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CMS-SMP-23-005 ; CERN-EP-2024-127
Observation of $ {\gamma\gamma\to\tau\tau} $ in proton-proton collisions and limits on the anomalous electromagnetic moments of the $ \tau $ lepton
Submitted to Reports on Progress in Physics
Abstract: The production of a pair of $ \tau $ leptons via photon-photon fusion, $ {\gamma\gamma\to\tau\tau} $, is observed for the first time in proton-proton collisions, with a significance of 5.3 standard deviations. This observation is based on a data set recorded with the CMS detector at the LHC at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 138 fb$ ^{-1} $. Events with a pair of $ \tau $ leptons produced via photon-photon fusion are selected by requiring them to be back-to-back in the azimuthal direction and to have a minimum number of charged hadrons associated with their production vertex. The $ \tau $ leptons are reconstructed in their leptonic and hadronic decay modes. The measured fiducial cross section of $ {\gamma\gamma\to\tau\tau} $ is $ \sigma^\text{fid}_\text{obs}= $ 12.4$ ^{+3.8}_{-3.1} $ fb. Constraints are set on the contributions to the anomalous magnetic moment ($ a_{\tau} $) and electric dipole moments ($ d_{\tau} $) of the $ \tau $ lepton originating from potential effects of new physics on the $ \gamma\tau\tau $ vertex: $ a_{\tau}= $ 0.0009$_{-0.0031}^{+0.0032} $ and $ |d_{\tau}| < $ 2.9$\times $10$^{-17}$ e.cm (95% confidence level), consistent with the standard model.
Figures & Tables Summary Additional Figures References CMS Publications
Figures

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Figure 1:
Feynman diagrams for the production of $ \tau $ lepton pairs by photon-photon fusion. The exclusive (left), single proton dissociation (middle), and double proton dissociation (right) topologies are shown.

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Figure 1-a:
Feynman diagrams for the production of $ \tau $ lepton pairs by photon-photon fusion. The exclusive (left), single proton dissociation (middle), and double proton dissociation (right) topologies are shown.

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Figure 1-b:
Feynman diagrams for the production of $ \tau $ lepton pairs by photon-photon fusion. The exclusive (left), single proton dissociation (middle), and double proton dissociation (right) topologies are shown.

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Figure 1-c:
Feynman diagrams for the production of $ \tau $ lepton pairs by photon-photon fusion. The exclusive (left), single proton dissociation (middle), and double proton dissociation (right) topologies are shown.

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Figure 2:
Schematic view of the 0.1 cm wide windows probed along the $ z $ axis to derive corrections to the pileup track density in simulation. Windows within 1 cm from the dimuon vertex, illustrated with the red box, are discarded so as not to count tracks from the hard-scattering interaction. The green curve indicates the probability distribution of z-coordinates for PU vertices in the beamspot.

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Figure 3:
Distribution of $ N_\text{tracks}^\text{PU} $ in windows of 0.1 cm width along the $ z $ axis for the observed events (black), uncorrected simulation (red), and beamspot-corrected simulation (blue) for data collected in 2017. The windows shown here are located at the beamspot center (upper left), and one (upper right) or two (lower) beamspot widths away from the center. The ratio of beamspot-corrected simulation to observation (lower plots) is taken as a residual correction to the simulations. The last bin includes the overflow. Similar distributions and corrections are derived independently for the other data-taking periods.

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Figure 3-a:
Distribution of $ N_\text{tracks}^\text{PU} $ in windows of 0.1 cm width along the $ z $ axis for the observed events (black), uncorrected simulation (red), and beamspot-corrected simulation (blue) for data collected in 2017. The windows shown here are located at the beamspot center (upper left), and one (upper right) or two (lower) beamspot widths away from the center. The ratio of beamspot-corrected simulation to observation (lower plots) is taken as a residual correction to the simulations. The last bin includes the overflow. Similar distributions and corrections are derived independently for the other data-taking periods.

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Figure 3-b:
Distribution of $ N_\text{tracks}^\text{PU} $ in windows of 0.1 cm width along the $ z $ axis for the observed events (black), uncorrected simulation (red), and beamspot-corrected simulation (blue) for data collected in 2017. The windows shown here are located at the beamspot center (upper left), and one (upper right) or two (lower) beamspot widths away from the center. The ratio of beamspot-corrected simulation to observation (lower plots) is taken as a residual correction to the simulations. The last bin includes the overflow. Similar distributions and corrections are derived independently for the other data-taking periods.

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Figure 3-c:
Distribution of $ N_\text{tracks}^\text{PU} $ in windows of 0.1 cm width along the $ z $ axis for the observed events (black), uncorrected simulation (red), and beamspot-corrected simulation (blue) for data collected in 2017. The windows shown here are located at the beamspot center (upper left), and one (upper right) or two (lower) beamspot widths away from the center. The ratio of beamspot-corrected simulation to observation (lower plots) is taken as a residual correction to the simulations. The last bin includes the overflow. Similar distributions and corrections are derived independently for the other data-taking periods.

