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| CMS-SMP-22-001 ; CERN-EP-2023-285 | ||
| Measurement of differential ZZ$+$jets production cross sections in pp collisions at $ \sqrt{s} = $ 13 TeV | ||
| CMS Collaboration | ||
| 3 April 2024 | ||
| JHEP 10 (2024) 209 | ||
| Abstract: Diboson production in association with jets is studied in the fully leptonic final states, $ \mathrm{p}\mathrm{p} \to (\mathrm{Z}/\gamma^*)(\mathrm{Z}/\gamma^*){+}$jets $\to 2\ell2\ell'{+}$jets, ($ \ell,\ell' = \mathrm{e} $ or $ \mu $) in proton-proton collisions at a center-of-mass energy of 13 TeV. The data sample corresponds to an integrated luminosity of 138 fb$ ^{-1} $ collected with the CMS detector at the LHC. Differential distributions and normalized differential cross sections are measured as a function of jet multiplicity, transverse momentum $ p_{\mathrm{T}} $, pseudorapidity $ \eta $, invariant mass and $ \Delta\eta $ of the highest-$ p_{\mathrm{T}} $ and second-highest-$ p_{\mathrm{T}} $ jets, and as a function of invariant mass of the four-lepton system for events with various jet multiplicities. These differential cross sections are compared with theoretical predictions that mostly agree with the experimental data. However, in a few regions we observe discrepancies between the predicted and measured values. Further improvement of the predictions is required to describe the ZZ$+$jets production in the whole phase space. | ||
| Links: e-print arXiv:2404.02711 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; Physics Briefing ; CADI line (restricted) ; | ||
| Figures | |
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Figure 1:
Example Feynman diagrams of ZZ production associated with jets via (left) quark-initiated production and (right) loop-induced gluon fusion production. |
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Figure 1-a:
Example Feynman diagrams of ZZ production associated with jets via (left) quark-initiated production and (right) loop-induced gluon fusion production. |
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Figure 1-b:
Example Feynman diagrams of ZZ production associated with jets via (left) quark-initiated production and (right) loop-induced gluon fusion production. |
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Figure 2:
Distribution of the number of jets with $ p_{\mathrm{T}} > $ 30 GeV (left) and of $ m_{\mathrm{Z}\mathrm{Z}} $ (right) for ZZ+jets events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV for the combined 4e, 4$ \mu $, and 2e2$ \mu $ decay channels. Points represent the data, vertical bars the statistical uncertainties, and shaded histograms represent the expected standard model predictions and reducible background estimated from data. The gray band represents the systematic uncertainties in the predictions, which includes systematic uncertainties associated with trigger efficiency, lepton efficiencies, jet energy correction and jet energy resolution, pileup, luminosity, Monte Carlo generator choice, $ \mathrm{g}\mathrm{g} \! \to \! \mathrm{Z}\mathrm{Z} $ cross section, and reducible background. The overflow is included in the last bin of the distributions. |
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Figure 2-a:
Distribution of the number of jets with $ p_{\mathrm{T}} > $ 30 GeV (left) and of $ m_{\mathrm{Z}\mathrm{Z}} $ (right) for ZZ+jets events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV for the combined 4e, 4$ \mu $, and 2e2$ \mu $ decay channels. Points represent the data, vertical bars the statistical uncertainties, and shaded histograms represent the expected standard model predictions and reducible background estimated from data. The gray band represents the systematic uncertainties in the predictions, which includes systematic uncertainties associated with trigger efficiency, lepton efficiencies, jet energy correction and jet energy resolution, pileup, luminosity, Monte Carlo generator choice, $ \mathrm{g}\mathrm{g} \! \to \! \mathrm{Z}\mathrm{Z} $ cross section, and reducible background. The overflow is included in the last bin of the distributions. |
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Figure 2-b:
Distribution of the number of jets with $ p_{\mathrm{T}} > $ 30 GeV (left) and of $ m_{\mathrm{Z}\mathrm{Z}} $ (right) for ZZ+jets events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV for the combined 4e, 4$ \mu $, and 2e2$ \mu $ decay channels. Points represent the data, vertical bars the statistical uncertainties, and shaded histograms represent the expected standard model predictions and reducible background estimated from data. The gray band represents the systematic uncertainties in the predictions, which includes systematic uncertainties associated with trigger efficiency, lepton efficiencies, jet energy correction and jet energy resolution, pileup, luminosity, Monte Carlo generator choice, $ \mathrm{g}\mathrm{g} \! \to \! \mathrm{Z}\mathrm{Z} $ cross section, and reducible background. The overflow is included in the last bin of the distributions. |
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Figure 3:
Distribution of the $ p_{\mathrm{T}} $ of the highest-$ p_{\mathrm{T}} $ jet (upper left) in events with at least one jet, and of the $ p_{\mathrm{T}} $ of the second-highest-$ p_{\mathrm{T}} $ jet (upper right) in events containing at least two jets. The $ |\eta| $ distribution of the highest-$ p_{\mathrm{T}} $ (lower left) and second-highest-$ p_{\mathrm{T}} $ (lower right) jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 3-a:
