CMS logoCMS event Hgg
Compact Muon Solenoid
LHC, CERN

CMS-PAS-FSQ-15-009
Femtoscopic Bose-Einstein correlations of charged hadrons in proton-proton collisions at $\sqrt{s}= $ 13 TeV
Abstract: Femtoscopic correlations between charged hadrons are measured over a broad multiplicity range, from a few particles up to about 250 reconstructed charged hadrons, in proton-proton collisions at $\sqrt{s}= $ 13 TeV. The results are based on data collected by the CMS detector at the CERN LHC. Three methods with different dependencies on simulations, when extracting and fitting the correlation functions, are compared and are found to give consistent results. The measured lengths of homogeneity are studied as functions of particle multiplicity and pair transverse mass. The results are compared with those from lower energies and other experiments, as well as with theoretical predictions.
Figures & Tables Summary References CMS Publications
Figures

png pdf
Figure 1:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-a:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-b:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-c:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-d:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-e:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 1-f:
Same-sign and opposite-sign SR employing PYTHIA6 (Z2*) in different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ with the respective Gaussian fit from Equation (4). The following $ {q_\text {inv}} $ ranges are excluded from the fits: 0.2 $ < {q_\text {inv}} < $ 0.3 GeV, 0.4 $ < {q_\text {inv}} < $ 0.9 GeV and 0.95 $ < {q_\text {inv}} < $ 1.2 GeV. Coulomb interactions are not included in the simulation.

png pdf
Figure 2:
Relations between same- and opposite-sign fit parameters from Eq. (4), as a function of $ {k_{\mathrm {T}}} $ and $N^\text {offline}_\text {track}$ for events in MB (i.e., higher $(\sigma _{B})^{-1}$ and lower $\text {Log}(B)$) and HM (i.e., lower $(\sigma _{B})^{-1}$ and higher $\text {Log}(B)$) ranges. For a given $ {k_{\mathrm {T}}} $ range, each point represents an $N^\text {offline}_\text {track}$ bin. The line in the left plot is the fit to all the data.

png pdf
Figure 2-a:
Relations between same- and opposite-sign fit parameters from Eq. (4), as a function of $ {k_{\mathrm {T}}} $ and $N^\text {offline}_\text {track}$ for events in MB (i.e., higher $(\sigma _{B})^{-1}$ and lower $\text {Log}(B)$) and HM (i.e., lower $(\sigma _{B})^{-1}$ and higher $\text {Log}(B)$) ranges. For a given $ {k_{\mathrm {T}}} $ range, each point represents an $N^\text {offline}_\text {track}$ bin. The line in the left plot is the fit to all the data.

png pdf
Figure 2-b:
Relations between same- and opposite-sign fit parameters from Eq. (4), as a function of $ {k_{\mathrm {T}}} $ and $N^\text {offline}_\text {track}$ for events in MB (i.e., higher $(\sigma _{B})^{-1}$ and lower $\text {Log}(B)$) and HM (i.e., lower $(\sigma _{B})^{-1}$ and higher $\text {Log}(B)$) ranges. For a given $ {k_{\mathrm {T}}} $ range, each point represents an $N^\text {offline}_\text {track}$ bin. The line in the left plot is the fit to all the data.

png pdf
Figure 3:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-a:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-b:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-c:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-d:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-e:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 3-f:
Same-sign and opposite-sign single ratios in data for different bins of $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $, with their respective fits. The label "(Exp. $\times $ Gauss) Fit'' refers to the same-sign data and is given by Eq. (7). The label "Gauss Fit'' corresponds to Eq. (4) applied to opposite-sign data and "Background'' is the component of Eq. (7) that is found from Gauss Fit by using Eqs. (5) and (6) to convert the fit parameters. Coulomb corrections are accounted for using the Gamow factor.

png pdf
Figure 4:
Results for $ {R_\text {inv}} $ (left) and $\lambda $ (right) from the three methods as a function of multiplicity (top) and $ {k_{\mathrm {T}}} $ (bottom). In the top plots, statistical and systematical uncertainties are represented by internal and external error bars, respectively. In the bottom plots, statistical and systematic uncertainties are shown as error bars and open boxes, respectively.

