Impact-parameter origin of multiplicity fluctuations in nondiffractive hadron-hadron collisions: quantitative establishment and experimental tests of the Chou-Yang geometrical picture

The charged-particle multiplicity in nondiffractive hadron-hadron collisions fluctuates widely from event to event, with a distribution that at ISR and SPS collider energies was approximately KNO-scaled, i.e. a function only of $z=n_{\mathrm{ch}}/\langle n_{\mathrm{ch}}\rangle$. Chou and Yang (1982) proposed that a major source of these fluctuations is the event-to-event fluctuation of the collision impact parameter $b$, constructed a mapping $b(z)$ by equating the scaled multiplicity density to the nondiffractive eikonal cross-section element, and derived from a two-dimensional uncertainty relation the prediction that the mean-squared angular spread of fragmentation particles conditional on central multiplicity increases with $z$ and scales as $1/k_0^2$ with the center-of-mass momentum scale. The paper left this geometrical program quantitatively unestablished: the conditional angular data needed to test the prediction and to constrain the width of $b$ at fixed $z$ were lacking, the diffraction subtraction was model-dependent, the sensitivity of $b(z)$ to the blackness function and to the KNO fit was unquantified, and long-range forward-backward rapidity correlations were explained only qualitatively. Later work built impact-parameter multiplicity models (Finkelstein 1988; Krasznovszky-Wagner 1989/1993; U-matrix 2024) and established that exact KNO scaling is violated at higher energies, but no published test of the conditional angular-spread prediction or data-driven extraction of the geometrical share of multiplicity variance is known. The problem remains open in a modernized form.

Background

In high-energy hadron-hadron collisions the number of charged particles produced per event, $n_{\mathrm{ch}}$ (the charged multiplicity), fluctuates strongly from event to event. Koba, Nielsen and Olesen (KNO) argued in 1972 that at asymptotically high energy the multiplicity distribution, plotted in terms of the scaled variable $z = n_{\mathrm{ch}}/\langle n_{\mathrm{ch}}\rangle$, becomes independent of the collision energy $\sqrt{s}$: $P(n,\sqrt{s})\simeq(1/\langle n\rangle)\,\psi(z)$, called KNO scaling. Data from the CERN Intersecting Storage Rings (ISR) and the first measurements at the CERN SPS (Super Proton Synchrotron) $p\bar{p}$ collider at $\sqrt{s}=540$ GeV showed an approximately scaling but very broad $\psi(z)$, i.e. very large multiplicity fluctuations.

The geometrical picture of Chou and Yang describes high-energy collisions in terms of the impact parameter $b$, the transverse separation between the centers of the two incoming hadrons. In the eikonal approximation (a high-energy approximation treating the collision as a sequence of independent, mostly forward scatterings), the elastic $S$-matrix is written $S(b)=e^{-\Omega(b)}$, where $\Omega(b)$ is the blackness or opacity of the interaction at separation $b$; the inelastic cross-section density in $b$-space is $(1-|S(b)|^2)\,\mathrm{d}^2b$. Part of the inelastic cross section is diffraction dissociation, quasi-elastic events in which one or both beam particles emerge as low-mass excited systems separated by large rapidity gaps (rapidity $y$ is the logarithm of longitudinal momentum; pseudorapidity $\eta$ is the corresponding angular variable $\eta=-\ln\tan(\theta/2)$). Such events have a distinct multiplicity structure and must be subtracted to define the nondiffractive event class.

Chou and Yang (1982) proposed that a major source of the broad, roughly energy-independent multiplicity distribution is the event-to-event fluctuation of $b$ itself: at small $b$ the collision is blacker and produces more particles; at large $b$ it is peripheral and produces fewer. Assuming a monotonic one-to-one association between $z$ and $b$ (with the angular momentum given by $L=b\,p_{\mathrm{beam}}$, where $p_{\mathrm{beam}}=270$ GeV is the beam momentum at $\sqrt{s}=540$ GeV), they constructed a numerical mapping $b(z)$ by equating the KNO probability $\psi(z)\,\mathrm{d}z$, with the ansatz $\psi(z)=\tfrac{\pi}{2}\,z\,e^{-\pi z^2/4}$, to the nondiffractive part of the eikonal inelastic cross-section element, using an $\Omega(b)$ determined from lower-energy elastic data and two alternative prescriptions for subtracting diffraction (a fixed fraction of the inelastic cross section, versus a probability $\propto\Omega e^{-\Omega}$ normalized to a total diffraction-dissociation cross section of $0.22\,\sigma_T$); the two prescriptions gave nearly indistinguishable $b(z)$ for the specific $\Omega(b)$ and $\psi(z)$ inputs used there. They used this picture to interpret the measured energy sharing between the fragmentation region (roughly $|y|>0.8\,y_{\mathrm{incoming}}$, where the leading remnants of the beam hadrons emerge) and the central region: large-$b$ events carry a larger fraction of the outgoing energy in the fragmentation region, while the average energy per particle in $|\eta|<5$ changes only by about 30% while the multiplicity varies by a factor of about 6.

