Incoherent multiple collisions and the anomalous mass-number dependence of high-transverse-momentum hadron production in proton-nucleus collisions

Good, Kazama, and Yang (1977) asked whether the anomalous mass-number dependence of high-transverse-momentum hadron production in proton-nucleus collisions — the Chicago-Princeton finding that the invariant cross section scales as $A^{\alpha(p_\perp)}$ with $\alpha(p_\perp) > 1$ at large $p_\perp$ (the Cronin effect) — can be accounted for entirely by incoherent multiple collisions, i.e., successive independent hits of the projectile and its secondaries on nucleons of the target. Because such showering depends only on the total path length traversed inside nuclear matter, yields from 'compacted' and 'separated' configurations of nuclear matter with identical path-length distributions must be equal. The authors proposed weight-integral experiments comparing thick-target yields with weighted combinations of single-nucleus cross sections, called for microscopic calculations anchored to measured single-collision data to explain the $p_\perp$ dependence of $\alpha(p_\perp)$, and flagged the caveats conditioning these tests: the assumption that intermediate resonances whose decay lengths span internucleonic distances do not appreciably invalidate the identities, the need to extrapolate single-nucleus cross sections beyond tungsten/uranium, and the finite domain of the power-law fit, which must fail as $A\to\infty$. The proposed identity experiments were never performed; multiple-collision models — first at the hadron level, later at the parton level — qualitatively reproduce $\alpha(p_\perp)$; a quantitative microscopic decomposition into initial-state transverse-momentum broadening, shadowing, energy loss, and final-state interactions remains contested. The problem — including the direct experimental separation of multiple-collision effects from residual anomalous nuclear effects — remains open.

Background

When a beam of high-energy protons strikes an atomic nucleus of mass number $A$, the yield of a given hadron $h$ produced at large transverse momentum $p_\perp$ (the momentum component perpendicular to the beam; large $p_\perp$ signals a hard, short-distance collision) is empirically larger, per target nucleus, than expected from a simple superposition of independent proton-nucleon collisions. In fixed-target experiments at Fermilab with 200-400 GeV protons, Cronin and collaborators found that the invariant production cross section (the Lorentz-invariant yield $E\,d\sigma/d^{3}p$) at fixed $p_\perp$ scales as a power of the mass number, $d\sigma/d^3p \propto A^{\alpha(p_\perp)}$, with the exponent $\alpha$ rising above unity at large $p_\perp$ (about 1.1 for $\pi^\pm$ and $K^+$ and about 1.3 for protons, antiprotons, and $K^-$ at $p_\perp \approx 5$ GeV/$c$, tending to saturate for $p_\perp \gtrsim 4$ GeV/$c$). This 'Cronin effect' was surprising: absorption of the projectile by nucleons downstream ('shadowing') naively suggests $\alpha < 1$, and the weak-coupling limit of quantum chromodynamics (QCD) suggests $\alpha \to 1$ at asymptotically large $p_\perp$.

Good, Kazama, and Yang proposed to determine whether the enhancement is due to 'incoherent multiple collisions' — a showering process in which the incident proton successively strikes several nucleons of the target, each hit being an independent elastic or inelastic event, with the observed particle descending from the projectile through a chain of forward-moving ancestors (the 'genealogical line' through the target). The structural property of this picture is that production yields depend only on the distribution $P(L)$ of the total path length $L$ that the genealogical line spends inside nuclear matter — not on how that matter is arranged along the line. Consequently, configurations with identical $P(L)$ but different arrangements — nuclear matter 'compacted' into one continuous bunch (a single nucleus) or 'separated' into several nuclei with macroscopic gaps in between — must give identical yields. The authors turned this invariance into proposed experiments: since any $P(L)$ can be written as a weighted superposition of single-nucleus path-length distributions, with weights $W(na,S)$ over simulating nuclei of radius $S$ at target thickness $na$ (the weights may be negative because they multiply different measurements rather than physical densities), the yield from a thick target can be compared with a weighted integral of measured single-nucleus cross sections. Equality would demonstrate that incoherent multiple collisions account for all of the observed $A$ dependence; a deficit would isolate and measure the residual anomalous nuclear effect of compacting nucleons together. Two variants were worked out: an identity relating the double-collision differential cross section (and the initial curvature of the counts-per-projectile-versus-thickness curve) to weighted single-nucleus yields, and a thick-target simulation identity for the full thickness-dependent yield.