Distribution of the $ p_{\mathrm{T}} $ of the highest-$ p_{\mathrm{T}} $ jet (upper left) in events with at least one jet, and of the $ p_{\mathrm{T}} $ of the second-highest-$ p_{\mathrm{T}} $ jet (upper right) in events containing at least two jets. The $ |\eta| $ distribution of the highest-$ p_{\mathrm{T}} $ (lower left) and second-highest-$ p_{\mathrm{T}} $ (lower right) jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 3-b:
Distribution of the $ p_{\mathrm{T}} $ of the highest-$ p_{\mathrm{T}} $ jet (upper left) in events with at least one jet, and of the $ p_{\mathrm{T}} $ of the second-highest-$ p_{\mathrm{T}} $ jet (upper right) in events containing at least two jets. The $ |\eta| $ distribution of the highest-$ p_{\mathrm{T}} $ (lower left) and second-highest-$ p_{\mathrm{T}} $ (lower right) jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 3-c:
Distribution of the $ p_{\mathrm{T}} $ of the highest-$ p_{\mathrm{T}} $ jet (upper left) in events with at least one jet, and of the $ p_{\mathrm{T}} $ of the second-highest-$ p_{\mathrm{T}} $ jet (upper right) in events containing at least two jets. The $ |\eta| $ distribution of the highest-$ p_{\mathrm{T}} $ (lower left) and second-highest-$ p_{\mathrm{T}} $ (lower right) jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 3-d:
Distribution of the $ p_{\mathrm{T}} $ of the highest-$ p_{\mathrm{T}} $ jet (upper left) in events with at least one jet, and of the $ p_{\mathrm{T}} $ of the second-highest-$ p_{\mathrm{T}} $ jet (upper right) in events containing at least two jets. The $ |\eta| $ distribution of the highest-$ p_{\mathrm{T}} $ (lower left) and second-highest-$ p_{\mathrm{T}} $ (lower right) jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 4:
The dijet mass (left) and $ |\Delta \eta| $ (right) between the two highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 4-a:
The dijet mass (left) and $ |\Delta \eta| $ (right) between the two highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 4-b:
The dijet mass (left) and $ |\Delta \eta| $ (right) between the two highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events are required to have 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. Other details are as in the caption of Fig. 2. |
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Figure 5:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 5-a:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 5-b:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 5-c:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 5-d:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 5-e:
The $ m_{4\ell} $ distributions for events with 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV and different jet multiplicities. Other details are as in the caption of Fig. 2. |
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Figure 6:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-a:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-b:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-c:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-d:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-e:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 6-f:
The $ m_{4\ell} $ distributions in the full four-lepton invariant mass range for events with different jet multiplicities, normalized by bin width. Other details are as in the caption of Fig. 2. |
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Figure 7:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ m_{4\ell} $, (right) the number of jets with $ p_{\mathrm{T}} > $ 30 GeV. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV is applied. Points represent the unfolded data, solid histograms the MadGraph-5_aMC@NLO $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions, and red dashed histograms the POWHEG $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions. MCFM $\mathrm{gg} \to \mathrm{ZZ} $, POWHEG $ \mathrm{H} \to \mathrm{Z}\mathrm{Z} $, and MadGraph-5_aMC@NLO EW ZZ predictions are included in these two sets of predictions. The purple dashed histograms represent the nNNLO+PS predictions, and the yellow dashed histogram represents the nNNLO+PS prediction with EW corrections applied. Vertical bars on both MC predictions represent the statistical uncertainties. The lower panels show the ratio of the measured to the predicted cross sections. The shaded areas represent the total uncertainties calculated as the sum in quadrature of the statistical and systematic uncertainties, whereas the crosses represent the statistical uncertainties only. The overflow is included in the last bin of the distributions. |
