Coherent pion emission in high-energy heavy-ion collisions: quantifying the coherent fraction, its effect on Bose-Einstein (HBT) correlations, and its separation from competing mechanisms

In his 2006 lecture 'Quantum Optics and Heavy Ion Physics', Roy J. Glauber advanced two ideas for relativistic heavy-ion collisions: a parametric 'used nucleon model' whose efficiency parameter $\mu$ interpolates particle production between the wounded-nucleon and collision-counting limits, and the quantum-optics argument that the pion field produced in the collision may retain a coherent component which survives even a violent, largely randomizing collision and could be 'one of several reasons' why measured like-pion Bose-Einstein (Hanbury Brown-Twiss, HBT) correlation functions fail to reach the chaotic limit of 2 at vanishing relative momentum. The open problem is to determine quantitatively the coherent fraction of pion emission and its effect on two- and multi-pion HBT observables, and to verify experimentally whether such coherence is present while separating it from competing mechanisms (long-lived resonance decays, multi-body Coulomb effects, momentum resolution, incomplete thermalization). Two decades of femtoscopy sharpened but did not settle the question: genuine three- and four-pion correlations in Pb-Pb collisions at the LHC are significantly suppressed relative to chaotic expectations with an origin that remains unclear; coherent-fraction estimates are model-dependent, of order 20-35%; and partially coherent source models can reproduce the measured two-, three- and four-pion correlations, but so can alternative explanations.

Background

Relativistic heavy-ion collisions smash two nuclei (gold at the Relativistic Heavy Ion Collider, RHIC, at Brookhaven; lead at the Large Hadron Collider, LHC, at CERN) at high energy to create hot, dense strongly interacting matter. Because collisions sample a broad range of nuclear overlap, every measurement is binned in centrality, and the standard geometric characterization uses Glauber-type models: high-energy nucleons traverse nuclei along nearly straight eikonal trajectories, so one can count the participating or 'wounded' nucleons $N_{\text{part}}$ (those struck at least once; the basis of the wounded nucleon model of Bialas, Bleszynski and Czyz) and the number of binary nucleon-nucleon collisions $N_{\text{coll}}$. The wounded nucleon model assumes each struck nucleon radiates a fixed share of the multiplicity independent of how many times it is hit; it undershoots the multiplicity growth in central collisions, and experiments therefore fit a two-component mixture for the charged-particle pseudorapidity density, $dN/d\eta = n_{pp}[(1-x)N_{\text{part}} + xN_{\text{coll}}]$ (here $\eta$ is the standard polar-angle variable of high-energy experiments, $n_{pp}$ the average yield per inelastic proton-proton collision, and the mixing fraction $x$ is taken from data). In that lecture, Glauber proposed a smooth alternative he jokingly called the 'used nucleon model': the first inelastic collision of a nucleon produces one unit of whatever is being counted, the second a fraction $\mu$, the third $\mu^2$, and so on, so that after $k$ collisions the yield is the geometrical series $1+\mu+\dots+\mu^{k-1}$. Averaging over the collision-number distribution reproduces the wounded nucleon model at $\mu=0$ and approaches collision counting as $\mu\to 1$, leaving $\mu$ a free parameter associated with 'whatever sort of process interests you'.

Interferometry with the emitted pions is the second ingredient. Hanbury Brown-Twiss (HBT) interferometry, originally introduced to measure stellar angular diameters, exploits the Bose-Einstein symmetrization of the wave function of two identical pions: the normalized two-pion correlation function $C_2(Q)$, with $Q$ the relative momentum, rises as $Q\to 0$ and reaches 2 for a fully chaotic (thermal) source whose emission events are mutually uncorrelated; the measurement of source space-time properties through these correlations is called femtoscopy. The extrapolated intercept is conventionally written $C_2(0)=1+\lambda$, and $\lambda<1$ has several standard explanations: dilution of primordial ('core') pions by long-lived resonance decay products (the 'halo': long-lived neutral mesons such as the $\eta$, $\eta'$ and $K_S$ whose decays feed the pion sample), instrumental momentum resolution, particle misidentification, multi-body Coulomb repulsion, and partial coherence of the source. Genuine multi-pion correlations - the three- and four-pion cumulants, from which all lower-order contributions have been subtracted - obey known factorial-moment relations for a chaotic source and are suppressed by a coherent component, which makes them the canonical probe for coherence.

Coherence itself comes from quantum optics, a field founded in large part on Glauber's own work. A coherent state is an eigenstate of the annihilation operator; it is radiated by a prescribed classical current, exhibits Poisson counting statistics, and has correlation functions of all orders that factorize, so a pure coherent field shows no HBT enhancement at all, while chaotic (thermal) radiation shows maximal bunching. A coherent amplitude superposed on chaotic noise gives a displaced chaotic field whose counting statistics interpolate between the two; even a small admixture of noise strongly deforms a Poisson distribution. Glauber's transfer of this picture to heavy ions is elementary but suggestive: the ground state of a meson field coupled to a fixed source density is a coherent state, and if the source is suddenly switched off or replaced by a random one, the coherent excitation is set free with random excitations superposed on it - a state from which, in his words, 'the coherent excitation remains, and you can't get rid of it'. The coherent component could be swamped by the noise of an energetic collision, but it could be one reason 'why the measured correlation functions for like pions do not rise to the value 2 for vanishing relative pion momenta'. The lecture offered this mechanism only qualitatively: no estimate of the coherent fraction, no calculation of its effect on HBT correlation functions, no experimental strategy for separating it from resonance decays or Coulomb effects, and no connection between the nucleon-collision geometry of the first part and the coherence discussion of the second.

Problem Statement

For pion production in high-energy nucleus-nucleus collisions (and the proton-proton (pp) and proton-nucleus (p-A) systems measured alongside them), determine quantitatively whether the emitted pion field contains a coherent component, specifically: (1) the magnitude of the coherent fraction - the fraction of the emitted pion intensity described by a coherent amplitude rather than chaotic (thermal) radiation - with its uncertainty, and its dependence on centrality, charged-particle multiplicity and system size; (2) the quantitative effect of that coherent component on the like-pion Bose-Einstein observables, namely the two-pion correlation function (its intercept $\lambda$ and its dependence on pair momentum) and the genuine three- and four-pion correlation cumulants; and (3) an experimental discrimination that establishes the presence, or a bounded absence, of coherence and separates it from the competing mechanisms that produce the same qualitative signatures: long-lived resonance decays (core-halo dilution), multi-body Coulomb final-state interactions, momentum resolution and misidentification, and incomplete thermalization. All three parts are required and jointly constitute the determination; a partial answer to any one part alone does not resolve the task. A quantitative determination must be simultaneously consistent with the measured single-pion spectra and the two- and multi-pion correlation data. The determination of the multiplicity-interpolation parameter $\mu$ for integrated yields (Glauber's 'used nucleon model') was already carried out for RHIC midrapidity data and is not part of the surviving open core.

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