⚛️ phenomenology

Entropy from inclusive scattering

This paper investigates entropy derived from inclusive scattering probabilities within non-interacting unitary pomeron exchange models, revealing a high-energy discrepancy between inelastic cross-sections that resolves at asymptotic limits and demonstrating that entropy initially rises with energy before peaking at rapidities of 12–17 and subsequently declining.

Original authors: M. A. Braun

Published 2026-08-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: M. A. Braun

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, chaotic dance floor where subatomic particles are the dancers. When two particles crash into each other at nearly the speed of light, they don't just bounce off; they explode into a shower of new particles, like confetti bursting from a cannon. Physicists call this "scattering." For decades, scientists have tried to understand the rules of this dance. They use a mathematical tool called a "cross-section" to measure how likely a crash is to happen and how many new particles (the confetti) are created. But there's a tricky problem: to calculate something called "entropy"—a measure of how messy or disordered the final party is—you need to count specific, distinct outcomes. However, in the real world of particle physics, the outcomes are a continuous blur, not a neat list of numbers. It's like trying to count the exact number of grains of sand in a shifting dune; if you try to count them too precisely, the numbers get infinite and break your calculator.

To solve this, scientists often use a simplified model involving "pomerons." Think of a pomeron not as a particle you can hold, but as a ghostly, invisible thread that connects the two crashing particles. When they collide, these threads stretch and snap, creating new particles. The big question this paper tackles is: If we assume these threads don't talk to each other (no "pomeron-pomeron interaction"), can we perfectly predict the number of particles created and the resulting messiness (entropy) of the collision? The author, working with a team from St. Petersburg State University, sets out to test this specific, simplified scenario to see if the math holds up or if the universe has a few hidden tricks up its sleeve.

The Ghostly Threads and the Broken Promise

In this study, the author, M.A. Braun, decided to play a game of "what if." They imagined a world where the ghostly threads (pomerons) involved in particle collisions never interact with each other. They just cross paths and snap independently. This is a much simpler version of reality than what usually happens, but it serves as a perfect "laboratory" to test the math. The goal was to see if they could take the data from these simple collisions and calculate the probability of creating exactly $n$ particles, and from there, calculate the entropy (the measure of disorder).

The researchers looked at two different types of these ghostly threads: the "Regge-Gribov" (RG) pomeron, which is like a standard, slightly wobbly thread, and the "BFKL" pomeron, which is a more complex, high-energy version. They ran the numbers to see if the total number of particles created matched what the laws of physics (specifically, a rule called "unitarity") demanded.

The Big Surprise: The Math Doesn't Add Up (Yet)
The author found a significant discrepancy. When they calculated the number of particles created based on how the threads snap, the total didn't quite match the total number of particles the model should produce according to the rules of unitarity. It was like baking a cake where the recipe says you should have 100 grams of flour, but when you weigh the batter, you only have 98 grams.

However, this isn't a disaster for the theory. The author discovered that this "missing flour" gets smaller and smaller as the energy of the collision increases. At low energies, the mismatch is noticeable. But as the energy gets higher and higher, the error shrinks exponentially. In the limit of infinite energy, the math finally works perfectly. The inconsistency vanishes, and the model becomes consistent.

The Entropy Rollercoaster

Once they sorted out the probabilities (even with the small error at lower energies), the team calculated the entropy. Here is where things get really interesting and defy what many people might expect.

In many simple models, scientists predicted that as you crank up the energy, the entropy (the messiness) would just keep growing forever, like a balloon inflating endlessly. The author checked this against their calculations.

The Result: The entropy does rise at first, but it doesn't go up forever. Instead, it hits a peak and then starts to slowly fall.

For the RG pomeron model, the entropy reaches its maximum "messiness" at a rapidity (a measure of energy) of about 12 to 17. After that, it begins to decrease.
For the BFKL pomeron model, the trend is similar: the entropy rises, peaks around 12, and then slowly declines at higher energies.

