Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum

TL;DR

This work develops and analyzes one-dimensional Mueller dipole cascades that include splitting, recombination, and transitions to vacuum to describe high-energy hadron production. Analytically and numerically, the authors show that both the β-cascade (splitting plus recombination) and the γ-cascade (with vacuum transitions) yield Negative Binomial-like multiplicity distributions, with characteristic saturation or decay depending on the regime and parameters; the γ-cascade, in particular, addresses forward-rapidity LHCb entropy data by incorporating vacuum transitions and a proper normalization. They map the cascades to quantum-information metrics (entropy, purity, complexity) and demonstrate distinct signatures for each cascade, notably a monotonic entropy saturation for β and a non-monotonic entropy behavior for γ. Fitting to LHCb entropy data suggests production and vacuum-transition processes dominate with small recombination, highlighting the importance of vacuum transitions in describing forward hadron production and entanglement-like measures in high-energy QCD cascades.

Abstract

We study one dimensional dipole cascade models in the high-energy limit of QCD. Motivated by data on hadron multiplicities in the LHCb kinematical range, we generalize existing cascade models for splitting and recombination to account also for transitions to the vacuum. This modification allows us to describe the data. Furthermore, we perform both analytical and numerical studies of the cascades and find that the cascade including loop corrections, as well as the one accounting for transitions to the vacuum, can both be related to the Negative Binomial Distribution. In the latter case, however, one must properly normalize the cascade to account for transitions to the vacuum. We also study the scaling properties of the cascades and identify new regimes, which we call "focal" and "parallel." Finally, we investigate the Quantum Information (QI) measures of the cascades, which allow us to highlight their distinctive properties.

Figures (14)

• Figure 1: Probabilities $p_n(y)$ ($n=1,\ldots ,10$) for the $\beta$ branching (\ref{['eq:EquationSat1']}) for $r=0.1$ (left) and $r=0.5$ (right), and $\alpha=0.5$. One can see that for small $r$ probabilities cross at $y_{\rm cross}\simeq 6.5$, whereas for larger $r$ they reach asymptotic values without crossing.
• Figure 2: Mean multiplicities of the $\beta$ cascade (solid lines) as functions of $y$ for $\alpha=0.2$ (orange), $\alpha=0.5$ (green) and $\alpha=0.8$ (red). Values of $r$: 0.05 (left panel) and 0.5 (right panel). For comparison dashed-dotted line corresponds to $\beta = 0$. Dashed blue line: $1/r$.
• Figure 3: Asymptotic values of $r \,\bar{n}_{\rm sat}$ and $r \, \sigma^2_{\rm sat}$ as functions of $r$. Dashed orange lines correspond to $1/r$ dependence discussed in the text.
• Figure 4: Variance of the $\beta$ cascade (solid lines) as a function of $y$ for for $\alpha=0.2$ (orange), $\alpha=0.5$ (green) and $\alpha=0.8$ (red). Values of $r$: 0.05 (left panel) and 0.5 (right panel). Dashed blue line: $1/r$.
• Figure 5: Probability distributions $p_n(y)$ for the $\beta$-cascade (blue circles) with $\alpha=0.5$ and the corresponding NBD (orange squares) for selected values of $y=3$ and 15. In the upper row we have plotted cascade for $r=0.05$, whereas the lower row corresponds to $r=0.5$. Lines are drawn to guide the eyes.