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| CMS-TOP-18-009 ; CERN-EP-2019-124 | ||
| Measurement of top quark pair production in association with a Z boson in proton-proton collisions at $\sqrt{s} = $ 13 TeV | ||
| CMS Collaboration | ||
| 25 July 2019 | ||
| JHEP 03 (2020) 056 | ||
| Abstract: A measurement of the inclusive cross section of top quark pair production in association with a Z boson using proton-proton collisions at a center-of-mass energy of 13 TeV at the LHC is performed. The data sample corresponds to an integrated luminosity of 77.5 fb$^{-1}$, collected by the CMS experiment during 2016 and 2017. The measurement is performed using final states containing three or four charged leptons (electrons or muons), and the Z boson is detected through its decay to an oppositely charged lepton pair. The production cross section is measured to be $\sigma({\mathrm{t\bar{t}}\mathrm{Z}} )=$ 0.95 $\pm$ 0.05 (stat) $\pm$ 0.06 (syst) pb. For the first time, differential cross sections are measured as functions of the transverse momentum of the Z boson and the angular distribution of the negatively charged lepton from the Z boson decay. The most stringent direct limits to date on the anomalous couplings of the top quark to the Z boson are presented, including constraints on the Wilson coefficients in the framework of the standard model effective field theory. | ||
| Links: e-print arXiv:1907.11270 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; | ||
| Figures & Tables | Summary | Additional Figures | References | CMS Publications |
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| Figures | |
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Figure 1:
The observed (points) and predicted (shaded histograms) event yields versus lepton flavor (upper left), and the reconstructed transverse momentum of the Z boson candidates (upper right) in the WZ-enriched data control event category, and versus lepton flavor (lower left) and ${N_{\mathrm{b}}}$ (lower right) in the ZZ-enriched event category. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panels show the ratio of the event yields in data to the predictions. |
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Figure 1-a:
The observed (points) and predicted (shaded histograms) event yields versus lepton flavor in the WZ-enriched data control event category. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the predictions. |
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Figure 1-b:
The observed (points) and predicted (shaded histograms) event yields versus the reconstructed transverse momentum of the Z boson candidates in the WZ-enriched data control event category. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the predictions. |
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Figure 1-c:
The observed (points) and predicted (shaded histograms) event yields versus lepton flavor in the ZZ-enriched event category. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the predictions. |
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Figure 1-d:
The observed (points) and predicted (shaded histograms) event yields versus ${N_{\mathrm{b}}}$ in the ZZ-enriched event category. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the predictions. |
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Figure 2:
The observed (points) and predicted (shaded histograms) event yields in regions enriched with nonprompt lepton backgrounds in $ {\mathrm{t} \mathrm{\bar{t}}} $-like processes as a function of the lepton flavors (upper left), the ${p_{\mathrm {T}}}$ of the lowest-${p_{\mathrm {T}}}$ (trailing) lepton (upper right), and ${N_{\mathrm{b}}}$ (bottom). The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panels show the ratio of the event yields in data to the background predictions. |
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Figure 2-a:
The observed (points) and predicted (shaded histograms) event yields in regions enriched with nonprompt lepton backgrounds in $ {\mathrm{t} \mathrm{\bar{t}}} $-like processes as a function of the lepton flavors. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the background predictions. |
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Figure 2-b:
The observed (points) and predicted (shaded histograms) event yields in regions enriched with nonprompt lepton backgrounds in $ {\mathrm{t} \mathrm{\bar{t}}} $-like processes as a function of the ${p_{\mathrm {T}}}$ of the lowest-${p_{\mathrm {T}}}$ (trailing) lepton. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the background predictions. |
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Figure 2-c:
The observed (points) and predicted (shaded histograms) event yields in regions enriched with nonprompt lepton backgrounds in $ {\mathrm{t} \mathrm{\bar{t}}} $-like processes as a function of ${N_{\mathrm{b}}}$. The vertical lines on the points show the statistical uncertainties in the data, and the band the total uncertainty in the predictions. The lower panel shows the ratio of the event yields in data to the background predictions. |