This is a crucial finding because it contradicts the "simplified" idea that more energy always equals more disorder in a simple, linear way. The universe, it seems, has a limit to how chaotic it gets in these specific collisions before it starts to organize itself again.

The Two Models: A Tale of Two Threads

The paper also highlights a difference between the two types of threads they tested:

The RG Pomeron (The Local Thread): This model behaves somewhat nicely. The "missing flour" (the inconsistency) disappears very quickly as energy goes up. By the time the rapidity reaches 10, the error is tiny (about 0.01%). The probability of creating particles shows a clear pattern: at lower energies, you mostly get a few particles, but as energy rises, the peak of the probability shifts to higher numbers of particles.
The BFKL Pomeron (The High-Energy Thread): This one is trickier. Because the "production intensity" (how many particles a single thread snaps into) is much smaller here, the inconsistency lasts much longer. Even at a rapidity of 20, there is still a 2% error. The probability of finding particles doesn't show a nice peak; instead, it just smoothly drops off as the number of particles increases.

Why This Matters

The author concludes that while the simple model of non-interacting threads is a great tool for understanding the basics, it has a flaw at finite energies. The math only works perfectly when the energy is infinitely high. This suggests that in the real world, where energy is high but not infinite, the interactions between the threads (which the model ignored) must play a role to fix the math.

The study also rules out the idea that entropy simply grows linearly with energy in these scenarios. Instead, it suggests a more complex behavior where entropy rises, peaks, and then falls. The author notes that to get a truly complete picture, future work needs to include the interactions between the threads themselves, a task that is mathematically "formidable" and not easily solved.

In short, the paper shows us that even in a simplified world of ghostly threads, the universe has a limit to its chaos, and the math only clicks into perfect place when we look at the highest possible energies.

Technical Summary: Entropy from Inclusive Scattering

Problem Statement
The paper addresses the challenge of constructing a probabilistic interpretation and calculating the entropy of scattering processes within unitary pomeron exchange models. While scattering cross-sections are fundamentally probability densities in continuous phase space, entropy requires a discrete set of probabilities associated with elementary events. Previous approaches have attempted to discretize phase space via orbital momenta or particle multiplicity ( $n$ ). A critical consistency issue arises when probabilities are derived from two different sources:

Partial cross-sections ( $\sigma_n$ ): The sum of contributions from $n$ emitted particles.
Inclusive cross-sections ( $\nu(k)$ ): Integrated $k$ -fold inclusive cross-sections, which act as probability moments.

In a consistent theory, the probabilities $P(n)$ derived from $\sigma_n$ must match those reconstructed from the moments $\nu(k)$ via the inverse relation. The paper investigates whether this consistency holds in simplified pomeron models where pomerons do not interact with each other (only sequential exchanges occur), specifically examining the relationship between the total inelastic cross-section and the sum of probabilities derived from particle emission.

Methodology
The author employs a simplified model of multiple pomeron exchanges (Fig. 1) without interactions between the pomerons themselves. The study covers two distinct pomeron types:

Regge-Gribov (RG) Pomeron: Treated as a pole in the complex orbital momentum plane with a linear trajectory $\alpha(t) = 1 + \Delta + \alpha't$ .
BFKL Pomeron: Characterized by a fixed intercept $\alpha(t) = 1 + \Delta$ (no slope) and specific emission properties derived from QCD.