The paper itself flagged what would have to be established for such tests to be conclusive. First, the equalities assume that resonances and intermediate particles — short-lived excited hadrons such as $N^{*}$ (excited nucleon states), $\eta$ and $\pi^{0}$ (neutral mesons) — whose decay lengths exceed typical internucleonic spacings but remain far below macroscopic distances do not materially change the counts when nuclear matter is compacted or separated; if such states propagate and reinteract between collisions, the identities fail. Second, even if the tests succeed, they demonstrate only that multiple collisions produce the $A$ dependence; understanding the observed $p_\perp$ dependence of the exponent $\alpha(p_\perp)$ requires explicit microscopic calculations starting from measured single-collision data — calculations the authors announced as forthcoming but never published. Third, the single-nucleus power-law fit $d\sigma/d^3p = C\,S^{3\alpha}$ ($S$ the nuclear radius, so that $S^{3\alpha}$ is proportional to $A^{\alpha}$ for a uniform sphere) was verified experimentally only up to tungsten, so tests relying on larger radii need extrapolation; and fourth, this power law cannot remain valid as $A\to\infty$, since it would predict unphysical growth of yields for very thick targets where damping must set in.

The problem reappeared in transformed language in modern QCD-based proton-nucleus physics, where the same anomalous enhancement is analyzed through partonic mechanisms — initial-state multiple parton scattering and transverse-momentum broadening, modifications of nuclear parton distributions ('nuclear shadowing'), coherent parton energy loss, gluon saturation, and final-state reinteractions — and is compared with measured nuclear modification factors $R_{pA}$ (the proton-nucleus yield divided by the proton-proton yield scaled by the number of binary nucleon-nucleon collisions) at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). Whether any of these mechanisms accounts quantitatively for the effect, singly or in combination, remains contested, which is the environment in which the original question survives.

Problem Statement

The problem, stated source-faithfully after Good, Kazama, and Yang (1977), is to determine whether incoherent multiple collisions account for the anomalous mass-number dependence of high-$p_\perp$ hadron production in proton-nucleus collisions, i.e., for the observed exponent $\alpha(p_\perp) > 1$ and its dependence on $p_\perp$. It comprises two linked parts that jointly address this question; neither part alone settles it:

(1) Experimental test of the simulation identities: the experiment was never performed and no separated-configuration data exist, so the question is whether the proposed test is realizable with measured single-nucleus cross sections and the weight functions $W(na,S)$, and whether it would return the identity or a deficit. The test measures hadron yields per incident proton from 'separated' nuclear-matter configurations (thick targets of a light nucleus at chosen thickness $na$, or two nuclei of radius $R$ with macroscopic gaps) and compares them with the weighted integrals $\int W(na,S)\,(d\sigma(S)/d^3p)\,dS$ of measured single-nucleus cross sections over simulating radii $S$ ('compacted' matter), the two configurations having identical total path-length distributions $P(L)$; it includes the companion test on the initial curvature of the counts-per-projectile-versus-thickness curve. Agreement demonstrates that multiple collisions account for all of the $A$ dependence; a deficit on the separated side isolates and measures the residual anomalous nuclear effect of squeezing nucleons together.

(2) The microscopic calculation: quantitatively account for the $p_\perp$ dependence of $\alpha(p_\perp)$ from calculations of incoherent-multiple-collision effects starting from measured single-collision (proton-nucleon) data — quantitatively reproducing the measured $\alpha(p_\perp)$ across the fixed-target $p_\perp$ range, or establishing quantitatively why the incoherent-multiple-collision input cannot reproduce it. Per the source, such calculations remain necessary even if the identities of part (1) are experimentally verified, since verification alone demonstrates that the observed $\alpha(p_\perp)$ is explainable by incoherent-multiple-collision effects without explaining its $p_\perp$ dependence.

The source attaches caveats to these tests rather than further goals: (i) the identities rest on essentially one assumption — that resonances and intermediate particles ($N^{*}$, $\eta$, $\pi^{0}$) whose decay lengths exceed typical internucleonic spacings but remain far below macroscopic distances do not appreciably invalidate them; (ii) the single-nucleus cross sections entering the thick-target identity can be measured only up to radii of order uranium and beyond that must be extrapolated, so the target thickness $n$ must be chosen such that the weight function is not large beyond uranium; and (iii) if multiple-collision effects are the reason for the observed $A$ dependence, the single-power parametrization $d\sigma/d^3p = C\,S^{3\alpha}$ cannot remain valid as $A\to\infty$, and must be used with caution beyond tungsten.

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