent with existing constraints (since $p\bar{p}$ data is scarce above 2 TeV).

Significance and Claims
The paper claims to provide a "modest" but significant contribution to the Odderon debate:

Necessity of Odderon: The study demonstrates that standard parametrizations of total cross-sections cannot reproduce the 13 TeV $\rho_{pp}$ data without a modification corresponding to the Odderon.
Geometric Scaling of Odderon: This work represents the first application of geometric scaling to $C$ -odd amplitudes. The authors argue that the success of the simple scaling ansatz in describing the data suggests the Odderon amplitude likely exhibits geometric scaling.
Evidence for Discovery: The authors conclude that their minimalistic assumptions provide "further hint – if not proof – of the discovery of the Odderon at the LHC." They emphasize that the modification required to fit the data is small and physically plausible within the framework of the Maximal Odderon hypothesis, even if the theoretical foundations of the Maximal Odderon are debated in the literature.

The authors explicitly state that their goal was not to perform a full global fit including the Odderon parameters (which has been done by other groups) but to demonstrate that the introduction of an Odderon term is possible and necessary within the geometric scaling framework to resolve the $\rho_{pp}$ discrepancy.

Technical Summary: Geometric Scaling and the Odderon

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