png pdf
Figure 4-a:
Results for $ {R_\text {inv}} $ (left) and $\lambda $ (right) from the three methods as a function of multiplicity (top) and $ {k_{\mathrm {T}}} $ (bottom). In the top plots, statistical and systematical uncertainties are represented by internal and external error bars, respectively. In the bottom plots, statistical and systematic uncertainties are shown as error bars and open boxes, respectively.

png pdf
Figure 4-b:
Results for $ {R_\text {inv}} $ (left) and $\lambda $ (right) from the three methods as a function of multiplicity (top) and $ {k_{\mathrm {T}}} $ (bottom). In the top plots, statistical and systematical uncertainties are represented by internal and external error bars, respectively. In the bottom plots, statistical and systematic uncertainties are shown as error bars and open boxes, respectively.

png pdf
Figure 4-c:
Results for $ {R_\text {inv}} $ (left) and $\lambda $ (right) from the three methods as a function of multiplicity (top) and $ {k_{\mathrm {T}}} $ (bottom). In the top plots, statistical and systematical uncertainties are represented by internal and external error bars, respectively. In the bottom plots, statistical and systematic uncertainties are shown as error bars and open boxes, respectively.

png pdf
Figure 4-d:
Results for $ {R_\text {inv}} $ (left) and $\lambda $ (right) from the three methods as a function of multiplicity (top) and $ {k_{\mathrm {T}}} $ (bottom). In the top plots, statistical and systematical uncertainties are represented by internal and external error bars, respectively. In the bottom plots, statistical and systematic uncertainties are shown as error bars and open boxes, respectively.

png pdf
Figure 5:
Radius fit parameters as a function of particle level multiplicities using the HCS method in pp collisions at 13 TeV compared to results for pp collisions at 7 TeV from CMS (left) and ATLAS (right). Both the ordinate and abscissa for the CMS data in the right plot have been adjusted for compatibility with the ATLAS analysis procedure. See text for details. The error bars in the CMS [4] case represent systematic uncertainties (statistical ones are smaller than the marker size) and in the ATLAS [14] case, statistical and systematic uncertainties added in quadrature.

png pdf
Figure 5-a:
Radius fit parameters as a function of particle level multiplicities using the HCS method in pp collisions at 13 TeV compared to results for pp collisions at 7 TeV from CMS (left) and ATLAS (right). Both the ordinate and abscissa for the CMS data in the right plot have been adjusted for compatibility with the ATLAS analysis procedure. See text for details. The error bars in the CMS [4] case represent systematic uncertainties (statistical ones are smaller than the marker size) and in the ATLAS [14] case, statistical and systematic uncertainties added in quadrature.

png pdf
Figure 5-b:
Radius fit parameters as a function of particle level multiplicities using the HCS method in pp collisions at 13 TeV compared to results for pp collisions at 7 TeV from CMS (left) and ATLAS (right). Both the ordinate and abscissa for the CMS data in the right plot have been adjusted for compatibility with the ATLAS analysis procedure. See text for details. The error bars in the CMS [4] case represent systematic uncertainties (statistical ones are smaller than the marker size) and in the ATLAS [14] case, statistical and systematic uncertainties added in quadrature.

png pdf
Figure 6:
Comparison of $ {R_\text {inv}} $ obtained with the HCS method with theoretical expectations. Left: Values of $ {R_\text {inv}} $ as a function of $ < N_\text {tracks}> ^{1/3}$, with two fit functions: a single linear function and a linear function plus a constant for $N_\text {tracks}^{1/3} > 4.8$. Right: Comparison of $ {R_\text {inv}} $ with the predictions from CGC, as a function of $(dN_\text {tracks}/d\eta)^{1/3}$. The fits for $(dN_\text {tracks}/d\eta)^{1/3} > 1.7$ are performed using Eq. (8), with parameter values in the second row of Table 2. A linear function is fitted for $(dN_\text {tracks}/d\eta)^{1/3} < 1.7$. Statistical uncertainties only are considered in all fits.

png pdf
Figure 6-a:
Comparison of $ {R_\text {inv}} $ obtained with the HCS method with theoretical expectations. Left: Values of $ {R_\text {inv}} $ as a function of $ < N_\text {tracks}> ^{1/3}$, with two fit functions: a single linear function and a linear function plus a constant for $N_\text {tracks}^{1/3} > 4.8$. Right: Comparison of $ {R_\text {inv}} $ with the predictions from CGC, as a function of $(dN_\text {tracks}/d\eta)^{1/3}$. The fits for $(dN_\text {tracks}/d\eta)^{1/3} > 1.7$ are performed using Eq. (8), with parameter values in the second row of Table 2. A linear function is fitted for $(dN_\text {tracks}/d\eta)^{1/3} < 1.7$. Statistical uncertainties only are considered in all fits.

