How transverse momentum conservation breaks azimuthal correlation factorization

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.

💡 Research Summary

The paper addresses a long‑standing puzzle in small‑system (p‑Pb) collisions: the two‑particle azimuthal correlation factorization ratios, (r_{2}) and (r_{3}), measured by CMS show opposite trends—(r_{2}<1) while (r_{3}>1). Conventional hydrodynamic calculations predict (r_{n}\le 1) for all harmonics, so the data cannot be reconciled within a pure flow picture. The authors propose that transverse‑momentum conservation (TMC) is the dominant mechanism responsible for the observed factorization breaking.

Starting from the exact many‑particle distribution that enforces global transverse‑momentum conservation, \