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Figure 7-a:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ m_{4\ell} $, (right) the number of jets with $ p_{\mathrm{T}} > $ 30 GeV. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV is applied. Points represent the unfolded data, solid histograms the MadGraph-5_aMC@NLO $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions, and red dashed histograms the POWHEG $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions. MCFM $\mathrm{gg} \to \mathrm{ZZ} $, POWHEG $ \mathrm{H} \to \mathrm{Z}\mathrm{Z} $, and MadGraph-5_aMC@NLO EW ZZ predictions are included in these two sets of predictions. The purple dashed histograms represent the nNNLO+PS predictions, and the yellow dashed histogram represents the nNNLO+PS prediction with EW corrections applied. Vertical bars on both MC predictions represent the statistical uncertainties. The lower panels show the ratio of the measured to the predicted cross sections. The shaded areas represent the total uncertainties calculated as the sum in quadrature of the statistical and systematic uncertainties, whereas the crosses represent the statistical uncertainties only. The overflow is included in the last bin of the distributions. |
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Figure 7-b:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ m_{4\ell} $, (right) the number of jets with $ p_{\mathrm{T}} > $ 30 GeV. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV is applied. Points represent the unfolded data, solid histograms the MadGraph-5_aMC@NLO $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions, and red dashed histograms the POWHEG $ \mathrm{q}\overline{\mathrm{q}} \to \mathrm{Z}\mathrm{Z} $ predictions. MCFM $\mathrm{gg} \to \mathrm{ZZ} $, POWHEG $ \mathrm{H} \to \mathrm{Z}\mathrm{Z} $, and MadGraph-5_aMC@NLO EW ZZ predictions are included in these two sets of predictions. The purple dashed histograms represent the nNNLO+PS predictions, and the yellow dashed histogram represents the nNNLO+PS prediction with EW corrections applied. Vertical bars on both MC predictions represent the statistical uncertainties. The lower panels show the ratio of the measured to the predicted cross sections. The shaded areas represent the total uncertainties calculated as the sum in quadrature of the statistical and systematic uncertainties, whereas the crosses represent the statistical uncertainties only. The overflow is included in the last bin of the distributions. |
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Figure 8:
Differential cross sections normalized to the fiducial cross section as a function of $ p_{\mathrm{T}} $ and $ |\eta| $ of the highest- and the second-highest-$ p_{\mathrm{T}} $ jet in events containing at least one or two jets, respectively. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, $\mathrm{Z}_2 < $ 120 GeV is applied. Other details are as in the caption of Fig. 7. |
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Figure 8-a:
Differential cross sections normalized to the fiducial cross section as a function of $ p_{\mathrm{T}} $ and $ |\eta| $ of the highest- and the second-highest-$ p_{\mathrm{T}} $ jet in events containing at least one or two jets, respectively. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, $\mathrm{Z}_2 < $ 120 GeV is applied. Other details are as in the caption of Fig. 7. |
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Figure 8-b:
Differential cross sections normalized to the fiducial cross section as a function of $ p_{\mathrm{T}} $ and $ |\eta| $ of the highest- and the second-highest-$ p_{\mathrm{T}} $ jet in events containing at least one or two jets, respectively. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, $\mathrm{Z}_2 < $ 120 GeV is applied. Other details are as in the caption of Fig. 7. |
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Figure 8-c:
Differential cross sections normalized to the fiducial cross section as a function of $ p_{\mathrm{T}} $ and $ |\eta| $ of the highest- and the second-highest-$ p_{\mathrm{T}} $ jet in events containing at least one or two jets, respectively. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, $\mathrm{Z}_2 < $ 120 GeV is applied. Other details are as in the caption of Fig. 7. |
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Figure 8-d:
Differential cross sections normalized to the fiducial cross section as a function of $ p_{\mathrm{T}} $ and $ |\eta| $ of the highest- and the second-highest-$ p_{\mathrm{T}} $ jet in events containing at least one or two jets, respectively. The on-shell Z requirement 60 $ < m_{\mathrm{Z}_1}$, $\mathrm{Z}_2 < $ 120 GeV is applied. Other details are as in the caption of Fig. 7. |
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Figure 9:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ |\Delta \eta| $ and (right) dijet mass between highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events with 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV requirement. Other details are as in the caption of Fig. 7. |
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Figure 9-a:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ |\Delta \eta| $ and (right) dijet mass between highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events with 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV requirement. Other details are as in the caption of Fig. 7. |
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Figure 9-b:
Differential cross sections normalized to the fiducial cross section as a function of (left) $ |\Delta \eta| $ and (right) dijet mass between highest-$ p_{\mathrm{T}} $ jets in events with at least two jets. Events with 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV requirement. Other details are as in the caption of Fig. 7. |
