Entropy and mean multiplicity from dipole models in the high energy limit

This paper proposes entropy as a function of the logarithm of average multiplicity to resolve pseudorapidity ambiguities in high-energy proton-proton collisions, demonstrating that a generalized dipole model provides a significantly better description of experimental data than the standard 1D Mueller dipole model.

The Gist

Imagine you are at a massive, chaotic party where guests keep arriving, meeting, and splitting into smaller groups. In the world of particle physics, this "party" is a collision between two protons at nearly the speed of light. The "guests" are subatomic particles, and the "groups" they form are called dipoles.

This paper is like a detective story where two physicists, Krzysztof and Sándor, try to figure out which mathematical rule best explains how this party gets more crowded as the energy (the "loudness" of the music) increases.

Here is the breakdown of their investigation using simple analogies:

1. The Problem: Counting the Guests

For decades, scientists have counted how many particles come out of these collisions. But there's a catch: different experiments count them differently.

• Experiment A might only count guests in a small corner of the room.
• Experiment B might count guests in the whole room, but the room keeps getting bigger as the party gets louder.

Because they count differently, it's hard to compare their results or see if a theory is actually working. It's like trying to compare the "crowdedness" of a small coffee shop to a massive stadium just by looking at the number of people, without realizing the stadium is much bigger.

2. The Solution: A Universal "Vibe" Meter

To fix this, the authors propose a new way to measure the party: Entropy.
Think of entropy not as a complex physics term, but as a measure of chaos or disorder.

• If everyone stands in a neat line, the entropy is low.
• If everyone is dancing wildly in every corner, the entropy is high.

The authors suggest that instead of just counting the number of guests ($n$), we should look at the relationship between the average number of guests and the level of chaos. They call this relationship $S(\ln\langle n\rangle)$.

The Analogy: Imagine you are trying to describe how wild a party is. Instead of just saying "There are 100 people," you say, "For every 10 people, the chaos level goes up by X." This ratio becomes a "universal language" that works whether you are in a coffee shop or a stadium.

3. The Two Theories: The Old Map vs. The New GPS

The authors tested two different "maps" (mathematical models) to see which one predicts the party's chaos correctly.

• Map A: The 1D Mueller Model (The Old Map)
  This model assumes that as the party gets louder, the guests split apart in a very predictable, simple way. It's like a tree growing branches where every branch splits into exactly two new branches.

  • The Result: This map works okay for small parties, but when the energy gets high, it fails to predict the wild chaos. It thinks the party is too orderly.

• Map B: The Generalized Dipole Model (The New GPS)
  This is a newer, more complex map. It adds a "wiggle room" parameter (called $h$) that allows the guests to split in more unpredictable, messy ways. It accounts for the fact that sometimes one guest splits into three, or five, or interacts with the "vacuum" (empty space) in weird ways.

  • The Result: This map is a perfect match. It predicts that the party gets much more chaotic and crowded than the old map thought, and it fits the real data from experiments like ALICE, CMS, and ATLAS perfectly.

4. The Big Discovery

When they compared their maps to the actual data from particle colliders (like the Large Hadron Collider), they found:

• The Old Map (Mueller model) was too simple. It underestimated how messy the high-energy collisions get.
• The New Map (Generalized model) was spot on. It showed that at high energies, the "entanglement" (the quantum connection between particles) is at its maximum, creating a state of maximum chaos.

Why Does This Matter?

This isn't just about counting particles. The authors suggest that this "maximum chaos" might be a sign of something deeper in the universe, related to quantum entanglement (where particles are linked across space) and Krylov complexity (a way to measure how complicated a quantum state is).

In a nutshell:
The authors found a better way to measure the "wildness" of particle collisions. By using a new "chaos meter" (Entropy vs. Multiplicity), they proved that a more complex mathematical model is needed to understand the universe at its highest energies. The old, simple rules don't work anymore; the universe is messier and more connected than we thought.

Technical Summary

1. Problem Statement

The paper addresses two primary challenges in high-energy particle physics regarding charged particle multiplicity distributions:

• Experimental Ambiguity: There is a significant discrepancy in how different experiments (e.g., HERA, LHCb vs. ALICE, ATLAS, CMS, UA5) define pseudorapidity ($\eta$) windows. Some use narrow, shifting windows, while others use wide, centered windows. This makes direct comparisons of raw multiplicity distributions difficult.
• Theoretical Limitations: Existing theoretical models, specifically the 1D Mueller dipole model, struggle to accurately describe the shape of multiplicity distributions and the associated entropy across a wide range of energies and pseudorapidities, particularly at lower multiplicities.

The authors aim to resolve these issues by proposing a universal observable and testing it against dipole cascade models to see which best describes proton-proton ($pp$) collision data.

2. Methodology

A. The Universal Observable: $S(\ln\langle n \rangle)$

To bypass the ambiguity of pseudorapidity definitions, the authors propose using Shannon entropy ($S$) as a function of the logarithm of the average multiplicity ($\ln\langle n \rangle$).

