Bose-Einstein Correlation Functions Research Articles

Principles and prospects of Bose-Einstein correlation study at BESIII* *This study was supported by the National Key R&D Program of China (2020YFA0406403) and the National Natural Science Foundation of China (11275211, 11335008, 12035013)

One method for determining the characteristic parameters of a hadron production source is to measure the Bose-Einstein correlation functions. In this study, we present fundamental concepts and formulas related to the Bose-Einstein correlations, focusing on the measurement principles and the Lund model from an experimental perspective. We perform Monte Carlo simulations using the Lund model generator in the 2–3 GeV energy range. Through these feasibility studies, we identify key features of the Bose-Einstein correlations that offer valuable insights for experimental measurements. Utilizing data samples collected at BESIII, we perform measurements of the Bose-Einstein correlation functions, with an expected experimental precision of a few percent for the hadron source radius and incoherence parameter.

A novel method for calculating Bose–Einstein correlation functions with Coulomb final-state interaction

Measurements of Bose–Einstein correlations played a crucial role in the discovery and the subsequent detailed exploration of the Quark-Gluon-Plasma (QGP) created in high-energy collisions of heavy nuclei. Such measurements gave rise to femtoscopy, a flourishing sub-field of high-energy physics, and there are important new directions to explore and discoveries to be made in the near future. One of these important current topics is the precise investigation of the shape of the correlation functions utilizing Lévy-stable sources. In this paper, we utilize Lebesgue’s dominated convergence theorem and Fubini’s theorem to present a novel method of calculating the shape of the two-particle correlation functions, including the Coulomb final-state interaction. This method relies on an exact calculation of a large part of the necessary integrals of the Coulomb wave function and can be utilized to calculate the correlation function for any source function with an easily accessible Fourier transform. In this way, it is eminently applicable to Lévy-stable source functions. In this particular case, we find that the new method is more robust and allows the investigation of a wider parameter range than the previously utilized techniques. We present an easily applicable software package that is ready to use in experimental studies.

Centrality Dependent Lévy Analysis of two-pion BEC Functions at PHENIX

We present most recent PHENIX preliminary data on centrality dependence of two-pion Bose-Einstein correlation functions measured in √SNN = 200 GeV Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC). The data are well described by assuming the source to be a Lévy-stable distribution. The Lévy parameters, R, α, λ are measured in 18 bins of transverse mass (mT) for 4 centrality intervals. The Lévy scale parameter R(mT) decreases with mT and exhibits a clear centrality ordering which supports its geometrical interpretation. The Lévy exponent α(mT) is independent of mT in every centrality bin but shows some centrality dependence. At all centralities α is significantly different from that of a Gaussian (α = 2) or Cauchy (α = 1) source distribution. The Lévy strength parameter λ was also investigated. We observed that λ(mT) decreases at low mT and saturates around mT = 0.6 GeV/c2. In the already published 0-30% centrality class, Monte Carlo simulations are found to be inconsistent with the measurements, unless a significant reduction of the in-medium mass of the η′ meson is included.

Two-pion Bose-Einstein Correlations data analysis for the CMS Pb-Pb nucleon collisions

From the data collected in the Compact Muon Solenoid (CMS) Pb-Pb nucleon collisions experiment, the two-pion Bose-Einstein correlation functions for different combination of same charges and different charges are plotted. The influence of repulsion and attraction through Coulomb interaction between charged pions is reduced after applying the standard Gamow-factor Coulomb correction on Gaussian function C(Qinv). According to the Yano-koonin-Podgoretski parametrization, the five-dimensional components of the invariant momentum difference between two pions are calculated. One of the five components, the momentum difference in the transverse plane QT, can be further divided into Qside and Qout. Q0, Qlong, Qside and Qout were then separately plotted and fitted with the Gaussian function. The sizes of pion source or the effective interferometric source can be extracted from the fitting finally.