png pdf
Figure 6-b:
Comparison of $ {R_\text {inv}} $ obtained with the HCS method with theoretical expectations. Left: Values of $ {R_\text {inv}} $ as a function of $ < N_\text {tracks}> ^{1/3}$, with two fit functions: a single linear function and a linear function plus a constant for $N_\text {tracks}^{1/3} > 4.8$. Right: Comparison of $ {R_\text {inv}} $ with the predictions from CGC, as a function of $(dN_\text {tracks}/d\eta)^{1/3}$. The fits for $(dN_\text {tracks}/d\eta)^{1/3} > 1.7$ are performed using Eq. (8), with parameter values in the second row of Table 2. A linear function is fitted for $(dN_\text {tracks}/d\eta)^{1/3} < 1.7$. Statistical uncertainties only are considered in all fits.

png pdf
Figure 7:
$1/R_{\mathrm {inv}}^2$ as a function of $m_{\mathrm {T}} = \sqrt {m_{\pi}^2 + < {k_{\mathrm {T}}} > ^2}$ for the HCS method. The MB range corresponds to 0 $ \le N_\text {trk}^\text {offline} \le $ 79 and the HM one, to 80 $ \le N_\text {trk}^\text {offline} \le $ 250. Statistical uncertainties are represented by error bars, systematic uncertainties related to the HCS method are shown as open boxes and the relative uncertainties from the intramethods variation are represented by the shaded bands. Only statistical uncertainties are considered in all the fits.

png pdf
Figure 7-a:
$1/R_{\mathrm {inv}}^2$ as a function of $m_{\mathrm {T}} = \sqrt {m_{\pi}^2 + < {k_{\mathrm {T}}} > ^2}$ for the HCS method. The MB range corresponds to 0 $ \le N_\text {trk}^\text {offline} \le $ 79 and the HM one, to 80 $ \le N_\text {trk}^\text {offline} \le $ 250. Statistical uncertainties are represented by error bars, systematic uncertainties related to the HCS method are shown as open boxes and the relative uncertainties from the intramethods variation are represented by the shaded bands. Only statistical uncertainties are considered in all the fits.

png pdf
Figure 7-b:
$1/R_{\mathrm {inv}}^2$ as a function of $m_{\mathrm {T}} = \sqrt {m_{\pi}^2 + < {k_{\mathrm {T}}} > ^2}$ for the HCS method. The MB range corresponds to 0 $ \le N_\text {trk}^\text {offline} \le $ 79 and the HM one, to 80 $ \le N_\text {trk}^\text {offline} \le $ 250. Statistical uncertainties are represented by error bars, systematic uncertainties related to the HCS method are shown as open boxes and the relative uncertainties from the intramethods variation are represented by the shaded bands. Only statistical uncertainties are considered in all the fits.

png pdf
Figure 8:
Illustration of the steps in the DR method. Left: The SR in data is constructed, followed by a similar procedure with MC (PYTHIA6 $\mathrm {Z2}^{*}$) generated events. Right: The ratio of the two curves on the left taken to define the DR. The reference sample is obtained with the $\eta $-mixing procedure. All results correspond to integrated values in $N^\text {offline}_\text {trk}$ and $k_T$.

png pdf
Figure 8-a:
Illustration of the steps in the DR method. Left: The SR in data is constructed, followed by a similar procedure with MC (PYTHIA6 $\mathrm {Z2}^{*}$) generated events. Right: The ratio of the two curves on the left taken to define the DR. The reference sample is obtained with the $\eta $-mixing procedure. All results correspond to integrated values in $N^\text {offline}_\text {trk}$ and $k_T$.

png pdf
Figure 8-b:
Illustration of the steps in the DR method. Left: The SR in data is constructed, followed by a similar procedure with MC (PYTHIA6 $\mathrm {Z2}^{*}$) generated events. Right: The ratio of the two curves on the left taken to define the DR. The reference sample is obtained with the $\eta $-mixing procedure. All results correspond to integrated values in $N^\text {offline}_\text {trk}$ and $k_T$.