• Definitions:
  • Average multiplicity: $\langle n \rangle = \sum n P(n)$
  • Entropy: $S = -\sum P(n) \ln(P(n))$
• Rationale: This relationship is derived directly from experimental data without dependence on specific $\eta$ window definitions, serving as a "universal" observable to compare different experiments and theoretical models.

B. Theoretical Models

The authors investigate and compare two dipole cascade models describing parton evolution in rapidity ($y$):

1. 1D Mueller Dipole Model:

   • Based on the Balitsky–Fadin–Kuraev–Lipatov (BFKL) evolution.
   • Governed by a cascade equation where the probability distribution $P_n(y)$ follows a geometric distribution.
   • Predicts a specific entropy relation: $S \approx \ln\langle n \rangle + 1$ (in the sub-asymptotic limit).
   • Characterized by parameters: $\alpha$ (emission kernel speed) and $C$ (normalization). The conformal weight is fixed at $h = 0.5$.

2. Generalized Mueller Model:

   • An extension of the Mueller model incorporating Krylov complexity and conformal $SL(2,R)$ group theory.
   • Introduces an additional parameter, conformal weight $h$, which controls the dispersion of the distribution.
   • The probability distribution follows a Negative Binomial Distribution (NBD).
   • The mean multiplicity is given by $\langle n \rangle = 2h(Ce^{\alpha y} - 1)$.
   • This model allows for a vacuum contribution ($p_0$) and offers more flexibility in fitting the data's dispersion.

C. Data Analysis

• Data Source: Charged particle multiplicity distributions from $pp$ collisions collected by ALICE, UA5, CMS, LHCb, and ATLAS.
• Range: Center-of-mass energies ($\sqrt{s}$) from 0.2 TeV to 13 TeV; pseudorapidity ranges from $|\eta| < 0.5$ to $4.5 < \eta < 5$.
• Procedure: The authors calculated $S$ and $\langle n \rangle$ from the experimental data, plotted $S$ vs. $\ln\langle n \rangle$, and fitted the model parameters ($\alpha, C, h$) to minimize the $\chi^2$.

3. Key Contributions

• Proposal of a Universal Observable: The introduction of $S(\ln\langle n \rangle)$ as a robust metric to compare experimental results across different detector acceptances and theoretical frameworks, effectively neutralizing the "pseudorapidity window" problem.
• Application of Krylov Complexity: The paper maps concepts from quantum information theory (Krylov complexity and conformal weights) to QCD parton cascades, providing a theoretical basis for the generalized dipole model.
• Systematic Model Comparison: A rigorous quantitative comparison between the standard 1D Mueller model and its generalized NBD-based extension using high-precision LHC and Tevatron data.

4. Results

• Model Performance:
  • The Generalized Model provides a significantly superior description of the data compared to the 1D Mueller model.
  • Fit Quality: The $\chi^2$ per degree of freedom (NDF) for the Generalized Model is 24/63, whereas the 1D Mueller model yields 309/63, indicating a poor fit for the standard model.
• Parameter Extraction:
  • Generalized Model: Best fit obtained with conformal weight $h \approx 0.92 \pm 0.05$ and $\alpha = 0.32$.
  • 1D Mueller Model: Requires fixed $h=0.5$, which fails to capture the dispersion observed in the data.
• Behavioral Differences:
  • The 1D Mueller model predicts an exponential decay in multiplicity distributions, leading to deviations at lower multiplicities.
  • The Generalized model (NBD) produces more dispersed distributions that closely match the experimental data across the entire energy range.
  • The Generalized model predicts higher average parton numbers and higher entropy values in the investigated rapidity range compared to the 1D model.

5. Significance and Outlook

• Validation of Entanglement Conjecture: The results support the conjecture that the initial state in high-energy limits is maximally entangled, with the entropy scaling as $S \approx \ln\langle n \rangle$. The generalized model's success suggests that the underlying quantum state spread (Krylov complexity) is a critical factor in hadronic collisions.
• Methodological Resolution: The study demonstrates that using $S(\ln\langle n \rangle)$ resolves the ambiguity caused by different experimental acceptance cuts, allowing for a unified analysis of global collider data.
• Future Directions:
  • Saturation Physics: The authors suggest extending these models to include parton recombination effects to better study gluon saturation.
  • Complexity: The framework could be refined to handle the more complex colliding environment of $pp$ collisions compared to Deep Inelastic Scattering (DIS).
  • Experimental Refinement: Future detailed analyses using this universal observable could further refine our understanding of the initial state of high-energy collisions.

In conclusion, the paper establishes that the Generalized Dipole Model, rooted in conformal field theory and Krylov complexity, is the superior theoretical framework for describing charged particle multiplicities and entropy in high-energy $pp$ collisions, outperforming the traditional 1D Mueller model significantly.