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Figure 3:
Observed event yields in data for different values of ${N_\text {j}}$ and ${N_{\mathrm{b}}}$ for events with 3 and 4 leptons, compared with the signal and background yields, as obtained from the fit. The lower panel displays the ratio of the data to the predictions of the signal and background from simulation. The inner and outer bands show the statistical and total uncertainties, respectively. |
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Figure 4:
Kinematic distributions from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The distributions include the lepton flavor (upper left), number of b-tagged jets (upper right), jet multiplicity (middle left), dilepton invariant mass ${m(\ell \ell)}$ (middle right), ${{p_{\mathrm {T}}} (\mathrm{Z})}$ (lower left), and $\cos\theta ^\ast _{\mathrm{Z}}$ (lower right). The lower panels in each plot give the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-a:
Distribution of the lepton flavor, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-b:
Distribution of the number of b-tagged jets, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-c:
Distribution of the jet multiplicity, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-d:
Distribution of the dilepton invariant mass ${m(\ell \ell)}$, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-e:
Distribution of ${{p_{\mathrm {T}}} (\mathrm{Z})}$, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 4-f:
Distribution of $\cos\theta ^\ast _{\mathrm{Z}}$, from a ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ signal-enriched subset of events for data (points), compared to the contributions of the signal and background yields from the fit (shaded histograms). The lower panel gives the ratio of the data to the sum of the signal and background from the fit. The band shows the total uncertainty in the signal and background yields, as obtained from the fit. |
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Figure 5:
Measured differential ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ production cross sections in the full phase space as a function of the transverse momentum ${{p_{\mathrm {T}}} (\mathrm{Z})}$ of the Z boson (upper row) and $\cos\theta ^\ast _{\mathrm{Z}}$, as defined in the text (lower row). Shown are the absolute (left) and normalized (right) cross sections. The data are represented by the points. The inner (outer) vertical lines indicate the statistical (total) uncertainties. The solid histogram shows the prediction from the MadGraph 5\_aMC@NLO MC simulation, and the dashed histogram shows the theory prediction at NLO+NNLL accuracy. The hatched bands indicate the theoretical uncertainties in the predictions, as defined in the text. The lower panels display the ratios of the predictions to the measurement. |
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Figure 5-a:
Measured differential ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ production cross sections in the full phase space as a function of the transverse momentum ${{p_{\mathrm {T}}} (\mathrm{Z})}$ of the Z boson Shown are the absolute cross sections. The data are represented by the points. The inner (outer) vertical lines indicate the statistical (total) uncertainties. The solid histogram shows the prediction from the MadGraph 5_aMC@NLO MC simulation, and the dashed histogram shows the theory prediction at NLO+NNLL accuracy. The hatched bands indicate the theoretical uncertainties in the predictions, as defined in the text. The lower panel displays the ratios of the predictions to the measurement. |
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Figure 5-b:
Measured differential ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ production cross sections in the full phase space as a function of the transverse momentum ${{p_{\mathrm {T}}} (\mathrm{Z})}$ of the Z boson. Shown are the normalized cross sections. The data are represented by the points. The inner (outer) vertical lines indicate the statistical (total) uncertainties. The solid histogram shows the prediction from the MadGraph 5_aMC@NLO MC simulation, and the dashed histogram shows the theory prediction at NLO+NNLL accuracy. The hatched bands indicate the theoretical uncertainties in the predictions, as defined in the text. The lower panel displays the ratios of the predictions to the measurement. |
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Figure 5-c:
Measured differential ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ production cross sections in the full phase space as a function of the transverse momentum ${{p_{\mathrm {T}}} (\mathrm{Z})}$ of $\cos\theta ^\ast _{\mathrm{Z}}$. Shown are the absolute cross sections. The data are represented by the points. The inner (outer) vertical lines indicate the statistical (total) uncertainties. The solid histogram shows the prediction from the MadGraph 5_aMC@NLO MC simulation, and the dashed histogram shows the theory prediction at NLO+NNLL accuracy. The hatched bands indicate the theoretical uncertainties in the predictions, as defined in the text. The lower panel displays the ratios of the predictions to the measurement. |
