Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in $pp$ collisions

This paper investigates the reciprocal symmetry of charged-particle multiplicity distributions in proton-proton collisions by deriving higher-order local constraints and an entanglement entropy formula, finding that while the symmetry holds approximately near the mean multiplicity, it breaks down globally at 13 TeV due to residual deviations exposed by high-precision data.

The Gist

Imagine you are watching a high-energy particle collision, like two protons smashing together at nearly the speed of light. When they crash, they don't just bounce off; they explode into a shower of new particles. Physicists count how many particles come out in each crash. This number varies wildly from crash to crash.

This paper is like a detective story where the authors are looking for a hidden "rule of symmetry" in how these particle showers behave. They found a strange pattern: if you look at the data in a specific way, the pattern looks the same whether you zoom in or zoom out, or even if you flip the data upside down.

Here is a breakdown of their findings using simple analogies:

1. The "Mirror" Mystery

The authors noticed that the distribution of particles follows a rule called reciprocal symmetry. Imagine you have a mirror placed right in the middle of a crowd. If you look at the people on the left side, their arrangement looks exactly like the reflection of the people on the right side.

In this physics world, the "mirror" isn't a physical object but a mathematical flip. If you take the number of particles in a collision and compare it to its "inverse" (like flipping a fraction upside down), the shape of the data looks identical. The authors call this function $f_s(z)$, and they found that $f_s(z) = f_s(1/z)$.

2. The "Staircase" of Clues

Because this mirror symmetry exists, it creates a "tower" of clues. Think of the data as a smooth hill.

• The First Clue (Level 0): The very top of the hill (the average number of particles) has a specific slope. The authors had already confirmed this in previous work.
• The Second Clue (Level 1): This paper derives a new, more complex clue. It's like checking not just the slope of the hill, but how the slope itself is curving. They created a specific mathematical test (a formula involving the third derivative of the data) to see if this second clue holds true.

3. The Experiment: Does the Mirror Hold?

The team tested these clues using real data from the ATLAS detector at the Large Hadron Collider (LHC), looking at collisions at three different energy levels: 7, 8, and 13 TeV.

• At lower energies (7 and 8 TeV): The data was a bit "fuzzy" (like a low-resolution photo). The clues were consistent with the mirror symmetry, but the picture wasn't sharp enough to be 100% sure.
• At the highest energy (13 TeV): The data was crystal clear (high resolution).
  • The Good News: Right in the center of the data (the average), the mirror symmetry held up perfectly. The new "Level 1" clue passed the test.
  • The Bad News: When they looked at the entire range of the data (not just the center), the mirror started to crack. The symmetry wasn't perfect everywhere; it was just an approximation that worked best near the center.

The Verdict: The symmetry is like a well-made but slightly imperfect mirror. It works great right in the middle, but if you look too far to the edges, the reflection gets distorted.

4. Why Isn't It Perfect? (The "Noisy Machine" Test)

The authors asked: Could this symmetry be caused by a simple random error in the process?

Imagine a machine that shoots out particles. If the machine's speed fluctuates randomly (like a car engine sputtering), the authors calculated what the data should look like. They found that this simple "random noise" model produces a shape that does not have the mirror symmetry. The curve it produces is lopsided.

This means the symmetry isn't just a lucky accident of random noise. It suggests something deeper and more complex is happening in the laws of physics that govern these collisions, something that a simple "noisy machine" model can't explain.

5. The "Entanglement" Connection

Finally, the paper connects this particle counting to a concept called entanglement entropy. In quantum physics, "entanglement" is like a spooky connection between particles where they share information.

The authors derived a new formula to calculate this "quantum connection" based on the particle counts.

• They found that the main part of this connection depends on the average number of particles.
• The "correction" (the fine-tuning) depends on how much the data deviates from a simple exponential curve.
• When they plugged in the real ATLAS data, their new formula matched the direct calculation of the entropy almost perfectly (to within 0.1%).

Summary

The paper discovers a beautiful, mirror-like symmetry in how particles are created in high-energy crashes. They proved that this symmetry creates a specific mathematical rule that holds true near the average number of particles. However, they also showed that this symmetry isn't perfect across the whole board—it's an approximation. Furthermore, this symmetry is too complex to be explained by simple random errors, hinting at deeper, more intricate rules of nature that we are just beginning to understand.

Technical Summary

Technical Summary: Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in pp collisions

Problem Statement
The paper investigates the reciprocal symmetry $f_s(z) = f_s(1/z)$ observed in proton-proton ($pp$) charged-multiplicity distributions at $\sqrt{s} = 7, 8,$ and $13$ TeV. Here, $z = n/\langle n \rangle$ is the KNO scaling variable, and $f_s(z)$ quantifies the deviation of the scaled probability $\langle n \rangle P_n$ from the leading exponential behavior ($e^{-z}$) predicted by Mueller's color-dipole cascade. While previous work established this symmetry in the window $1/3 < z < 3$ and verified the first-order local constraint ($k=0$) at $z=1$, the origin of this symmetry and its validity at higher orders remain unclear. The authors aim to derive higher-order local constraints, test them against ATLAS data, determine if the symmetry arises from simple extensions of Mueller's Markov branching process, and evaluate its impact on the entanglement entropy of the multiplicity distribution.

