A Large-$N$ Expansion for Minimum Bias

Despite being the overwhelming majority of events produced in hadron or heavy ion collisions, minimum bias events do not enjoy a robust first-principles theoretical description as their dynamics are dominated by low-energy quantum chromodynamics. In this paper, we present a novel expansion scheme of the cross section for minimum bias events that exploits an ergodic hypothesis for particles in the events and events in an ensemble of data. We identify power counting rules and symmetries of minimum bias from which the form of the squared matrix element can be expanded in symmetric polynomials of the phase space coordinates. This expansion is entirely defined in terms of observable quantities, in contrast to models of heavy ion collisions that rely on unmeasurable quantities like the number of nucleons participating in the collision, or tunes of parton shower parameters to describe the underlying event in proton collisions. The expansion parameter that we identify from our power counting is the number of detected particles $N$ and as $N\to\infty$ the variance of the squared matrix element about its mean, constant value on phase space vanishes. With this expansion, we show that the transverse momentum distribution of particles takes a universal form that only depends on a single parameter, has a fractional dispersion relation, and agrees with data in its realm of validity. We show that the constraint of positivity of the squared matrix element requires that all azimuthal correlations vanish in the $N\to\infty$ limit at fixed center-of-mass energy, as observed in data. The approach we follow allows for a unified treatment of small and large system collective behavior, being equally applicable to describe, e.g., elliptic flow in PbPb collisions and the "ridge" in pp collisions.