⚛️ phenomenology

Geometric scaling in elastic $pp$ collisions

This paper argues that geometric scaling persists in elastic proton-proton collisions from 23 GeV to 13 TeV, evidenced by a constant ratio of bump-to-dip positions in differential cross-sections, while also identifying the scattering amplitude's real and imaginary parts to compute the $\rho$ parameter and discussing the scaling's violation outside the dip-bump region at LHC energies.

Original authors: Michał Praszałowicz

Published 2026-07-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Michał Praszałowicz

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 the smallest particles in existence, like protons, zoom around and occasionally bump into each other. This isn't a clumsy collision like two cars crashing; it's more like a delicate, ghostly handshake where they bounce off without ever actually touching. This field of study is called particle physics, and it tries to understand the rules of this dance. To make sense of these tiny bounces, scientists use a concept called "geometric scaling." Think of it like a magical zoom lens. If you take a photo of a proton collision at low energy and another at high energy, they look different because the protons are moving faster. But geometric scaling suggests that if you shrink or stretch the second photo by just the right amount, the two pictures would look exactly the same. It's as if the universe has a secret rulebook that says, "No matter how fast you run, the shape of your shadow stays the same, you just get bigger or smaller." Scientists care about this because finding these patterns helps them understand the fundamental forces that hold matter together and how the universe behaves at its most extreme speeds.

Now, let's zoom in on a specific dance move that has puzzled physicists for decades: the "dip and bump." When scientists measure how protons scatter, they don't just see a smooth curve. Instead, the data looks like a rollercoaster track with a deep valley (a dip) followed immediately by a high hill (a bump). For a long time, researchers wondered if this shape changed as the protons got faster. Some data from the Large Hadron Collider (LHC), the world's biggest particle accelerator, seemed to suggest that the old rules might be breaking down at these super-high speeds.

However, Michał Praszałowicz, a physicist from Poland, argues that the old rules are still holding strong, just in a specific part of the dance floor. In this paper, the author revisits the idea of geometric scaling and shows that it still works remarkably well for proton collisions, even at the massive energies of the LHC. The key discovery is a hidden regularity in that rollercoaster shape. The author points out that while the height of the hill and the depth of the valley change with energy, the distance between them stays perfectly constant. If you measure the position of the dip and the position of the bump, the ratio of the bump's position to the dip's position is always 1.355 ± 0.011. This number doesn't change whether the collision happens at 23 GeV (from experiments decades ago) or at 13 TeV (the current record-breaking speeds). It's like if you were stretching a rubber band with a dot and a star on it; no matter how much you stretch it, the star always stays exactly 1.355 times further away from the start than the dot does.

To figure out why this happens, the author uses some clever mathematical tricks involving "crossing symmetry" and the "optical theorem." Imagine the proton collision as a wave. The paper separates this wave into two parts: a big, imaginary part that does most of the work, and a tiny, real part that is usually ignored. The author shows that the "dip" in the data happens exactly where the big imaginary wave cancels itself out to zero. At that exact moment, the tiny real part of the wave becomes the only thing left to create a signal. This allows the author to calculate a specific number called the $\rho$ parameter, which measures how strong that tiny real part is compared to the big imaginary part. The paper predicts this number based entirely on how the total collision area grows with energy, and the predictions match the experimental data from the LHC and cosmic rays very well.

The paper also looks at the ratio of the height of the bump to the depth of the dip. While this ratio changes at lower energies, the author finds that at the LHC, it settles down and stays steady, which supports the idea that the geometric scaling is still in charge. However, the author is careful to note that this perfect scaling isn't the whole story. The paper argues that while the dip and bump region follows these strict rules, the rest of the collision data—especially at very small angles—does not. The author suggests that the reason the total amount of elastic scattering doesn't perfectly follow the old scaling laws at the LHC is because the "magic zoom lens" breaks down outside of that specific dip-and-bump zone. So, while the universe still follows a beautiful, predictable pattern in the most dramatic part of the collision, there is still some messy, unexplained behavior happening on the edges that the current model doesn't quite capture yet.

Technical Summary: Geometric Scaling in Elastic $pp$ Collisions

Problem Statement
The paper addresses the validity of Geometric Scaling (GS) in proton-proton ($pp$) elastic scattering across a vast energy range, from the Intersecting Storage Rings (ISR) to the Large Hadron Collider (LHC). Originally conjectured by Dias de Deus in 1973, GS posits that the elastic scattering amplitude depends on a single scaling variable $\tau = -t R^2(s)$ (or $b/R(s)$ in impact parameter space), where $R(s)$ is an energy-dependent interaction radius. While ISR data supported this hypothesis, preliminary fits to LHC data suggested a breakdown of GS at TeV energies, indicated by differing power-law exponents for the total cross-section ( $\sigma_{tot}$ ) and the elastic cross-section ( $\sigma_{el}$ ). The paper investigates whether GS persists at LHC energies if the analysis is restricted to specific kinematic regions, specifically the dip-bump structure of the differential cross-section.