L\'evy-stable Two-pion Bose--Einstein Correlation Functions Measured with PHENIX in $\sqrt {s_{NN}} =$ 200 GeV Au+Au Collisions

Measurement of quantumstatistical correlation functions in high energy nuclear physics is an important tool to investigate the QCD phase diagram. It may be used to search for the critical point, and also to understand underlying processes such as in-medium mass modifications or partially coherent particle production. Furthermore, the measurements of the femtoscopic correlation functions shed light on the space-time structure of particle production. Consequently, the precise measurements and description of the correlation functions are essential. The shape of two-pion Bose-Einstein correlation functions were often assumed to be Gaussian, but the recent precision of the experiments reveals that the statistically correct assumption is the more general Levy-distribution. In this paper we present the recent results of the measurements of two-pion Levy-stable Bose-Einstein correlation functions in Au+Au collisions at PHENIX

A New and Finite Family of Solutions of Hydrodynamics: Part III: Advanced Estimate of the Life-time Parameter

We derive a new formula for the longitudinal HBT-radius of the two particle Bose-Einstein correlation function from a new family of finite and exact, accelerating solution of relativistic perfect fluid hydrodynamics for a temperature independent speed of sound. The new result generalizes the Makhlin-Sinyukov and Herrmann-Bertsch formulae and leads to an advanced life-time estimate of high energy heavy ion and proton-proton collisions.

Fluctuations of collective flows for event-by-event hydrodynamic evolution sources

Through investigating the space-time evolution of the hydrodynamic particle-emitting sources with the fluctuating initial conditions generated by Heavy Ion Jet Interaction Generator (HIJING), we discussed the space-time and velocity evolution of quark gluon plasma (QGP). In order to detect the event-by-event inhomogeneity of the sources, we examine the distribution of the error-inverse-weighted fluctuations, between the two-pion Bose-Einstein correlation functions of single events. We find that the distribution becomes wide for the fluctuating initial sources with velocity v = 0, which means the sources evolution faster than nonzero initial conditions. We also discussed the elliptic flow effect, velocity and temperature evolution in Heavy Ion Reactions, and get the same results.

Centrality Dependent Lévy-Stable Two-Pion Bose-Einstein Correlations in \( {\sqrt{s_{NN}}} \) = 200 GeV Au+Au Collisions at the PHENIX Experiment

Investigation of femtoscopic correlation functions in relativistic heavy ion reactions is an important tool to access the space-time structure of particle production in the sQGP. The shape of the correlation functions is often assumed to be Gaussian, but a detailed analysis reveals that the statistically correct assumption could be the so-called L\'evy distribution. The detailed analysis of correlation functions in various systems may shed light on the location of the critical endpoint on the QCD phase diagram. It could also reveal if there is partially coherent pion production or could indicate the possible in-medium mass modification of the $\eta'$ meson due to the (partial) restoration of the $U_A(1)$ axial symmetry. In this paper we present the status of the centrality dependent measurements of two-pion L\'evy Bose-Einstein correlation functions in Au+Au collisions at PHENIX.

Quantum canonical ensemble and correlation femtoscopy at fixed multiplicities

Identical particle correlations at fixed multiplicity are considered by means of quantum canonical ensemble of finite systems. We calculate one-particle momentum spectra and two-particle Bose-Einstein correlation functions in the ideal gas by using a recurrence relation for the partition function. Within such a model we investigate the validity of the thermal Wick's theorem and its applicability for decomposition of the two-particle distribution function. The dependence of the Bose-Einstein correlation parameters on the average momentum of the particle pair is also investigated. Specifically, we present the analytical formulas that allow one to estimate the effect of suppressing the correlation functions in a finite canonical system. The results can be used for the femtoscopy analysis of the A+A and p+p collisions with selected (fixed) multiplicity.