png pdf
Figure 9:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-a:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-b:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-c:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-d:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-e:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 9-f:
Same-sign and opposite-sign SR correlation functions are shown in different $N^\text {offline}_\text {track}$ and $ {k_{\mathrm {T}}} $ bins, together with the full fits (continuous curves) given in Eq. (10) and Eq. (13), for minimum-bias (top panel) and high-multiplicity (bottom panel) events.

png pdf
Figure 10:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-a:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-b:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-c:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-d:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-e:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 10-f:
Correlation functions from the DR technique, integrated in the range 0 $ < k_{\mathrm {T}} < $ 1 GeV, in six multiplicity bins. The results are zoomed along the vertical axis.

png pdf
Figure 11:
Left: The depth of the anticorrelation $\Delta $ as a function of multiplicity (for $ {k_{\mathrm {T}}} $-integrated values) Right: $\Delta $ in finer bins of $N_\text {tracks}$ and $ {k_{\mathrm {T}}} $. The statistical uncertainties are represented by error bars, while the systematic ones are represented by open boxes.

png pdf
Figure 11-a:
Left: The depth of the anticorrelation $\Delta $ as a function of multiplicity (for $ {k_{\mathrm {T}}} $-integrated values) Right: $\Delta $ in finer bins of $N_\text {tracks}$ and $ {k_{\mathrm {T}}} $. The statistical uncertainties are represented by error bars, while the systematic ones are represented by open boxes.

png pdf
Figure 11-b:
Left: The depth of the anticorrelation $\Delta $ as a function of multiplicity (for $ {k_{\mathrm {T}}} $-integrated values) Right: $\Delta $ in finer bins of $N_\text {tracks}$ and $ {k_{\mathrm {T}}} $. The statistical uncertainties are represented by error bars, while the systematic ones are represented by open boxes.
Tables

png pdf
Table 1:
Total systematic uncertainties in different $ {k_{\mathrm {T}}} $ bins for the Hybrid Cluster Subtraction\ technique. The ranges in the uncertainties indicate the variation with $N^\text {offline}_\text {track}$.

png pdf
Table 2:
Parameter values refering to Eq. (8) from CGC calculations [59] in pp collsions at 7 TeV and from the fit to the data in Fig. 6 from pp collisions at 13 TeV.

png pdf
Table 3:
Values of the fit parameters from Eqs. (11) and (12), describing the cluster contribution in the data opposite-sign correlation function.
Summary
A femtoscopic analysis of Bose-Einstein correlations has been performed using data from the CMS detector for proton-proton collisions at $\sqrt{s}= $ 13 TeV covering a broad range of charged particle multiplicity, from a few particles up to about 250 reconstructed charged hadrons. Three analysis methods, each with a different dependence on MC simulations, were used to generate correlation functions and were found to give consistent results. One dimensional studies of the lengths-of-homogeneity, $R_\text {inv}$, and the intercept parameter, $\lambda$, have been carried out for both inclusive events and high-multiplicity events selected using a dedicated online trigger. For multiplicities in the range 0 $ < N^\text {offline}_\text {track} < 250$ and average pair transverse momentum 0 $ < {k_{\mathrm{T}}} < 1$ GeV, values of the lengths of homogeneity and intercept are found to be in the range 0.8 $ < R_\text {inv} < $ 3.0 fm and 0.5 $ < \lambda < $ 1.0, respectively.

Over most of the multiplicity range studied, the values of $R_\text {inv}$ increase with increasing event multiplicities and are observed to be proportional to $ < N_{\text{tracks}} > ^{1/3}$, a trend which is predicted by hydrodynamical calculations. For high-multiplicity events with more than $\approx$100 charged particles, the observed dependence of $R_\text {inv}$ suggests a possible saturation, with the lengths of homogeneity also consistent with a constant value. Comparisons of the multiplicity dependence are made with predictions of the CGC effective theory, by means of a parametrization of the radius of the system formed in pp collisions. The values of the radius parameters in the model are much lower than those in the data, although the general shape of the dependence on multiplicity is similar in both cases. The radius fit parameter $R_\text {inv}$ is also observed to decrease with increasing ${k_{\mathrm{T}}}$, a behavior that is consistent with emission from a system that is expanding prior to its decoupling.

