CMSHIN21011 ; CERNEP2023094  
Twoparticle BoseEinstein correlations and their Lévy parameters in PbPb collisions at $ \sqrt{\smash[b]{s_{_{\mathrm{NN}}}}} = $ 5.02 TeV  
CMS Collaboration  
20 June 2023  
Phys. Rev. C 109 (2024) 024914  
Abstract: Twoparticle BoseEinstein momentum correlation functions are studied for chargedhadron pairs in leadlead collisions at a centerofmass energy per nucleon pair of $ \sqrt{\smash[b]{s_{_{\mathrm{NN}}}}} = $ 5.02 TeV. The data sample, containing 4.27 10$^{9}$ minimum bias events corresponding to an integrated luminosity of 0.607 nb$^{1}$, was collected by the CMS experiment in 2018. The experimental results are discussed in terms of a Lévytype source distribution. The parameters of this distribution are extracted as functions of particle pair average transverse mass and collision centrality. These parameters include the Lévy index or shape parameter ($ \alpha $), the Lévy scale parameter ($ R $), and the correlation strength parameter ($ \lambda $). The source shape, characterized by $ \alpha $, is found to be neither Cauchy nor Gaussian, implying the need for a full Lévy analysis. Similarly to what was previously found for systems characterized by Gaussian source radii, a hydrodynamical scaling is observed for the Lévy $ R $ parameter. The $ \lambda $ parameter is studied in terms of the corehalo model.  
Links: eprint arXiv:2306.11574 [hepex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; 
Figures  
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Figure 1:
An example of a long range background fit to the correlation function $ C_2(q) $ of negatively charged hadron pairs with 1.00 $ < K_{\mathrm{T}} < $ 1.05 GeV/$c$ in the 1020% centrality bin. The BoseEinstein peak is below $ q=$ 0.2 GeV/$c$, therefore the 0.2 $ < q < $ 8.0 GeV/$c$ region was used for the long range background fit. 
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Figure 2:
An example fit to the doubleratio correlation function $ DR(q) $ of negatively charged hadron pairs with 1.30 $ < K_{\mathrm{T}} < $ 1.35 GeV/$c$ in the 2030% centrality bin. The error bars show the statistical uncertainties. The fitted function is shown in blue, while the red overlay indicates the range used for the fit. The size of the Coulomb correction is indicated in magenta. The lower panel shows the deviation of the fit from the data in each bin in units of the standard deviation in that bin. 
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Figure 3:
The twoparticle correlation function of negatively charged hadron pairs with 0.9 $ < K_{\mathrm{T}} < $ 1.3 GeV/$c$ in the 0100% centrality range, calculated using Monte Carlo events with (Reco) and without (Gen) detector reconstruction effects. The error bars show the statistical uncertainties. The detector effects are most significant below approximately 50 MeVc. The quantumstatistical effects are not present in the simulations, hence there is no BoseEinstein peak. 
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Figure 4:
The Lévy scale parameter $ R $ versus the transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. 
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Figure 5:
The 1/$ R^2 $ distribution vs. transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. A linear fit to the data is shown for each centrality bin. The fit parameters are tabulated in Table 3. 
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Figure 6:
The slope $ A $ (left) and the intercept $ B $ (right) linearfit parameters versus $ \langle N_{\text{part}}\rangle $, for positively and negatively charged hadron pairs. The error bars show the statistical uncertainties. The correlated systematic uncertainty is also indicated. The points are slightly shifted along the horizontal axes for better visibility. 
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Figure 6a:
The slope $ A $ linearfit parameter versus $ \langle N_{\text{part}}\rangle $, for positively and negatively charged hadron pairs. The error bars show the statistical uncertainties. The correlated systematic uncertainty is also indicated. The points are slightly shifted along the horizontal axes for better visibility. 
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Figure 6b:
The intercept $ B $ linearfit parameter versus $ \langle N_{\text{part}}\rangle $, for positively and negatively charged hadron pairs. The error bars show the statistical uncertainties. The correlated systematic uncertainty is also indicated. The points are slightly shifted along the horizontal axes for better visibility. 
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Figure 7:
The Lévy scale parameter $ R $ versus $ \langle N_{\text{part}}\rangle^{1/3} $ in different $ m_{\mathrm{T}} $ classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. The correlated systematic uncertainty is also indicated. A linear fit to the data is shown for each $ m_{\mathrm{T}} $ class. The fit parameters are tabulated in Table 4. 
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Figure 8:
The Lévy stability index $ \alpha $ versus the transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. 
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Figure 9:
The average Lévy stability index $ \langle \alpha \rangle $ versus $ \langle N_{\text{part}}\rangle $, for both positively and negatively charged hadron pairs. The error bars show the statistical uncertainties. The correlated systematic uncertainty is also indicated. The points are slightly shifted along the horizontal axes for better visibility. 
