Signals for fluctuating constituent numbers in small systems

We propose an extension of the initial condition model TRENTo for sampling the number of partons inside the nucleons that participate in a heavy-ion collision. This sampling method is based on parton distribution functions (PDFs) and therefore has a natural dependence on the momentum transferred in the collision and the scale being probed during the collision. We examine the resulting distributions and their dependence on the momentum transfer. Additionally, we explore the sensitivity of different observables on the number of partons using the TRENTo framework and the estimators available therein for final-state observables.

💡 Research Summary

The authors present an extension of the TRENTo initial‑state model that incorporates event‑by‑event fluctuations of the number of partons inside each nucleon, using parton distribution functions (PDFs) as the underlying probability distributions. In the conventional TRENTo framework, sub‑nucleonic structure is modeled by placing a fixed number m of Gaussian “hot spots” at each nucleon–nucleon collision point. While this captures some geometric fluctuations, it ignores the well‑established fact that the partonic content of a proton or neutron is not fixed: the number of sea quarks and gluons depends on the probing scale (the momentum transfer Q²) and fluctuates from collision to collision.

The new sampling algorithm proceeds as follows. First, the three valence quarks (two up, one down) are always included; their momentum fractions x are drawn from the corresponding valence PDFs. Then, based on the chosen Q², a parton species (gluon, up, down, strange, charm, bottom) is selected with probabilities reflecting the relative PDFs at that scale. A single gluon or a quark–antiquark pair is sampled, its x drawn from the appropriate distribution, and the total momentum fraction x_tot is updated. This loop repeats until x_tot exceeds unity, at which point the event’s parton multiplicity m is defined as three (the valence quarks) plus the number of sea partons drawn.

A crucial physical cutoff is introduced: the minimal momentum fraction x_min = Q² / s, derived from kinematics (with s the center‑of‑mass energy squared). This prevents the PDFs’ divergence as x → 0 and imposes an upper bound m_max = 1 / x_min on the number of partons that can be sampled. For LHC energies, x_min is of order 10⁻⁶, so m_max can be large, but still finite. The authors use the UCL‑HEP PDF set (e.g., NNPDF) and illustrate the resulting m‑distributions for Q² = 1, 5, 10 GeV². All distributions are bounded below by m = 3 (the valence quarks). As Q² increases, the PDFs become more peaked at low x, leading to a larger average ⟨m⟩ (Fig. 3) and a smaller standard deviation σ(m) (Fig. 4). Interestingly, for very high Q² the average ⟨m⟩ begins to decrease because the kinematic cutoff x_min moves upward, excluding the very low‑x partons that would otherwise inflate the count.

Having established a physically motivated, scale‑dependent parton‑number distribution, the authors explore its impact on observable quantities. They argue that the total deposited energy per nucleon–nucleon collision is essentially independent of m, so bulk observables such as the charged‑particle multiplicity N_ch (∝ initial entropy S) and the mean transverse momentum ⟨p_T⟩ (∝ deposited energy E) are only weakly sensitive to the parton‑number fluctuations. In contrast, geometric quantities—initial eccentricities ε_n—are directly shaped by the spatial arrangement of the deposited “hot spots.” Since the number of hot spots now fluctuates event‑by‑event, the eccentricities acquire an additional source of variance, which propagates to the final‑state flow coefficients v_n (via the approximate linear response v_n ∝ ε_n).

The sensitivity is expected to be strongest in small collision systems (e.g., O+O, Ne+Ne) where the number of binary nucleon–nucleon collisions is of order a few hundred. In such systems the averaging over many independent parton‑sampling events is limited, so the fluctuations in m leave a noticeable imprint on the initial geometry and thus on v_n. In large systems (Pb+Pb), thousands of binary collisions effectively average out the parton‑number fluctuations, making their effect on final observables marginal.

The paper also discusses practical considerations. Sampling a large number of partons for high‑energy collisions can be computationally expensive. To mitigate this, the authors suggest fitting the m‑distribution’s cumulative function using the mean and standard deviation as functions of Q², allowing a fast parametrized sampling. They note that correlations among partons (e.g., momentum conservation, color connections) are neglected in the present implementation, and that future work could incorporate n‑parton joint distributions or transverse‑momentum dependent PDFs (TMDs) to generate initial transverse flow fields.

In summary, the work provides a theoretically grounded method to embed Q²‑dependent parton‑number fluctuations into the TRENTo initial‑state model, demonstrates how these fluctuations modify the initial eccentricities, and identifies small‑system flow observables as the most promising probes. The approach opens a path toward more realistic sub‑nucleonic modeling, potentially improving Bayesian calibrations of initial‑state parameters and offering new discriminants for the role of partonic structure in relativistic heavy‑ion collisions.