Inspired by hydrodynamic models, the dependence of $R_{\rm inv}^{-2}$ on the pair transverse mass was investigated and the two were observed to be proportional, a behavior similar to that seen in AA collisions. The proportionality constant between $R_{\rm inv}^{-2}$ and transverse mass can be related to the flow parameter of a Hubble type expansion of the system. For pp collisions at 13 TeV, this expansion was found to be slower for larger event multiplicity, a dependence which was also found in AA collisions. Therefore, the present analysis reveals additional similarities of the systems produced in high multiplicity pp collisions and those found using data for larger initial systems. These results may provide additional constrains for future attempts using hydrodynamical models and/or the CGC framework to explain the entire range of similarities between high multiplicity pp and heavy ion interactions.
References
1 W. L. G. Goldhaber, S. Goldhaber and A. Pais Influence of Bose-Einstein Statistics in the Antiproton-Proton Annihilation Process PR120 (1960) 300
2 CMS Collaboration First Measurement of Bose-Einstein Correlations in proton-proton Collisions at $ \sqrt{s} = $ 0.9 and 2.36 TeV at the LHC PRL 105 (2010) 032001 CMS-QCD-10-003
1005.3294
3 CMS Collaboration Measurement of Bose-Einstein Correlations in $ pp $ Collisions at $ \sqrt{s}= $ 0.9 and 7 TeV JHEP 05 (2011) 029 CMS-QCD-10-023
1101.3518
4 CMS Collaboration Bose-Einstein correlations in pp, pPb, and PbPb collisions at $ \sqrt{s} = $ 0.9 -- 7 TeV To appear in PRC (2018) CMS-FSQ-14-002
1712.07198
5 AFS Collaboration Bose-Esinstein correlations in $ \alpha \alpha $, $ pp $ and $ p\bar{p} $ interactions PLB 129 (1983) 269
6 AFS Collaboration Bose-Einstein correlations between kaons PLB 155 (1985) 128
7 AFS Collaboration Evidence of a directional dependence of Bose-Einstein correlations at the CERN Intersecting Storage Rings PLB 187 (1987) 420
8 UA1 Collaboration Bose-Einstein correlations in $ p \bar{p} $ interactions at $ \sqrt{s} = $ 0.2 to 0.9 TeV PLB 226 (1989) 410
9 E735 Collaboration Study of the source size in p pbar collisions at 1.8 TeV using pion interferometry PRD 48 (1993) 1931
10 PHOBOS Collaboration Transverse momentum and rapidity dependence of HBT correlations in Au + Au collisions at s(NN)**(1/2) = 62.4-GeV and 200-GeV PRC 73 (2006) 031901 nucl-ex/0409001
11 STAR Collaboration Pion interferometry in Au+Au collisions at S(NN)**(1/2) = 200-GeV PRC 71 (2005) 044906 nucl-ex/0411036
12 PHENIX Collaboration Source breakup dynamics in Au+Au Collisions at s(NN)**(1/2) = 200-GeV via three-dimensional two-pion source imaging PRL 100 (2008) 232301 0712.4372
13 ALICE Collaboration Two-pion Bose-Einstein correlations in $ pp $ collisions at $ \sqrt{s}= $ 900 GeV PRD 82 (2010) 052001 1007.0516
14 ATLAS Collaboration Two-Particle Bose-Einstein Correlations in pp collisions at $ \sqrt{s}= $ 0.9 and 7 TeV with the ATLAS detector EPJC 75 (2015) 466 1502.07947
15 LHCb Collaboration Bose-Einstein correlations of same-sign charged pions in the forward region in $ pp $ collisions at $ \sqrt{s} = $ 7 TeV JHEP 12 (2017) 025 1709.01769
16 CMS Collaboration Observation of Long-Range Near-Side Angular Correlations in Proton-Proton Collisions at the LHC JHEP 1009 (2010) 091 CMS-QCD-10-002
1009.4122
17 CMS Collaboration Observation of Long-Range Near-Side Angular Correlations in Proton-Proton Collisions at the LHC JHEP 09 (2010) 091 CMS-QCD-10-002
1009.4122
18 ATLAS Collaboration Observation of Long-Range Elliptic Azimuthal Anisotropies in $ \sqrt{s}= $ 13 and 2.76 TeV pp Collisions with the ATLAS Detector PRL 116 (2016), no. 17, 172301 1509.04776
