Why elliptic and triangular flow nearly match in ultra-central lead-lead collisions: UCC-DiffHIC event-by-event runs supply the response decomposition

In ultra-central lead-lead collisions the ratio of elliptic to triangular flow can approach unity without a breakdown of linear response and without extra physics such as nuclear deformation. Linear scaling from initial eccentricity, with viscous damping stronger for shorter-wavelength structure, generally predicts elliptic flow above triangular flow and a clear hierarchy; earlier multistage calculations retained that splitting in the ultra-central limit. In UCC-DiffHIC event-by-event simulations the ratio with the standard TRENTo parameter set moves steadily toward unity as centrality decreases toward the most central collisions, matching the CMS measurement, while two Bayesian parameter sets fail to reproduce the degeneracy.

The suppression of the flow ratio comes from imperfect event-by-event correlation between final-state flow and initial geometry, captured by an exact identity written as a ratio of two Pearson response correlations. The linear response relation writes the flow vector as a response coefficient times the eccentricity vector, but the residual fluctuation term is always present in event-by-event simulations and had not been analyzed separately in the ultra-central flow problem. For all three initial-condition parameter sets, linear scaling from eccentricity alone yields elliptic flow above triangular flow, while the event-by-event ratio is systematically reduced, with the dominant effect from the Pearson correlations; in the most central bins the spread of elliptic flow about the linear-response trend is markedly larger than for triangular flow.

UCC-DiffHIC hands the mapping from initial condition to final charged-particle spectrum of the full hybrid chain to a conditional diffusion model, making high-statistics simulation of ultra-central collisions computationally feasible. Ultra-central events occupy only a fraction of a percent of the inelastic nucleus-nucleus cross section, so the statistics required by conventional event-by-event hydrodynamic plus hadronic transport simulations were prohibitive, and most studies extrapolated the linear scaling into an untested regime. The baseline DiffHIC under-samples ultra-central events below a centrality threshold in its minimum-bias training set, so a geometry-conditioned residual network trained only on ultra-central events is added; on about 5000 events below that threshold, central values of multiplicity and of two- and four-particle cumulants agree with the hybrid chain within 2.5%.

The discussion is limited to 5.02 TeV lead-lead collisions in the ultra-central region with centrality defined by initial-state entropy, with transport coefficients and equation of state of the fluid expansion held fixed and only three TRENTo initial-condition parameter sets compared; the conclusions concern the charged-particle elliptic-to-triangular flow ratio and several higher-order cumulants.

The two Bayesian parameter sets give only partial compensation and retain larger flow ratios; the authors state that smaller or fluctuating nuclear-deformation contributions cannot be excluded, and the specific values of the Pearson correlations depend on initial-geometry model parameters and on the Knudsen number and system size of the fluid expansion.