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| CMS-PAS-HIN-21-011 | ||
| Measurement of two-particle Bose-Einstein momentum correlations and their Lévy parameters at ${\sqrt {\smash [b]{s_{_{\mathrm {NN}}}}}}=$ 5.02 TeV PbPb collisions | ||
| CMS Collaboration | ||
| April 2022 | ||
| Abstract: Charged hadron, two-particle Bose-Einstein momentum correlation functions in lead-lead (PbPb) collisions at an average center of mass energy per nucleon (${\sqrt {\smash [b]{s_{_{\mathrm {NN}}}}}}$) of 5.02 TeV are presented. The data were obtained in 2018 by the CMS experiment at the LHC. The experimental results are discussed in terms of theoretically derived correlation functions for Levy type source distributions. Using average pair transverse momentum and centrality binning, the source distribution parameters are extracted as a function of pair transverse mass $m_{\mathrm{T}}$ and collision centrality. These parameters include the correlation strength parameter $\lambda$, the Levy index or shape parameter $\alpha$, and the Levy scale parameter $R$. We find that the source shape, characterized by $\alpha$, is neither Cauchy nor Gaussian, implying the need for a full Levy analysis. A hydrodynamical scaling, previously shown to work for Gaussian radii, is also found to hold for $R$. | ||
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Links:
CDS record (PDF) ;
CADI line (restricted) ;
These preliminary results are superseded in this paper, Submitted to PRC. The superseded preliminary plots can be found here. |
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| Figures | |
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Figure 1:
An example fit to the double-ratio correlation function $DR(q)$. The fitted function is black, while the red overlay indicates the range used for the fit. The $K_{\mathrm {T}}$ and centrality class is visible in the legend. |
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Figure 2:
The Lévy scale parameter $R$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 2-a:
The Lévy scale parameter $R$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 2-b:
The Lévy scale parameter $R$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 3:
$1/R^2$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each centrality. The fit results are in Table 5. |
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Figure 3-a:
$1/R^2$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each centrality. The fit results are in Table 5. |
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Figure 3-b:
$1/R^2$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each centrality. The fit results are in Table 5. |
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Figure 4:
The two fit parameters from the linear fit: the slope $A$ (left) and the intercept $B$ (right) versus $\langle N_{\text {part}}\rangle$ for negative and positive hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the systematic uncertainties which originate from the correlated systematic uncertainty of $R$. |
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Figure 4-a:
The two fit parameters from the linear fit: the slope $A$ (left) and the intercept $B$ (right) versus $\langle N_{\text {part}}\rangle$ for negative and positive hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the systematic uncertainties which originate from the correlated systematic uncertainty of $R$. |
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Figure 4-b:
The two fit parameters from the linear fit: the slope $A$ (left) and the intercept $B$ (right) versus $\langle N_{\text {part}}\rangle$ for negative and positive hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the systematic uncertainties which originate from the correlated systematic uncertainty of $R$. |
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Figure 5:
The Lévy scale parameter $R$ versus $\langle N_{\text {part}}\rangle^{1/3}$ in different $m_{\mathrm {T}}$ classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each $m_{\mathrm {T}}$ class. The fit results are in Table 6. |
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Figure 5-a:
The Lévy scale parameter $R$ versus $\langle N_{\text {part}}\rangle^{1/3}$ in different $m_{\mathrm {T}}$ classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each $m_{\mathrm {T}}$ class. The fit results are in Table 6. |
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Figure 5-b:
The Lévy scale parameter $R$ versus $\langle N_{\text {part}}\rangle^{1/3}$ in different $m_{\mathrm {T}}$ classes for positive (top) and negative (bottom) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. We fitted a line to the data for each $m_{\mathrm {T}}$ class. The fit results are in Table 6. |
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Figure 6:
The Lévy stability index $\alpha $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 6-a:
The Lévy stability index $\alpha $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 6-b:
The Lévy stability index $\alpha $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 7:
The average Lévy stability index $\langle \alpha \rangle$ versus $\langle N_{\text {part}}\rangle$ for both positive and negative hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the systematic uncertainties which originate from the correlated systematic uncertaity of $\alpha $. |
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Figure 8:
The correlation strength $\lambda $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 8-a:
The correlation strength $\lambda $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 8-b:
The correlation strength $\lambda $ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 9:
The correlation strength re-scaled with the square of the pion ratio $\lambda ^*$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 9-a:
The correlation strength re-scaled with the square of the pion ratio $\lambda ^*$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. The correlated systematic uncertainty is also indicated. |
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Figure 9-b:
The correlation strength re-scaled with the square of the pion ratio $\lambda ^*$ versus the transverse mass $m_{\mathrm {T}}$ in different centrality classes for positive (left) and negative (right) hadron pairs. The error bars are the statistical uncertainties, while the boxes indicate the point-to-point systematic uncertainties. 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 other two settings. The meaning of the analysis parameters are given in Subsections 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 $R$ parameter (in percentage). We averaged over $K_{\mathrm {T}}$ and the two charge setups. $\uparrow$ ($\downarrow$) represents positive (negative) modification in the value of the parameter. |
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Table 3:
The relative effect of the different types of systematic sources in each centrality class for the $\alpha $ parameter (in percentage). We averaged over $K_{\mathrm {T}}$ and the two charge setups. $\uparrow$ ($\downarrow$) represents positive (negative) modification in the value of the parameter. |
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Table 4:
The relative effect of the different types of systematic sources in each centrality class for the $\lambda $ parameter (in percentage). We averaged over $K_{\mathrm {T}}$ and the two charge setups. $\uparrow$ ($\downarrow$) represents positive (negative) modification in the value of the parameter. |
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Table 5:
The fit parameters and the confidence levels of the linear fits to $1/R^2$ versus $m_{\mathrm {T}}$ for positive (left) and negative (right) hadron pairs. |
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Table 6:
The fit parameters and the confidence levels of the linear fits to $R$ versus $\langle N_{\text {part}}\rangle^{1/3}$ for positive (left) and negative (right) hadron pairs. |
| Summary |
|
In this note we presented two-particle Bose-Einstein correlation function measurements based on approximately 2.65$\times $10$^9$ lead-lead (PbPb) events with ${\sqrt {\smash [b]{s_{_{\mathrm {NN}}}}}} = $ 5.02 TeV, recorded by CMS at the LHC. The correlation functions, found in different centrality and average transverse momentum ($K_{\mathrm{T}}$) classes, are analyzed in terms of stable Lévy sources within the core-halo model including Coulomb effects. The values of the Lévy scale parameter $R$, the Lévy stability index $\alpha$, correlation strength $\lambda$ are determined. The geometrical interpretation of the $R$ parameter is investigated by determining its dependence on the average number of participating nucleons in the collision ($\langle N_{\text{part}}\rangle$). A linear dependence of $1/R^2$ with the transverse mass ($m_{\mathrm{T}}$) assuming a pion mass for the charged particles is observed, consistent with having a hydrodynamic scaling even in the case of Lévy sources. Based on the observed linear behavior, it is estimated that the Hubble constant of the QGP created in 5.02 TeV PbPb collisions is between 0.12 c/fm and 0.18 c/fm. The $1/R^2$ intercept of the linear fits is negative in all cases. This requires further phenomenological studies. We found the $\alpha$ parameter to have little, if any, $m_{\mathrm{T}}$ dependence and to range between 1.6 and 2, increasing with centrality. We also measured a strong and decreasing trend for the $\lambda$ parameter as a function of $m_{\mathrm{T}}$. The $m_{\mathrm{T}}$ dependence is explained in terms of the lack of particle identification. After rescaling the $\lambda$ values to account for the pion fraction in each analysis range, a nearly constant trend of the mass fraction corrected $\lambda^*$ value with $m_{\mathrm{T}}$ is observed. The $\lambda^*$ values are 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 in a statistically acceptable manner, making the above discussed interpretation of the Lévy source parameters possible. |
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