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Figure 5-d:
Measured differential ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ production cross sections in the full phase space as a function of the transverse momentum ${{p_{\mathrm {T}}} (\mathrm{Z})}$ of $\cos\theta ^\ast _{\mathrm{Z}}$. Shown are the normalized cross sections. The data are represented by the points. The inner (outer) vertical lines indicate the statistical (total) uncertainties. The solid histogram shows the prediction from the MadGraph 5_aMC@NLO MC simulation, and the dashed histogram shows the theory prediction at NLO+NNLL accuracy. The hatched bands indicate the theoretical uncertainties in the predictions, as defined in the text. The lower panel displays the ratios of the predictions to the measurement. |
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Figure 6:
The observed (points) and predicted (shaded histograms) post-fit yields for the combined 2016 and 2017 data sets in the control and signal regions. In the $ {N_{\ell}} =$ 3 control and signal regions (bins 1-12), each of the four ${{p_{\mathrm {T}}} (\mathrm{Z})}$ categories is further split into three $\cos\theta ^\ast _{\mathrm{Z}}$ bins. The horizontal bars on the points give the statistical uncertainties in the data. The lower panel displays the ratio of the data to the predictions and the hatched regions show the total uncertainty. The solid line shows the best-fit prediction from the SMEFT fit. |
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Figure 7:
Results of scans in two 2D planes for the SMEFT interpretation. The shading quantified by the gray scale on the right reflects the negative log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. |
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Figure 7-a:
Result of a scan in a two 2D plane for the SMEFT interpretation. The shading quantified by the gray scale on the right reflects the negative log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. |
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Figure 7-b:
Result of a scan in a two 2D plane for the SMEFT interpretation. The shading quantified by the gray scale on the right reflects the negative log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. |
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Figure 8:
Results of scans in the axial-vector and vector current coupling plane (left) and the electroweak dipole moment plane (right). The shading quantified by the gray scale on the right of each plot reflects the log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. The area between the dot-dashed ellipses in the axial-vector and vector current coupling plane corresponds to the observed 68% CL area from the previous CMS result [89]. |
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Figure 8-a:
Result of a scan in the axial-vector and vector current coupling plane. The shading quantified by the gray scale on the right of each plot reflects the log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. The area between the dot-dashed ellipses in the plane corresponds to the observed 68% CL area from the previous CMS result [89]. |
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Figure 8-b:
Result of a scan in the electroweak dipole moment plane. The shading quantified by the gray scale on the right of each plot reflects the log-likelihood ratio $q$ with respect to the best-fit value, designated by the diamond. The solid and dashed lines indicate the 68 and 95% CL contours from the fit, respectively. The cross shows the SM prediction. |
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Figure 9:
1D scans of two Wilson coefficients, with the value of the other Wilson coefficients set to zero. The shaded areas correspond to the 68 and 95% CL intervals around the best fit value, respectively. The downward triangle indicates the SM value. Previously excluded regions at 95% CL [3] (if available) are indicated by the hatched band. Indirect constraints from Ref. [90] are shown as a cross-hatched band. |
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Figure 9-a:
1D scan of two Wilson coefficients, with the value of the other Wilson coefficients set to zero. The shaded areas correspond to the 68 and 95% CL intervals around the best fit value, respectively. The downward triangle indicates the SM value. Indirect constraints from Ref. [90] are shown as a cross-hatched band. |
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Figure 9-b:
1D scan of two Wilson coefficients, with the value of the other Wilson coefficients set to zero. The shaded areas correspond to the 68 and 95% CL intervals around the best fit value, respectively. The downward triangle indicates the SM value. Previously excluded regions at 95% CL [3] are indicated by the hatched band. Indirect constraints from Ref. [90] are shown as a cross-hatched band. |
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Figure 9-c:
1D scan of two Wilson coefficients, with the value of the other Wilson coefficients set to zero. The shaded areas correspond to the 68 and 95% CL intervals around the best fit value, respectively. The downward triangle indicates the SM value. Indirect constraints from Ref. [90] are shown as a cross-hatched band. |
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Figure 9-d:
1D scan of two Wilson coefficients, with the value of the other Wilson coefficients set to zero. The shaded areas correspond to the 68 and 95% CL intervals around the best fit value, respectively. The downward triangle indicates the SM value. Indirect constraints from Ref. [90] are shown as a cross-hatched band. |
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Figure 10:
Log-likelihood ratios for 1D scans of anomalous couplings. For the scan of $ {C_{1,\mathrm {A}}}$ (upper left), $ {C_{1,\mathrm {V}}}$ was set to the SM value of 0.24, and for the scan of $ {C_{1,\mathrm {V}}}$ (upper right), $ {C_{1,\mathrm {A}}}$ was set to the SM value of $-0.60$. For the scans of $ {C_{2,\mathrm {A}}}$ (lower left) and $ {C_{2,\mathrm {V}}}$ (lower right), which correspond to the top quark electric and magnetic dipole moments, respectively, both $ {C_{1,\mathrm {V}}}$ and $ {C_{1,\mathrm {A}}}$ are set to the SM values. The shaded areas correspond to the 68 and 95% CL intervals around the best-fit value, respectively. The downward triangle indicates the SM value. |
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Figure 10-a:
Log-likelihood ratio for the 1D scan of the $ {C_{1,\mathrm {A}}}$ anomalous coupling. The value of $ {C_{1,\mathrm {V}}}$ was set to the SM value of 0.24. The shaded areas correspond to the 68 and 95% CL intervals around the best-fit value, respectively. The downward triangle indicates the SM value. |
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Figure 10-b:
Log-likelihood ratio for the 1D scan of the $ {C_{1,\mathrm {V}}}$ anomalous coupling. The value of $ {C_{1,\mathrm {A}}}$ was set to the SM value of $-0.60$. The shaded areas correspond to the 68 and 95% CL intervals around the best-fit value, respectively. The downward triangle indicates the SM value. |
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Figure 10-c:
Log-likelihood ratio for the 1D scan of the $ {C_{2,\mathrm {A}}}$ anomalous coupling, which correspond to the top quark electric dipole moment. The values of $ {C_{1,\mathrm {V}}}$ and $ {C_{1,\mathrm {A}}}$ were set to the SM values. The shaded areas correspond to the 68 and 95% CL intervals around the best-fit value, respectively. The downward triangle indicates the SM value. |
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Figure 10-d:
Log-likelihood ratio for the 1D scan of the $ {C_{2,\mathrm {V}}}$ anomalous coupling, which correspond to the top quark magnetic dipole moment. The values of $ {C_{1,\mathrm {V}}}$ and $ {C_{1,\mathrm {A}}}$ were set to the SM values. The shaded areas correspond to the 68 and 95% CL intervals around the best-fit value, respectively. The downward triangle indicates the SM value. |
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Figure 11:
The observed 95% CL intervals for the Wilson coefficients from this measurement, the previous CMS result based on the inclusive ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ cross section measurement [3], and the most recent ATLAS result [4]. The direct limits within the SMEFiT framework [87] and from the TopFitter Collaboration [88], and the 68% CL indirect limits from electroweak data are also shown [90]. The vertical line displays the SM prediction. |
| Tables | |
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Table 1:
Event generators used to simulate events for the various processes. For each of the simulated processes shown, the order of the cross section normalization, the event generator used, the perturbative order of the generator calculation, and the NNPDF versions at NLO and at next-to-next-to-leading order (NNLO) used in simulating samples for the 2016 (2017) data sets. |
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Table 2:
Summary of the sources, magnitudes, treatments, and effects of the systematic uncertainties in the final ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ cross section measurement. The first column indicates the source of the uncertainty, the second column shows the corresponding input uncertainty range for each background source and the signal. The third column indicates how correlations are treated between the uncertainties in the 2016 and 2017 data, where xx means fully correlated and ${\times}$ uncorrelated. The last column gives the corresponding systematic uncertainty in the ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ cross section from each source. The total systematic uncertainty, the statistical uncertainty and the total uncertainty in the ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ cross section are shown in the last three lines. |
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Table 3:
The measured ${{\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z}}$ cross section for events with with 3 and 4 leptons and the combined measurement. |
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Table 4:
The observed number of events for three- and four-lepton events in a signal-enriched sample of events, and the predicted yields and total uncertainties from the fit for each process. |
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Table 5:
Definition of the signal regions (SRs) and control regions (CRs). For signal regions SR13, SR14, and SR15 and control regions CR13, CR14, and CR15, there is no requirement on $\cos\theta ^\ast _{\mathrm{Z}}$. |
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Table 6:
Expected and observed 68 and 95% CL intervals from this measurement for the listed Wilson coefficients. The expected and observed 95% CL intervals from a previous CMS measurement [3] and indirect 68% CL constraints from precision electroweak data [90] are shown for comparison. |
| Summary |
|
A measurement of top quark pair production in association with a Z boson using a data sample of proton-proton collisions at $\sqrt{s} = $ 13 TeV, corresponding to an integrated luminosity of 77.5 fb$^{-1}$, collected with the CMS detector at the LHC has been presented. The analysis was performed in the three- and four-lepton final states using analysis categories defined with jet and b jet multiplicities. Data samples enriched in background processes were used to validate predictions, as well as to constrain their uncertainties. The larger data set and reduced systematic uncertainties such as those associated with the lepton identification, helped to substantially improve the precision on the measured cross section with respect to previous measurements reported in Refs. [3,4]. The measured inclusive cross section $\sigma({\mathrm{t\bar{t}}\mathrm{Z}} )=$ 0.95 $\pm$ 0.05 (stat) $\pm$ 0.06 (syst) pb is in good agreement with the standard model prediction of 0.84 $\pm$ 0.10 pb [30,31,32]. This is the most precise measurement of the ${\mathrm{t\bar{t}}\mathrm{Z}}$ cross section to date, and the first measurement with a precision competing with current theoretical calculations. Absolute and normalized differential cross sections for the transverse momentum of the Z boson and for ${\cos\theta^\ast_{\mathrm{Z}}} $, the angle between the direction of the Z boson and the direction of the negatively charged lepton in the rest frame of the Z boson, are measured for the first time. The standard model predictions at next-to-leading order are found to be in good agreement with the measured differential cross sections. The measurement is also interpreted in terms of anomalous interactions of the t quark with the Z boson. Confidence intervals for the anomalous vector and the axial-vector current couplings and the dipole moment interactions are presented. Constraints on the Wilson coefficients in the standard model effective field theory are also presented. |
| Additional Figures | |
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Additional Figure 1:
Covariance matrix of the measured absolute differential cross section as a function of $ {p_{\mathrm {T}}} (\mathrm{Z})$. |
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Additional Figure 2:
Correlation matrix of the measured absolute differential cross section as a function of $ {p_{\mathrm {T}}} (\mathrm{Z})$. |
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Additional Figure 3:
Covariance matrix of the measured normalized differential cross section as a function of $ {p_{\mathrm {T}}} (\mathrm{Z})$. |
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Additional Figure 4:
Correlation matrix of the measured normalized differential cross section as a function of $ {p_{\mathrm {T}}} (\mathrm{Z})$. |
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Additional Figure 5:
Covariance matrix of the measured absolute differential cross section as a function of $\cos\theta _{\mathrm{Z}}^{*} $. |
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Additional Figure 6:
Correlation matrix of the measured absolute differential cross section as a function of $\cos\theta _{\mathrm{Z}}^{*} $. |
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Additional Figure 7:
Covariance matrix of the measured normalized differential cross section as a function of $\cos\theta _{\mathrm{Z}}^{*} $. |
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Additional Figure 8:
Correlation matrix of the measured normalized differential cross section as a function of $\cos\theta _{\mathrm{Z}}^{*} $. |
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Additional Figure 9:
Response matrix describing the probability for a ${\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z} $ event with a certain true value of the Z boson ${p_{\mathrm {T}}}$ to be selected with a certain reconstructed value of the Z boson ${p_{\mathrm {T}}}$. |
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Additional Figure 10:
Response matrix describing the probability for a ${\mathrm{t} \mathrm{\bar{t}}} \mathrm{Z} $ event with a certain true value of $\cos\theta _{\mathrm{Z}}^{*} $ to be selected with a certain reconstructed value of $\cos\theta _{\mathrm{Z}}^{*} $. |
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