Abstract
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 → ∞ 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 → ∞ 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. We also briefly comment on power counting and symmetries for minimum bias events in other collider environments and show that a possible ridge in e+e− collisions is highly suppressed as a consequence of its symmetries.
Highlights
Despite significant research efforts, a first-principles understanding of minimum bias events is lacking precisely because it exists at the scale for which QCD becomes strongly interacting
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
Summary
We will use the power counting and symmetries to provide an effective description of the scattering cross section for these minimum bias events exclusively in terms of properties of the measured particles. We implicitly write the cross section as the integral of the squared matrix element over N + NB1 + NB2-body phase space, and in the remaining lines, expand out the massless Lorentz-invariant integration measure and momentum-conserving δ-functions. The squared matrix element can only depend on a function of the total plus or minus lightcone momentum that goes down the beampipes:. For some function f (k+k−) and our power counting assumes that the momentum lost down the beam regions (or, the z-axis boost of the detected particles), is of a comparable size to the center-of-mass energy, Q2. With the power counting and symmetries enforced on the form of the matrix element, in the cross section we can integrate over the momentum lost down the beams. We note that the topology of the smeared phase space is that of a (3N − 2)-ball, found by integrating over two of the dimensions of the N -body phase space manifold [30]
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