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Figure 10:
The correlation strength $ \lambda $ (upper panel) and $ \lambda^* $, which is rescaled with the square of the pion fraction (lower panel), versus the transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. 
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Figure 10a:
The correlation strength $ \lambda $ versus the transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. 
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Figure 10b:
$ \lambda^* $, which is rescaled with the square of the pion fraction, versus the transverse mass $ m_{\mathrm{T}} $ in different centrality classes, for negatively (left) and positively (right) charged hadron pairs. The error bars show the statistical uncertainties, while the boxes indicate the pointtopoint systematic uncertainties. These boxes are slightly shifted along the horizontal axes for better visibility. The correlated systematic uncertainty is also indicated. 
Tables  
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Table 1:
The sources of the systematic uncertainties with their values in the default and lower/upper settings. The meaning of the analysis parameters are given in Sections 4.1 and 4.2. 
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Table 2:
The relative effect of the different types of systematic sources in each centrality class for the $ R $ (upper Table), $ \alpha $ (middle Table) and $ \lambda $ (lower Table) parameters (in percentage). The values were averaged over $ K_{\mathrm{T}} $ and the two charge signs; the upwards (downwards) arrow, $ \uparrow $ ($ \downarrow $) represents positive (negative) uncertainty in the value of the final fit parameters. 
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Table 3:
Fit parameters and the corresponding confidence levels (CL) of the linear fits to 1/$ R^2 $ versus $ m_{\mathrm{T}} $, for positively (left) and negatively (right) charged hadron pairs. 
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Table 4:
Fit parameters and the corresponding confidence levels (CL) of the linear fits to $ R $ versus $ \langle N_{\text{part}}\rangle^{1/3} $, for positively (left) and negatively (right) charged hadron pairs. 
Summary 
Twoparticle BoseEinstein momentum correlation function measurements are presented. The data sample consists of 4.27 10$^{9}$ minimum bias leadlead (PbPb) events, corresponding to an integrated luminosity of 0.607 nb$^{1}$, at a centerofmass energy per nucleon pair of $ \sqrt{\smash[b]{s_{_{\mathrm{NN}}}}} = $ 5.02 TeV, recorded by the CMS experiment at the LHC. The correlation functions found in different centrality and average transverse momentum classes are analyzed in terms of Lévy sources, including Coulomb effects. The values of the Lévy scale parameter $ R $, the Lévy stability index $ \alpha $, and the correlation strength $ \lambda $ are determined. A geometric interpretation of the $ R $ parameter is suggested by its dependence on the average number of participating nucleons in the collision. Assuming a pion mass for the charged particles, a linear dependence of 1/$ R^2 $ on the transverse mass ($ m_{\mathrm{T}} $) is observed, consistent with a hydrodynamic scaling behavior, even for the case of Lévy sources. Based on the observed linear behavior, it is estimated that the Hubble constant of the quarkgluon plasma created in 5.02 TeV PbPb collisions increases from 0.11\ensuremath{\unit $ c $/fm}} to 0.18\ensuremath{\unit $ c $/fm}} when moving from most central to most peripheral collisions. The intercept of the 1/$ R^2 $ versus $ m_{\mathrm{T}} $ linear fits is negative in all cases, requiring further studies for its interpretation. The $ \alpha $ parameter is found to have little, if any, $ m_{\mathrm{T}} $ dependence and to range between 1.62.0, increasing with centrality. The $ \alpha $ values found in this paper are approximately 4560% larger than those reported for goldgold collisions at $ \sqrt{\smash[b]{s_{_{\mathrm{NN}}}}} = $ 200 GeV at RHIC. This increase, while not fully understood, may result from the greater energy densities achieved at the LHC. As a function of $ m_{\mathrm{T}} $, a strong and decreasing trend is observed for the $ \lambda $ parameter, which can be explained by the lack of particle identification. After rescaling the $ \lambda $ values to account for the fraction of pions among the charged hadrons, a nearly constant trend of the pionfractioncorrected $ \lambda^* $ with $ m_{\mathrm{T}} $ is observed. The $ \lambda^* $ values are found to be smaller than unity, which can be interpreted on the basis of the corehalo model as a nonnegligible halo contribution. Furthermore, $ \lambda^* $ is found to decrease as the collisions become more central. Altogether, these results imply that the hadron emitting source in $ \sqrt{\smash[b]{s_{_{\mathrm{NN}}}}} = $ 5.02 TeV PbPb collisions can be described by Lévy distributions. This allows for a new and precise characterization of this source in highenergy heavy ion collisions. 
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Compact Muon Solenoid LHC, CERN 