Methodology
The author employs a theoretical framework combining Geometric Scaling with fundamental S-matrix principles: crossing symmetry, analyticity, and the optical theorem.

Scaling Hypothesis: The elastic amplitude $\tilde{T}_{el}(s, t)$ is assumed to depend on the scaling variable $\tau = -t/R^2(s)$ . In the impact parameter representation, the amplitude is scaled by $R(s)$ , leading to dimensionless cross-sections that should exhibit uniform energy dependence if GS holds.
Crossing Symmetry and Real Part Identification: To account for the non-zero real part of the amplitude (crucial near the diffraction dip), the author utilizes the crossing relation $\tilde{T}_{el}(-s, t) = \tilde{T}^*_{el}(s, t)$ $\tilde{T}_{e l} (- s, t) = \tilde{T}_{e l}^{*} (s, t)$ . By expanding the amplitude around $-is = e^{y - i\pi/2}$ $- i s = e^{y - iπ /2}$ , the real and imaginary parts are derived as:
- $\text{Im } \tilde{T}_{el} \propto s R^2(y) \Phi(\tau)$
- $\text{Re } \tilde{T}_{el} \propto s \frac{\pi}{2} \frac{dR^2}{dy} \frac{d}{d\tau}(\tau \Phi(\tau))$
  Here, $\Phi(\tau)$ is a universal function derived from the impact parameter profile.
Phenomenological Analysis: The study utilizes experimental data for $\sigma_{tot}$ and the positions of the dip ( $t_{dip}$ ) and bump ( $t_{bump}$ ) in the differential cross-section $d\sigma_{el}/dt$ . Two analytic parametrizations of $\sigma_{tot}$ (COMPETE/PDG and Donnachie-Landshoff) are used to determine the energy dependence of $R^2(s)$ .

Key Contributions and Results

Universality of the Dip-Bump Ratio: The paper establishes that the ratio of the bump position to the dip position, $T_{bd} = |t_{bump}|/|t_{dip}|$ , is constant ( $1.355 \pm 0.011$ ) across energies from 23 GeV to 13 TeV. This constancy strongly supports the existence of Geometric Scaling in the dip-bump region, even at LHC energies, as the energy dependence of $R^2(s)$ cancels out in the ratio.
Prediction of the $\rho$ Parameter: Using the derived relation $\rho = \frac{\pi}{2} \frac{1}{R^2(y)} \frac{dR^2(y)}{dy}$ , the paper provides a parameter-free prediction for the ratio of the real to imaginary parts of the forward scattering amplitude. The calculated $\rho$ values align well with experimental data from ISR and LHC, including the rapid decrease observed at high energies (attributed to the odderon).
Prediction of the Bump-to-Dip Cross-Section Ratio ( $R_{bd}$ ): A one-parameter prediction for $R_{bd}$ is derived. The model successfully reproduces the experimental saturation of this ratio at LHC energies ( $\sim 1.8$ ), which was previously difficult to explain under strict GS assumptions.
Resolution of the $\sigma_{el}$ vs. $\sigma_{tot}$ Discrepancy: The paper explains why the energy exponents for $\sigma_{el}$ and $\sigma_{tot}$ differ at the LHC. While GS holds for the dip-bump region, the total elastic cross-section integrates over all $t$ . The discrepancy arises because Geometric Scaling is violated outside the dip-bump region, particularly at small $t$ (forward scattering), where the energy dependence deviates from the scaling law.

Significance and Claims
The author argues that the main properties of total and differential cross-sections across all energies can be explained by a simple, intuitive picture based on Geometric Scaling, provided the analysis focuses on the dip-bump region. The constancy of the $T_{bd}$ ratio serves as a robust indicator that GS is not broken at the LHC, but rather that its domain of validity is restricted.

The paper modestly concludes that while the GS approach successfully predicts the $\rho$ parameter and the $R_{bd}$ ratio with high accuracy, it is an approximation. It fails to reproduce the energy dependence of the total elastic cross-section at the LHC because the scaling hypothesis does not hold universally across the entire $t$ -range, specifically failing at small momentum transfers. Thus, the work refines the understanding of GS, delineating where it applies (dip-bump region) and where it breaks down (forward region), rather than claiming it is a complete description of elastic scattering at all scales.

Technical Summary: Geometric Scaling in Elastic $pp$ Collisions

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