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Figure 10:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range. Other details are as in the caption of Fig. 7. |
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Figure 11:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV and for different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 11-a:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV and for different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 11-b:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV and for different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 11-c:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV and for different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 11-d:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for 60 $ < m_{\mathrm{Z}_1}$, ${\mathrm{Z}_2} < $ 120 GeV and for different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 12:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range and different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 12-a:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range and different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 12-b:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range and different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 12-c:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range and different jet multiplicities. Other details are as in the Fig. 7 caption. |
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Figure 12-d:
Differential cross sections normalized to the fiducial cross section as a function of $ m_{4\ell} $ for the full four-lepton invariant mass range and different jet multiplicities. Other details are as in the Fig. 7 caption. |
| Tables | |
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Table 1:
Particle-level selections used to define the fiducial phase space |
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Table 2:
Contributions of each source of systematic uncertainty to the normalized differential cross section measurements of jet variables. Uncertainties depend on the distributions and are listed as a range. |
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Table 3:
The contributions of each source of systematic uncertainty in the normalized differential cross sections measurements as a function of $ m_{4\ell} $ with jet multiplicity from 0 to 3 and more, in events satisfying 60 $ < m_{\mathrm{Z}_1, \mathrm{Z}_2} < $ 120 GeV. |
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Table 4:
The contributions of each source of systematic uncertainty in the normalized differential cross sections measurements as a function of $ m_{4\ell} $ with jet multiplicity from 0 to 3 and more, in events from the full $ m_{4\ell} $ range. |
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Table 5:
The observed and expected yields of ZZ events in different mass ranges, and estimated yields of background events, shown for each final state and for the sum. The first uncertainty is statistical, and the second one is systematic. (Due to rounding, the sum of individual entries may not match the total value shown.) |
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Table 6:
The observed and expected yields of ZZ events in different mass ranges, and estimated yields of background events, shown for each jet multiplicity. The first uncertainty is statistical, and the second one is systematic. (Due to rounding, the sum of individual entries may not match the total value shown.) |
| Summary |
| The four-lepton production in association with jets, $ \mathrm{p}\mathrm{p} \to (\mathrm{Z}/\gamma^*)(\mathrm{Z}/\gamma^*) {+} \text{jets} \to 2\ell2\ell'{+}$jets, where $ \ell,\ell' = \mathrm{e} $ or $ \mu $, was studied in proton-proton collisions at a center-of-mass energy of 13 TeV. The data sample corresponds to an integrated luminosity of 138 fb$ ^{-1} $ collected with the CMS detector at the LHC during 2016--2018. Differential distributions and differential cross sections normalized to the ZZ fiducial cross section were measured with respect to various kinematic variables: number of jets, jet transverse momentum ($ p_{\mathrm{T}} $) and pseudorapidity ($ \eta $), invariant mass of the dijet system and $ \eta $ difference between the highest-$ p_{\mathrm{T}} $ and second-highest-$ p_{\mathrm{T}} $ jets, and invariant mass of the four leptons ($ m_{4\ell} $) for different jet multiplicities. Tabulated results are provided in HEPData [59]. In general, predictions of theoretical models agree with the data, but in some regions significant discrepancies between predicted and measured values were observed. The recent nNNLO+PS prediction improves the data/prediction agreement in the 1-jet and high jet multiplicity regions, and describes the distribution of jet multiplicities better than NLO samples generated with the event generators MadGraph-5_aMC@NLO and POWHEG. The inclusion of electroweak corrections improves the description of the $ m_{4\ell} $ distribution. These measurements demonstrate the necessity for better Monte Carlo modeling in events with complex multiboson final states and extra jets. Further improvement of the predictions is required to describe the ZZ+jets production in the whole phase space. |
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