The methodology involves:

Eikonal Approximation: Calculating the total, elastic, and inelastic cross-sections using the eikonal approximation for multiple pomeron exchanges.
Probability Construction:
- Deriving the distribution of "cut pomerons" ( $P_{pom}(n)$ ) using AGK cutting rules.
- Convoluting the cut pomeron distribution with the probability of particle emission from a single pomeron (assumed to be Poissonian with mean multiplicity $j = cY$).
- Alternatively, calculating probabilities directly from $k$ -fold inclusive cross-sections ( $\nu(k)$ ) using the inverse relation $P(n) = \frac{1}{n!} \sum_{k \ge n} \frac{(-1)^{k-n}}{(k-n)!} \nu(k)$ .
Analytical and Numerical Analysis:
- Deriving analytical expressions for the ratio of the inelastic cross-section derived from emission to the unitary inelastic cross-section.
- Investigating the asymptotic behavior of the entropy $E(Y) = -\sum P(n) \ln P(n)$ .
- Performing numerical calculations for $Y \le 30$ using standard parameters (e.g., $\Delta = 0.13$ , $\alpha' = 0.2$ GeV $^{-2}$ for RG; $\alpha_s = 0.15$ for BFKL).

Key Contributions and Results

Inconsistency at Finite Energies:
The study rigorously demonstrates a discrepancy between the inelastic cross-section calculated from the sum of emission probabilities and the inelastic cross-section required by unitarity (eikonalization).
- The ratio $R = \sigma_{in}^{emission} / \sigma_{in}^{eikonal}$ deviates from unity at finite energies.
- Consistency is only restored in the limit of infinite energy ( $Y \to \infty$ ), where the number of emitted particles per pomeron becomes infinite.
- The magnitude of this inconsistency diminishes exponentially with energy ( $\sim e^{-cY}$ ).
- RG Pomeron: With a production intensity $c \sim 1$ , the inconsistency is negligible ( $\sim 10^{-4}$ ) at $Y=10$ .
- BFKL Pomeron: Due to a much smaller production intensity ( $c \approx 0.057$ ), the inconsistency is significant even at high rapidities, remaining around 2% at $Y=20$ .
Entropy Behavior:
Contrary to predictions and results from one-dimensional models which suggest a linear or logarithmic growth of entropy with energy, the numerical calculations reveal a different trend:
- The entropy $S(Y)$ initially rises with energy.
- It reaches a maximum at rapidity $Y \approx 12 \div 17$ .
- For higher energies ( $Y > 17$ ), the entropy slowly decreases.
- This behavior is observed in both RG and BFKL models, despite their differences in particle production intensity.
Probability Distributions $P(n)$ :
- RG Model: $P(n)$ exhibits a clear structure with a maximum that shifts to higher $n$ as energy increases.
- BFKL Model: Due to the low emission intensity, $P(n)$ decreases smoothly with $n$ without a distinct maximum in the studied range, and shows little dependence on energy above $Y=20$ (excluding normalization).
BFKL Emission Complications:
The paper analyzes the inclusion of a non-trivial exponential factor in the BFKL emission cross-section (arising from rapidity integration). It finds that while integrated cross-sections can be calculated, the resulting terms grow factorially ( $\sim e^{\gamma n^2}$ ), leading to severe divergence in the sum required to reconstruct probabilities. Consequently, reasonable probabilities cannot be constructed if this exponential factor is retained, forcing the study to rely on the simplified case where this factor is dropped.

Significance and Claims
The paper claims to provide a rigorous proof that the customary transition from cut pomeron probabilities to particle production probabilities is only consistent in the high-energy limit. It highlights that while pomeron models are theoretically oriented toward this limit, the violation of consistency at phenomenologically relevant energies (e.g., $Y < 20$ ) is non-negligible, particularly for BFKL dynamics.

The most significant finding is the counter-intuitive behavior of the scattering entropy. The paper explicitly states that the expected logarithmic or linear growth of entropy with energy is not observed in these unitary models. Instead, the entropy peaks and then slowly falls, a result attributed to the specific constraints of the unitary eikonal model and the finite nature of the particle production intensity at accessible energies. The author concludes that further developments require analyzing multigluon production with interactions between pomerons, a task noted as formidable and not easily achieved.

The Ghostly Threads and the Broken Promise

The Entropy Rollercoaster

The Two Models: A Tale of Two Threads

Why This Matters

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