From the two-dimensional Fourier relation between the elastic momentum transfer $t$ and $b$, they derived the uncertainty relation $\bar{t}\,\overline{(\Delta b)^2}\geqslant 1$; combined with $\bar{t}=k_0^2\,\overline{\theta^2}$ — the standard small-angle elastic-kinematics relation $|t|\simeq k_0^2\theta^2$, in which $k_0$ is the center-of-mass beam momentum, $k_0=\sqrt{s}/2$ (equal to the $p_{\mathrm{beam}}$ entering $L=b\,p_{\mathrm{beam}}$), and $\theta$ is the polar angle — this gives $\overline{(\Delta\theta)^2}\,\overline{(\Delta L)^2}\geqslant 1$. Observing that fragmentation particles are strongly forward-peaked (small $\overline{(\Delta\theta)^2}$) in low-central-multiplicity events, they inferred that the angular-momentum spread shrinks as $z$ grows, and, assuming the spread of $b$ at fixed $z$ is energy-independent, predicted $\overline{(\Delta\theta)^2}\propto 1/k_0^2$ for the angular spread of fragmentation particles conditional on central multiplicity. They also noted that $b$-fluctuations would naturally generate the observed long-range forward-backward multiplicity correlations (more particles on one side of the detector accompany more particles on the other, across large rapidity separations), but left a quantitative stochastic model of this correlation undeveloped.

Problem Statement

For nondiffractive hadron-hadron ($pp$ and $p\bar{p}$) collisions, determine quantitatively whether and how much of the event-by-event fluctuation of the charged multiplicity $n_{\mathrm{ch}}$ is geometrical in origin, i.e. attributable to the fluctuation of the impact parameter $b$, by completing and testing the Chou-Yang geometrical program. The four steps below must be addressed jointly. Where the required conditional forward (fragmentation-region) distributions are not tabulated in the cited works, they are to be obtained from published collider measurements or open data releases for nondiffractive $pp$ and $p\bar{p}$ collisions, with the chosen source identified and its uncertainties propagated:

1. Construct the mapping $b(z)$ between the scaled multiplicity $z=n_{\mathrm{ch}}/\langle n_{\mathrm{ch}}\rangle$ and $b$ from the measured scaled multiplicity distribution $\psi(z)$ and the eikonal inelastic cross section with $S(b)=e^{-\Omega(b)}$, subtracting the diffraction-dissociation contribution using experimentally constrained inputs (measured diffraction fractions rather than purely model assumptions), and quantify the sensitivity of $b(z)$ to the choices of the blackness function $\Omega(b)$ and of the fitted functional form of $\psi(z)$, including forms with $\psi(0)\neq 0$.

2. Extract or bound the width of the $b$-distribution at fixed $z$ from data, in particular from the angular distributions of fragmentation (forward) particles conditional on central multiplicity, and thereby quantify the fraction of the multiplicity variance at fixed energy attributable to $b$-fluctuations versus other, dynamical sources.

3. Test the uncertainty-relation consequence of the two-dimensional Fourier relation between momentum transfer and impact parameter: the prediction that the mean-squared angular spread of fragmentation particles at fixed $z$, $\overline{(\Delta\theta)^2}$, increases with increasing $z$ and scales as $1/k_0^2$ with the center-of-mass beam momentum $k_0=\sqrt{s}/2$, using conditional angular distributions of fragmentation particles.

4. Build a quantitative model of the long-range forward-backward rapidity correlations induced by $b$-fluctuations, specifying the stochastic multiplicity distribution at fixed $b$ and the conditional distribution of one-hemisphere multiplicity given $z$, and confront it with measured forward-backward correlation data.

References

1. T. T. Chou and Chen Ning Yang, "Remarks on multiplicity fluctuations and KNO scaling in $\bar{p}p$ collider experiments", Physics Letters B 116 (1982) 301-304. DOI: 10.1016/0370-2693(82)90348-3, https://doi.org/10.1016/0370-2693(82)90348-3 (source paper)
2. Z. Koba, H. B. Nielsen and P. Olesen, "Scaling of multiplicity distributions in high energy hadron collisions", Nuclear Physics B 40 (1972) 317-334. DOI: 10.1016/0550-3213(72)90551-2, https://doi.org/10.1016/0550-3213(72)90551-2 (the KNO-scaling proposal)
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