[论文解读] How transverse momentum conservation breaks azimuthal correlation factorization

Jia-Lin Pei, Guo-Liang Ma+1 more|arXiv (Cornell University)|Jan 27, 2026

High-Energy Particle Collisions Research被引用 0

一句话总结

论文表明横向动量守恒（TMC）驱动小系统中方位角两粒子相关性因式分解的破缺，能再现 CMS p-Pb 数据的 r2 和 r3，并揭示在 TMC 下 r_n 的符号规则。

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 ight )^{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.

研究动机与目标

• Motivate the study of azimuthal correlations in small systems (p+p and p+A) and the limitations of hydrodynamic explanations.
• Quantify the breakdown of two-particle azimuthal correlation factorization using r_n in the presence of transverse momentum conservation.
• Develop an analytical framework to relate TMC-induced correlations with flow fluctuations across pT and multiplicity.
• Compare analytical TMC-based predictions with CMS p-Pb data to identify the origin of collectivity in small systems.

提出的方法

• Define the factorization ratio r_n from two-particle azimuthal correlations V_nΔ and the single-particle v_n.
• Model the N-particle final state under strict transverse momentum conservation with a δ-function constraint and a common single-particle distribution f(p).
• Use a Gaussian approximation for the sum of transverse momenta (central limit theorem) to derive the two-particle distribution f2 and express V_nΔ.
• Expand the exponential constraints to second or third order to isolate pure TMC, pure flow, and their interplay contributions (Eqs. 24–28).
• Derive concise proxy expressions for r2 and r3 that combine pure TMC and pure flow terms with interaction terms (Eq. 32 and Eq. 35).
• Validate the model by comparing with CMS p-Pb data at 5.02 TeV for r2 and r3 across pT and multiplicity.]
• method translated text

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) Khach

实验结果

研究问题

• RQ1 Can transverse momentum conservation alone explain the observed breakdown of two-particle azimuthal factorization in small systems?
• RQ2 How do r2 and r3 depend on pT differences and event multiplicity under TMC, and how do they compare to CMS p-Pb measurements?
• RQ3 What is the relative importance of pure TMC, pure flow, and their interplay in shaping c_n{2} and r_n?
• RQ4 Is there a sign rule for r_n under TMC, and can it account for r3>1 in CMS p-Pb data?
• RQ5 Do pT-dependent event-plane fluctuations significantly modify the TMC-driven factorization breakdown?

主要发现

• TMC is identified as the key mechanism driving factorization breakdown in small systems for r2 and r3.
• The model reproduces CMS p-Pb data for both r2 (typically < 1) and r3 (observed > 1) across multiple pT and multiplicity ranges.
• A sign rule is established: under TMC, the deviation r_n − 1 follows (-1)^{n+1}, negative for even n and positive for odd n.
• The r2 and r3 dependencies on pT difference and multiplicity are consistent with stronger TMC effects at lower multiplicities and higher momenta.
• An efficient proxy combining pure TMC and pure flow terms (Eq. 32 for r2 and Eq. 35 for r3) closely matches the full calculations across broad kinematics.
• The work provides an analytical framework to quantify transverse-momentum-dependent flow fluctuations and informs the interpretation of collectivity in small colliding systems.

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) Khach