How transverse momentum conservation breaks azimuthal correlation factorization

TL;DR

This work addresses the breakdown of azimuthal factorization in small-system collisions by showing that transverse-momentum conservation (TMC) is the dominant mechanism shaping two-particle azimuthal correlations. The authors develop an analytical framework that combines TMC with flow fluctuations to derive expressions for the factorization ratios $r_2$ and $r_3$, and they validate these predictions by quantitatively reproducing CMS p-Pb data across multiple $p_T$ and multiplicity ranges. A key result is the sign rule that the deviation $r_n-1$ follows $(-1)^{n+1}$, yielding $r_2<1$ (even) and $r_3>1$ (odd), with a stronger effect for $r_3$ and lower multiplicities. The framework provides a principled way to quantify transverse-momentum-dependent flow fluctuations and enhances our understanding of collectivity in small collision systems, while suggesting future extensions to include longitudinal momentum conservation for a complete phase-space picture.

Abstract

The breakdown of azimuthal two-particle correlation factorization, quantified by the ratios $r_2$ and $r_3$, serves as a sensitive probe of transverse-momentum-dependent flow fluctuations. While hydrodynamic models predict $r_3 \leq 1$, experimental data from CMS in p-Pb collisions exhibit $r_3 > 1$, presenting a clear puzzle. We show that transverse momentum conservation (TMC) is the key mechanism dictating this factorization breakdown in small systems. We systematically calculate the effect of TMC as a function of the momentum difference between particles across various multiplicity and momentum ranges. Our results are in quantitative agreement with CMS p-Pb data for both $r_2$ and $r_3$. A central finding is a sign rule: under TMC, the deviation $r_n - 1$ follows $\left ( - 1 \right )^{n+1} $, being negative for even and positive for odd harmonic orders $n$. This work establishes an analytical framework to quantify transverse-momentum-dependent flow fluctuations and provides new insights into the origin of collectivity in small colliding systems.

Figures (9)

• Figure 1: Factorization ratio $r_2$ as a function of $p_{a}-p_{b}$ in four $p_a$ bins (columns) and four $N_{ch}$ ranges (rows), forming a 4$\times$4 array of panels. Curves: TMC calculations. Points: CMS p-Pb data at 5.02 TeV with statistical errors (systematic uncertainties negligible) cmsdata.
• Figure 2: Factorization ratio $r_3$ as a function of $p_{a}-p_{b}$ in four $p_a$ bins (columns) and four $N_{ch}$ ranges (rows), forming a 4$\times$4 array of panels. Curves: TMC calculations. Points: CMS p-Pb data at 5.02 TeV with statistical errors (systematic uncertainties negligible) cmsdata.
• Figure 3: $c_{2}\left \{ 2 \right \}$ from "pure TMC", "pure TMC+pure flow", and "pure TMC+pure flow+interplay" (labeled as "all") as a function of the number of particles $N$ for four $p_a$ bins.
• Figure 4: Factorization ratio $r_2$ (upper row) and $r_3$ (lower row) as a function of $p_{a}-p_{b}$ in four $p_{a}$ bins (columns) for $120\le N_{ch}\le 150$. Curves: calculations from TMC and flow (proxy and all). Points: CMS p-Pb data at 5.02 TeV with statistical errors (systematic uncertainties negligible) cmsdata.
• Figure 5: $c_{3}\left \{ 2 \right \}$ from "pure TMC", "pure flow+pure TMC", "pure flow+pure TMC+main interplay", and "pure flow+pure TMC+interplay"(labeled "all") as a function of the number of particles $N$ for four $p_a$ bins.