Methodology

1. Derivation of Local Constraints: The authors reformulate the symmetry condition by defining $h(u) \equiv f_s(e^u)$ where $u = \ln z$. The condition $f_s(z) = f_s(1/z)$ implies $h(u)$ is an even function. Consequently, all odd derivatives of $h(u)$ at $u=0$ must vanish ($h^{(2k+1)}(0) = 0$). Using Stirling numbers of the second kind and the chain rule, these conditions are translated into an infinite tower of algebraic constraints on the multiplicity distribution $P(n)$ and its derivatives at $n = \langle n \rangle$.
2. Data Analysis: The authors utilize ATLAS charged-plicity data at 7, 8, and 13 TeV. To test the $k=1$ constraint without imposing the symmetry by construction, they perform local polynomial fits (degree $N=4$) to the distribution $P(n)$ near $\langle n \rangle$. This allows for the extraction of derivatives $P^{(k)}(\langle n \rangle)$ and the calculation of dimensionless ratios $\rho_0$ and $\rho_1$, as well as the unconditional residual $\delta_3 \equiv 1 - 2\rho_0 + \rho_1$.
3. Model Discrimination: The paper examines whether the symmetry can be derived from a multiplicative-noise extension of Mueller's Markov branching process. They introduce event-by-event fluctuations in the cascade rate $\alpha$ and expand the resulting distribution to leading order in the noise variance.
4. Entanglement Entropy Calculation: A model-independent expression for the entanglement entropy $S$ is derived in terms of $f_s(z)$. The authors evaluate this numerically using the ATLAS data, comparing the result to the direct Shannon entropy calculated from the binned distributions.

Key Contributions and Results

• The $k=1$ Constraint: The authors derive the next member of the constraint tower ($k=1$), which yields the relation:
  $$\langle n \rangle^3 P'''(\langle n \rangle) + 6 \langle n \rangle^2 P''(\langle n \rangle) = 5 P(\langle n \rangle)$$
  This is equivalent to the condition $\delta_3 = 0$.
• Experimental Verification:
  • At 13 TeV, using the largest fit window ($|n - \langle n \rangle| \le 6$), the residual is found to be $\delta_3 = -0.02 \pm 0.11$, consistent with zero. However, smaller windows show larger deviations, indicating sensitivity to the fit range.
  • At 7 TeV, $\delta_3 = -0.32 \pm 0.18$ (for $W=6$), showing a $\sim 1.8\sigma$ deviation, though the authors caution this due to systematic spread across polynomial degrees.
  • At 8 TeV, uncertainties are too large to draw conclusions ($\delta_3 = -0.50 \pm 0.82$).
  • Global Symmetry Test: A $\chi^2$ test of the global symmetry ($f_s(z) = f_s(1/z)$) across $1/3 < z < 3$ is consistent with the data at 7 and 8 TeV but decisively rejects the symmetry at 13 TeV. The higher precision at 13 TeV reveals a residual departure from invariance away from $z=1$.
  • Conclusion on Symmetry: The symmetry holds at the leading two non-trivial orders near $z=1$ but breaks down globally at the precision afforded by 13 TeV data, suggesting $f_s(z) = f_s(1/z)$ is an approximate symmetry.
• Model Exclusion: The authors demonstrate that a multiplicative log-symmetric noise on the cascade rate produces a correction term $f_s \propto z^2 - 4z + 2$. This function is not invariant under $z \to 1/z$ (its roots are not reciprocal). Similarly, two-component geometric mixtures and the negative binomial distribution fail to reproduce the symmetry. Thus, the observed symmetry requires dynamics beyond simple extensions of the geometric cascade, potentially involving dipole merging (Pomeron loops).
• Entanglement Entropy: A model-independent formula is derived:
  $$S = \ln\langle n \rangle + I_0 - \frac{1}{2} \int_S e^{-z} f_s^2(z) dz + O(f_s^3)$$
  where $I_0 = \int_S e^{-z} dz$. The linear term in $f_s$ cancels due to normalization and the constraint $\langle z \rangle = 1$. Numerical evaluation on ATLAS data shows agreement with the direct Shannon entropy at the $10^{-3}$ level. The correction is quadratic in $f_s$, explaining why the leading-order approximation $S \simeq \ln\langle n \rangle + 1$ remains accurate despite significant deviations in $f_s$.

Significance
The paper establishes a rigorous framework for testing the reciprocal symmetry in multiplicity distributions through an infinite tower of local algebraic constraints. By deriving and testing the $k=1$ constraint, the authors provide a more stringent check than the previously verified $k=0$ condition. The results indicate that while the symmetry is a robust feature near the scaling point $z=1$, it is not an exact global symmetry at current experimental precisions. Furthermore, by ruling out simple multiplicative-noise extensions of Mueller's cascade, the work constrains the possible dynamical origins of the symmetry, pointing toward more complex QCD mechanisms like dipole recombination. Finally, the derived entropy formula offers a complementary, model-independent method to extract entanglement entropy from experimental data, linking the KNO-violating term $f_s$ directly to the information-theoretic properties of the partonic state.