Observables and initial conditions for rotating and expanding fireballs with spheroidal symmetry

Utilizing a recently found class of exact, analytic rotating solutions of non-relativistic fireball hydrodynamics, we calculate analytically the single-particle spectra, the elliptic flows and two-particle Bose-Einstein correlation functions for rotating and expanding fireballs with spheroidal symmetry. We demonstrate, that rotation generates final state momentum anisotropies even for a spatially symmetric, spherical initial geometry of the fireball. The mass dependence of the effective temperatures, as well as the HBT radius parameters and the elliptic flow are shown to be sensitive not only to radial flow effects but also to the magnitude of the initial angular momentum.

Model-independent Analysis of Nearly L\'evy Correlations

A model-independent method for the analysis of the two-particle short-range correlations is presented, that can be utilized to describe e.g. Bose-Einstein (HBT), dynamical (ridge) or other correlation functions, that have a nearly L\'evy or streched exponential shape. For the special case of L\'evy exponent alpha = 1, the earlier Laguerre expansions are recovered, for the alpha = 2 special case, a new expansion method is obtained for nearly Gaussian correlation functions. Multi-dimensional L\'evy expansions are also introduced and their potential application to analyze rigde correlation data is discussed.

Composite nature of hadrons and Bose-Einstein correlations

I am reporting results of two papers, written together with W.Florkowski and K.Zalewski [1, 2], discussing the consequences of the observation [3] that, due to their composite nature and thus finite size, hadrons observed in the HBT measurements must be correlated in space-time. Using the blast-wave model [4] adjusted [1] to ALICE data on the measured HBT radii in pp collisions at 7 TeV [5], the full Bose-Einstein correlation functions in three direction (out, side, long) are evaluated. The results are presented together with some additional comments.

Multiplicity, Jet, and Transverse Mass dependence of Bose-Einstein Correlations in e+e−- Annihilation

Bose-Einstein correlations of pairs of identical charged pions produced in hadronic Z decays are analyzed for both two- and three-jet events. A parametrization suggested by the tau-model is used to investigate the dependence of the Bose-Einstein correlation function on track multiplicity, number of jets, and transverse momentum.

Two-photon correlation in anisotropic quark-gluon plasma

The only way to obtain the space-time structure of heavy ion collision is through the study of two-particle momentum correlations. Thus we have studied the intensity correlation for the photons having fixed transverse momentum of one of the photons ($k_{1T}$= 2 GeV) in anisotropic Quark Gluon Plasma (aQGP) to have an idea about emission zone in presence of initial momentum anisotropy. The {\em free streaming interpolating} model with fixed initial condition has been used for the space-time evolution for most central collision at RHIC energy. The variation of Bose-Einstein correlation function (BECF), $C_2$, for two identical photons as a function of $q_{out}$, $q_{side}$ and $q_{long}$ is evaluated. We have restricted our analysis only to QGP phase to know the effect of anisotropy on the correlation function and HBT radii extracted. It is observed that the longitudinal dimension of the reaction zone is mostly affected due to the presence of momentum space anisotropy.

Correlation femtoscopy of small systems

The basic principles of the correlation femtoscopy, including its correspondence to the Hanbury Brown and Twiss intensity interferometry, are reexamined. The main subject of the paper is an analysis of the correlation femtoscopy when the source size is as small as the order of the uncertainty limit. It is about 1 fm for the current high energy experiments. Then the standard femtoscopy model of random sources is inapplicable. The uncertainty principle leads to the partial indistinguishability and coherence of closely located emitters that affect the observed femtoscopy scales. In thermal systems the role of corresponding coherent length is taken by the thermal de Broglie wavelength that also defines the size of a single emitter. The formalism of partially coherent phases in the amplitudes of closely located individual emitters is used for the quantitative analysis. The general approach is illustrated analytically for the case of the Gaussian approximation for emitting sources. A reduction of the interferometry radii and a suppression of the Bose-Einstein correlation functions for small sources due to the uncertainty principle are found. There is a positive correlation between the source size and the intercept of the correlation function. The peculiarities of the nonfemtoscopic correlations caused by minijets and fluctuations of the initial states of the systems formed in $pp$ and ${e}^{+}{e}^{\ensuremath{-}}$ collisions are also analyzed. The factorization property for the contributions of femtoscopic and nonfemtoscopic correlations into complete correlation function is observed in numerical calculations in a wide range of the model parameters.

