CMSPASSMP18010  
Measurement of the $ \tau $ lepton polarization in Z boson decays  
CMS Collaboration  
10 February 2023  
Abstract: The polarization of $ \tau $ leptons is measured using leptonic and hadronic $ \tau $ lepton decays in $ \mathrm{Z}\rightarrow\tau\tau $ events. A sample of pp collisions of an integrated luminosity of 36.3 fb$ ^{1} $ at $ \sqrt{s}= $ 13 TeV is used. The measured polarization at the Z pole is $ \mathcal{P}_{\tau}(\mathrm{Z}) =  $0.144 $ \pm $ 0.015 $ =  $0.144 $ \pm $ 0.006 (stat) $ \pm $ 0.014 (syst), in good agreement with the standard model value of the lepton asymmetry parameter of $ A_{\ell} = $ 0.1468 $ \pm $ 0.0003. The $ \tau $ polarization is used to determine the ratio of the vector to axialvector coupling of the $ \tau $ leptons in the neutral current and thus the weak mixing angle independently of the production process of the Z resonance. The obtained value is $ \sin^2\theta^{\mathrm{eff}}_W = $ 0.2319 $ \pm $ 0.0019 $ = $ 0.2319 $ \pm $ 0.0008 (stat) $ \pm $ 0.0018 (syst).  
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These preliminary results are superseded in this paper, Submitted to JHEP. The superseded preliminary plots can be found here. 
Figures  
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Figure 1:
Four possible helicity states of incoming quarks and outgoing $ \tau $ leptons. Thin arrows depict the direction of movement and the thick arrows show the helicity of the particles. The angle $ \theta_{\tau} $ is the scattering angle of the $ \tau^{} $ with respect to the quark momentum in the rest frame of the Z boson. 
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Figure 1a:
Four possible helicity states of incoming quarks and outgoing $ \tau $ leptons. Thin arrows depict the direction of movement and the thick arrows show the helicity of the particles. The angle $ \theta_{\tau} $ is the scattering angle of the $ \tau^{} $ with respect to the quark momentum in the rest frame of the Z boson. 
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Figure 1b:
Four possible helicity states of incoming quarks and outgoing $ \tau $ leptons. Thin arrows depict the direction of movement and the thick arrows show the helicity of the particles. The angle $ \theta_{\tau} $ is the scattering angle of the $ \tau^{} $ with respect to the quark momentum in the rest frame of the Z boson. 
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Figure 1c:
Four possible helicity states of incoming quarks and outgoing $ \tau $ leptons. Thin arrows depict the direction of movement and the thick arrows show the helicity of the particles. The angle $ \theta_{\tau} $ is the scattering angle of the $ \tau^{} $ with respect to the quark momentum in the rest frame of the Z boson. 
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Figure 1d:
Four possible helicity states of incoming quarks and outgoing $ \tau $ leptons. Thin arrows depict the direction of movement and the thick arrows show the helicity of the particles. The angle $ \theta_{\tau} $ is the scattering angle of the $ \tau^{} $ with respect to the quark momentum in the rest frame of the Z boson. 
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Figure 2:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2a:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2b:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2c:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2d:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2e:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 2f:
Definition of the angle $ \theta $ in the decays $ \tau \rightarrow h\nu \ \ (h = \pi,\ \ \rho,\ \ a_{1}) $ and $ \tau \rightarrow \mathrm{e}/\mu \nu\overline{\nu} $. Upper row for $ \tau_R $, bottom row for $ \tau_L $. 
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Figure 3:
Definitions of the angles $ \alpha $ in both $ \tau^{} \rightarrow \rho^{}\nu $ and $ \tau^{} \rightarrow a^{}_1\nu $ (a), the angle $ \beta $ in $ \tau^{} \rightarrow \rho^{}\nu $ (b) and for the decay $ \tau^{} \rightarrow a^{}_1\nu $ (d) and finally the angle $ \gamma $ (c) for the decay of $ a^{}_1 \rightarrow \pi^{}\pi^{+}\pi^{} $ 
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Figure 4:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 4a:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 4b:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 4c:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 4d:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 4e:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
png pdf 
Figure 4f:
Some representative templates for the channels $ \mu+\rho, \mu+a_1, \mu+\pi $ \it (upper row) and $ e+\rho, e+a_1, e+\pi $ \it (lower row), the blue and red lines indicate right and left handed $ \tau $ leptons, respectively. The error bars are statistical only and correspond to the limited MC samples after all selection cuts. 
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Figure 5:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
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Figure 5a:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
png pdf 
Figure 5b:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
png pdf 
Figure 5c:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
png pdf 
Figure 5d:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
png pdf 
Figure 5e:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
png pdf 
Figure 5f:
Variation of the template for negative (left) and positive (middle) $ \tau $ lepton helicities in the $ \mu+\rho $ and $ e+\rho $ channel under the change of the particle distribution function, here the variation with respect to cteq6l1 parametrization. The plot on the right shows the ratio of the ratios of the changes for negative and positive helicities, which is flat and centered exactly at 1.00 demonstrating that a possible effect on the polarization is much smaller than 1%. 