19 CMS Collaboration Observation of long-range near-side angular correlations in proton-lead collisions at the LHC PLB 718 (2013) 795 CMS-HIN-12-015
1210.5482
20 ALICE Collaboration Long-range angular correlations on the near and away side in p-Pb collisions at $ \sqrt{s_{NN}}= $ 5.02 TeV PLB 719 (2013) 29 1212.2001
21 ATLAS Collaboration Observation of Associated Near-Side and Away-Side Long-Range Correlations in $ \sqrt{s_{NN}}= $ 5.02 TeV Proton-Lead Collisions with the ATLAS Detector PRL 110 (2013), no. 18, 182302 1212.5198
22 LHCb Collaboration Measurements of long-range near-side angular correlations in $ \sqrt{s_{\text{NN}}}= $ 5 TeV proton-lead collisions in the forward region PLB 762 (2016) 473 1512.00439
23 CMS Collaboration Evidence for collectivity in pp collisions at the LHC PLB 765 (2017) 193 CMS-HIN-16-010
1606.06198
24 CMS Collaboration Measurement of long-range near-side two-particle angular correlations in pp collisions at $ \sqrt s = $ 13 TeV PRL 116 (2016), no. 17, 172302 CMS-FSQ-15-002
1510.03068
25 CMS Collaboration The CMS experiment at the CERN LHC JINST 3 (2008) S08004 CMS-00-001
26 T. Sjostrand, S. Mrenna, and P. Skands PYTHIA 6.4 physics and manual JHEP 05 (2006) 026 hep-ph/0603175
27 T. Sjostrand, S. Mrenna, and P. Z. Skands A brief introduction to PYTHIA 8.1 CPC 178 (2008) 852 0710.3820
28 R. Field Early LHC Underlying Event Data - Findings and Surprises 1010.3558
29 CMS Collaboration Event generator tunes obtained from underlying event and multiparton scattering measurements EPJC 76 (2016), no. 3, 155 CMS-GEN-14-001
1512.00815
30 R. Corke and T. Sjostrand Interleaved Parton Showers and Tuning Prospects JHEP 03 (2011) 032 1011.1759
31 T. Pierog et al. EPOS LHC: Test of collective hadronization with data measured at the CERN Large Hadron Collider PRC 92 (2015), no. 3, 034906 1306.0121
32 S. Agostinelli et al. Geant4---a simulation toolkit NIMA 506 (2003) 250
33 CMS Collaboration Description and performance of track and primary-vertex reconstruction with the CMS tracker JINST 9 (2014), no. 10, P10009 CMS-TRK-11-001
1405.6569
34 CMS Collaboration Measurement of transverse momentum relative to dijet systems in PbPb and pp collisions at $ \sqrt{s_{\mathrm{NN}}}= $ 2.76 TeV JHEP 01 (2016) 006 CMS-HIN-14-010
1509.09029
35 CMS Collaboration Study of the inclusive production of charged pions, kaons, and protons in pp collisions at $ \sqrt{s}= $ 0.9 , 2.76, and 7~TeV EPJC 72 (2012) 2164 CMS-FSQ-12-014
1207.4724
36 Y. Sinyukov et al. Coulomb corrections for interferometry analysis of expanding hadron systems PLB 432 (1998) 248
37 M. Gyulassy, S. K. Kauffman, L. W. Wilson Pion Interferometry o nuclear collisions. I. Theory PRC 20 (1979) 2267
38 S. Pratt Coherence and Coulomb effects on pion interferometry PRD 33 (1986) 72
39 M. Biyajima and T. Mizoguchi Coulomb wave function correction to Bose-Einstein correlations SULDP-1994-9
40 M. Bowler Coulomb corrections to Bose-Einstein correlations have been greatly exaggerated PLB 270 (1991) 69
41 Y. Sinyukov et al. Coulomb corrections for interferometry analysis of expanding hadron systems PLB 432 (1998) 248
42 A. N. Makhlin and Y. M. Sinyukov The hydrodynamics of hadron matter under a pion interferometric microscope Z. Phys. C 39 (1988) 69
43 T. Csorgo, S. Hegyi, and W. A. Zajc Bose-Einstein correlations for Levy stable source distributions EPJC 36 (2004) 67 nucl-th/0310042
44 T. Csorgo, A. T. Szerzo, and S. Hegyi Model independent shape analysis of correlations in one-dimension, two-dimensions or three-dimensions PLB 489 (2000) 15 hep-ph/9912220