Femtoscopic and Nonfemtoscopic Two-Particle Correlations inA+Aandp+pCollisions at RHIC and LHC Energies

The theoretical review of the last femtoscopy results for the systems created in ultrarelativisticA+A,p+p, andp+Pbcollisions is presented. The basic model, allowing to describe the interferometry data at SPS, RHIC, and LHC, is the hydrokinetic model. The model allows one to avoid the principal problem of the particlization of the medium at nonspace-like sites of transition hypersurfaces and switch to hadronic cascade at a space-like hypersurface with nonequilibrated particle input. The results for pion and kaon interferometry scales inPb+PbandAu+Aucollisions at LHC and RHIC are presented for different centralities. The new theoretical results as for the femtoscopy of small sources with sizes of 1-2 fm or less are discussed. The uncertainty principle destroys the standard approach of completely chaotic sources: the emitters in such sources cannot radiate independently and incoherently. As a result, the observed femtoscopy scales are reduced, and the Bose-Einstein correlation function is suppressed. The results are applied for the femtoscopy analysis ofp+pcollisions ats=7 TeV LHC energy andp+Pbones ats=5.02 TeV. The behavior of the corresponding interferometry volumes on multiplicity is compared with what is happening for centralA+Acollisions. In addition the nonfemtoscopic two-pion correlations in proton-proton collisions at the LHC energies are considered, and a simple model that takes into account correlations induced by the conservation laws and minijets is analyzed.

In-medium reduction of theη'mass √sNN= 200 GeV Au+Au collisions

A reduction of the mass of the \eta'(958) meson may indicate the restoration of the UA(1) symmetry in a hot and dense hadronic matter, corresponding to the return of the 9th, "prodigal" Goldstone boson. We report on an analysis of a combined PHENIX and STAR data set on the intercept parameter of the two-pion Bose-Einstein correlation functions, as measuremed in \sqrt{s_NN} = 200 GeV Au+Au collisions at RHIC. To describe this combined PHENIX and STAR dataset, an in-medium \eta' mass reduction of at least 200 MeV is needed, at the 99.9 % confidence level in a broad model class of resonance multiplicities. Energy, system size and centrality dependence of the observed effect is also discussed.

Indirect Observation of an In-Mediumη′Mass Reduction insNN=200 GeVAu+AuCollisions

PHENIX and STAR data on the intercept parameter of the two-pion Bose-Einstein correlation functions in sqrt(s_{NN}) = 200 GeV Au+Au collisions are analyzed in terms of different models of hadronic multiplicities. To describe this combined PHENIX and STAR dataset, an in-medium eta' mass reduction of at least 200 MeV is needed, at the 99.9 % confidence level in the considered model class. Such a significant eta' mass modification may indicate the restoration of the U_A(1) symmetry in a hot and dense hadronic matter and the return of the 9th "prodigal" Goldstone boson.

Significant in-medium [formula omitted] mass reduction in [formula omitted] Au+Au collisions

PHENIX and STAR data on the intercept parameter of the two-pion Bose-Einstein correlation functions in s N N = 200 GeV Au+Au collisions were analysed in terms of various models of hadronic abundances. To describe these data, an in-medium η ′ mass decrease of at least 200 MeV was needed in these models.

Bose-Einstein correlations of light hadrons and the stochastic scale of particle emitter source

Based on quantum field theory at finite temperature, we obtained new results for two-particle Bose-Einstein correlation (BEC) function C2(Q) in case of light hadrons. The important parameters of BEC function related to the size of emitting source, mean multiplicity, stochastic forces range and correlation radius with the particle energy and the mass dependence, and the temperature of the source are obtained in analytical form for the first time. Not only is the correlation between identical bosons explored but also the off-correlation between non-identical particles is proposed.