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Figure 6:
The final fit of templates to the data for the $ \tau_e\tau_\mu $ channel. 
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Figure 7:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7a:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7b:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7c:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7d:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7e:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 7f:
The final fits of templates to the data for the three $ \tau_e\tau_\mathrm{h} $ (left) and three $ \tau_\mu\tau_\mathrm{h} $ (right) categories. 
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Figure 8:
The final fits of the templates to the data for the $ \tau_\mathrm{h}\tau_\mathrm{h} $ channels. 
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Figure 8a:
The final fits of the templates to the data for the $ \tau_\mathrm{h}\tau_\mathrm{h} $ channels. 
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Figure 8b:
The final fits of the templates to the data for the $ \tau_\mathrm{h}\tau_\mathrm{h} $ channels. 
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Figure 8c:
The final fits of the templates to the data for the $ \tau_\mathrm{h}\tau_\mathrm{h} $ channels. 
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Figure 8d:
The final fits of the templates to the data for the $ \tau_\mathrm{h}\tau_\mathrm{h} $ channels. 
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Figure 9:
Fit results for the average $ \tau $ lepton polarization for the 11 event categories on the left and for the 4 channels separately and the combined fit to all channels and categories on the right. The inner error bars represent the statistical uncertainly, the outer bars includes the systematic uncertainty. 
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Figure 9a:
Fit results for the average $ \tau $ lepton polarization for the 11 event categories on the left and for the 4 channels separately and the combined fit to all channels and categories on the right. The inner error bars represent the statistical uncertainly, the outer bars includes the systematic uncertainty. 
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Figure 9b:
Fit results for the average $ \tau $ lepton polarization for the 11 event categories on the left and for the 4 channels separately and the combined fit to all channels and categories on the right. The inner error bars represent the statistical uncertainly, the outer bars includes the systematic uncertainty. 
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Figure 10:
Final result on the average $ \tau $ polarization in different bins of the Z boson rapidity: $ \eta  < $ 1.3, 1.3 $ < \eta  < $ 2.2 and $ \eta  > $ 2.2. 
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Figure 11:
A comparison of the $ \tau $ asymmetry, $ A_{\tau} $ measured from $ \tau $ polarization at CMS in this work and other experiments. The value of $ A_{\tau} $ for CMS is obtained from the result of Eq. 24 using Eq. 4. The green band indicates the $ \tau $ polarization value obtained by combining the SLD measurement [41] with the measurements performed by L3 [5], DELPHI [42], ALEPH [43], and OPAL [44]. The measurement performed by the ATLAS collaboration at a lower centerofmass energy of 8 TeV is documented in Ref. [6]. The CMS measurement refers to the result of the analysis presented in this note. The inner error bars represent the statistical uncertainly, the outer bars includes the systematic uncertainty. 
Tables  
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Table 1:
Selections applied in the data processing of this analysis. 
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Table 2:
Final choice of discriminators in the different event categories 
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Table 3:
Systematic uncertainties affecting only the normalization of templates. 
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Table 4:
Systematic uncertainties affecting the shapes of templates. The uncertainty magnitudes listed in the Table refer to modifications of the relevant quantity and their dependencies. The comment ''Eventdependent but negligible'' for the $ p_{\mathrm{T}}^\text{miss} $ entries indicate that these corrections are small and vary on an event by event basis due to the event selection. 
Summary 
We have used the CMS detector at the LHC to measure the polarization of $ \tau $ leptons in the decay of the Z boson produced in protonproton collisions at the LHC using nearly all possible decay channels of the $ \tau $ leptons. The measured value for the $ \tau $ polarization, $ \mathcal{P}_{\tau}(Z) =  $0.144 $ \pm $ 0.006 (stat) $ \pm $ 0.014 (syst) $ =  $0.144 $ \pm $ 0.015, is in agreement with values measured by the SLD experiment, at LEP, and by the ATLAS experiment and with the standard model value of the lepton asymmetry parameter $ A_{\ell} $. The uncertainty of the presented measurement is limited by systematic uncertainties. It is more precise than the ATLAS measurement and nearly as precise as single LEP experiments. The measured polarization constrains the effective couplings of $ \tau $ leptons to the Z boson and determines the effective weak mixing angle as $ \sin^{2}\theta_{W}^\mathrm{eff}= $ 0.2319 $ \pm $ 0.0019. This result has a precision of 0.8% and is independent of the production process of the Z boson. This measurement of the $ \tau $ lepton polarization in Z boson decays demonstrates that in the complicated LHC environment the spin of $ \tau $ lepton and spin correlations of $ \tau $ lepton pairs can be determined and be used to explore new physics, for example, for measurements of the CP properties of the Higgs Yukawa coupling to $ \tau $ leptons. 
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Compact Muon Solenoid LHC, CERN 