45 E802 Collaboration System, centrality, and transverse mass dependence of two pion correlation radii in heavy ion collisions at 11.6 A-GeV and 14.6 A-GeV PRC 66 (2002) 054906 nucl-ex/0204001
46 ATLAS Collaboration Femtoscopy with identified charged pions in proton-lead collisions at $ \sqrt{s{NN}} = $ 5.02 TeV with ATLAS PRC 96 (2017) 064908 1704.01621
47 PHOBOS Collaboration System size dependence of cluster properties from two-particle angular correlations in Cu+Cu and Au+Au collisions at $ \sqrt{s_{NN}} = $ 200~GeV PRC 81 (2010) 024904 0812.1172
48 Y. Hama and Sandra S. Padula Bose-Einstein correlation of particles produced by expanding sources PRD 37 (1988) 3237
49 J. C. Collins and M. J. Perry Superdense matter: Neutrons or asymptotically free quarks? PRL 34 (1975) 1353
50 N. Cabibbo and G. Parisi Exponential hadronic spectrum and quark liberation PLB 59 (1975) 67
51 B. A. Freedman and L. D. McLerran Fermions and gauge vector mesons at finite temperature and density. 1. Formal techniques PRD 16 (1977) 1130
52 E. V. Shuryak Theory of hadronic plasma Sov. Phys. JETP 47 (1978) 212.[Zh. Eksp. Teor. Fiz. 74, 408]
53 M. I. Khalatnikov Some questions onf the relativistic hydrodynamic Zh. Eksp. Teor. Fiz. 26 (1954) 529
54 L. D. Landau On the multiple production of particle sin high energy collisions Izv. Adak. Nauk SSSR Ser. Fiz. 17 (1953) 51
55 R. Campanini and G. Ferri Experimental equation of state in pp and $ p\bar{p} $ collisions and phase transition to quark gluon plasma Phys. Letter. B 703 (2011) 237 1106.2008
56 P. Bozek and W. Broniowski Size of the emission source and collectivity in ultra-relativistic p-Pb collisions PLB 720 (2013) 250 1301.3314
57 V. M. Shapoval, P. Braun-Munzinger, I. A. Karpenko, and \relax Yu. M. Sinyukov Femtoscopic scales in $ p+p $ and $ p+ $Pb collisions in view of the uncertainty principle PLB 725 (2013) 139 1304.3815
58 M. P. Larry McLerran and B. Schenke Transverse momentum of protons, pions and kaons in high multiplicity pp and pA collisions: Evidence for the color glass condensate? NP A 916 (2013) 210 1306.2350
59 P. T. A. Bzdak, B. Schenke and R. Venugopalan Initial-state geometry and the role of hydrodynamics in proton-proton, proton-nucleus, and deuteron-nucleus collisions PRC 87 (2013) 064906 1304.3403
60 T. Csorg\Ho and B. Lorstad Bose-Einstein correlations for three-dimensionally expanding, cylindrically symmetric, finite systems PRC 54 (1996) 1390 hep-ph/9509213
61 PHENIX Collaboration Bose-Einstein correlations of charged pion pairs in Au + Au collisions at s(NN)**(1/2) = 200-GeV PRL 93 (2004) 152302 nucl-ex/0401003
62 STAR Collaboration Pion interferometry in Au+Au collisions at S(NN)**(1/2) = 200-GeV PRC 71 (2005) 044906 nucl-ex/0411036
63 STAR Collaboration Pion Interferometry in Au+Au and Cu+Cu Collisions at RHIC PRC 80 (2009) 024905 0903.1296
64 PHENIX Collaboration Systematic study of charged-pion and kaon femtoscopy in Au + Au collisions at $ \sqrt{s_{_{NN}}} = $ 200 GeV PRC 92 (2015), no. 3, 034914 1504.05168
65 ZEUS Collaboration Bose-Einstein correlations in one and two-dimensions in deep inelastic scattering PLB 583 (2004) 231 hep-ex/0311030
66 L3 Collaboration Test of the $ \tau $-Model of Bose-Einstein Correlations and Reconstruction of the Source Function in Hadronic Z-boson Decay at LEP EPJC 71 (2011) 1648 1105.4788
67 T. Csorg\Ho and J. Zim\'anyi Pion Interferometry for Strongly Correlated Space-Time and Momentum Space NP A 517 (1990) 588
Compact Muon Solenoid
LHC, CERN