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Figure 4:
Distribution of the number of reconstructed tracks in a 0.1 cm wide window in the $ z $ direction, centered on the dimuon reconstructed vertex, for acoplanarity $ A < $ 0.015, in data collected in 2017. The DY simulation is split into several components based on the number of reconstructed tracks originating from the hard interaction. The red line shows the simulation before the correction. The black points show the observed data after subtracting the expected background contribution from the $ {\gamma\gamma\to\mu\mu} $ and $ {\gamma\gamma\to\mathrm{W}\mathrm{W}} $ processes (dashed orange line). The last bin includes the overflow. Similar distributions and corrections are derived independently for the other data-taking periods. The ratios between the observed data, from which the exclusive background contributions have been subtracted, and the DY prediction before (red) and after the corrections (black), are shown in the lower panel. The region with the selection requirement $ N_\text{tracks}= $ 0 or 1 used in the SR is highlighted with the orange shaded area in the lower panel.

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Figure 5:
Measurement of the scale factor for the elastic exclusive signal in $ \mu\mu $ events for $ N_\text{tracks}= $ 0 (left) or 1 (right), and $ A < $ 0.015. The shape of the inclusive background (blue) is estimated from the observed data in the 3 $ \leq N_\text{tracks}\leq $ 7 sideband, and rescaled to fit the observed data in 75 $ < m_{\mu\mu} < $ 105 GeV. The scale factor is fitted in the lower ratio panel with constant (red) and linear (blue) functions. The vertical error bars indicate the statistical uncertainty in the number of observed events.

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Figure 5-a:
Measurement of the scale factor for the elastic exclusive signal in $ \mu\mu $ events for $ N_\text{tracks}= $ 0 (left) or 1 (right), and $ A < $ 0.015. The shape of the inclusive background (blue) is estimated from the observed data in the 3 $ \leq N_\text{tracks}\leq $ 7 sideband, and rescaled to fit the observed data in 75 $ < m_{\mu\mu} < $ 105 GeV. The scale factor is fitted in the lower ratio panel with constant (red) and linear (blue) functions. The vertical error bars indicate the statistical uncertainty in the number of observed events.

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Figure 5-b:
Measurement of the scale factor for the elastic exclusive signal in $ \mu\mu $ events for $ N_\text{tracks}= $ 0 (left) or 1 (right), and $ A < $ 0.015. The shape of the inclusive background (blue) is estimated from the observed data in the 3 $ \leq N_\text{tracks}\leq $ 7 sideband, and rescaled to fit the observed data in 75 $ < m_{\mu\mu} < $ 105 GeV. The scale factor is fitted in the lower ratio panel with constant (red) and linear (blue) functions. The vertical error bars indicate the statistical uncertainty in the number of observed events.

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Figure 6:
Acoplanarity distribution for the observed events and in DY simulation before correction, in the 2018 data-taking period. The background prediction is normalized to match the observed yield and only the statistical uncertainty is shown. The data-to-simulation ratio is fitted with a polynomial to obtain the correction. The selection criterion $ A < $ 0.015 used in the SR is highlighted with the orange shaded area in the lower panel.

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Figure 7:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ MFs, $ \omega(N_\text{tracks}, \text{DM}^{\tau_{\mathrm{h}}}) $, in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state, in the high-$ m_{\mathrm{T}} $ (left) and SS (right) CRs, for the $ \mathrm{h}^\pm+\pi^{0} $(s) DM. The purple shaded area corresponds to the fit uncertainty. The vertical error bars indicate the statistical uncertainty in the MF correction factors measured in individual $ N_\text{tracks} $ ranges.

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Figure 7-a:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ MFs, $ \omega(N_\text{tracks}, \text{DM}^{\tau_{\mathrm{h}}}) $, in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state, in the high-$ m_{\mathrm{T}} $ (left) and SS (right) CRs, for the $ \mathrm{h}^\pm+\pi^{0} $(s) DM. The purple shaded area corresponds to the fit uncertainty. The vertical error bars indicate the statistical uncertainty in the MF correction factors measured in individual $ N_\text{tracks} $ ranges.

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Figure 7-b:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ MFs, $ \omega(N_\text{tracks}, \text{DM}^{\tau_{\mathrm{h}}}) $, in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state, in the high-$ m_{\mathrm{T}} $ (left) and SS (right) CRs, for the $ \mathrm{h}^\pm+\pi^{0} $(s) DM. The purple shaded area corresponds to the fit uncertainty. The vertical error bars indicate the statistical uncertainty in the MF correction factors measured in individual $ N_\text{tracks} $ ranges.

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Figure 8:
Postfit values of the nuisance parameters (black markers), shown as the difference of their best-fit values, $ \hat{\theta} $, and prefit values, $ \theta_0 $, relative to the prefit uncertainties $ \Delta\theta $. The horizontal error bars indicate the uncertainties in these measured postfit values. The impact $ \Delta\hat{\mu} $ of the nuisance parameter on the signal strength is computed as the difference of the nominal best fit value of $ \mu $ and the best fit value obtained when fixing the nuisance parameter under scrutiny to its best fit value $ \hat{\theta} $ plus/minus its postfit uncertainty (blue shaded area). The nuisance parameters are ordered by their impact, and only the 20 highest ranked parameters are shown.

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Figure 9:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 0, the lower panels showing the observed/expected ratio. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The minor inclusive diboson background contribution is drawn together with the DY background in the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, and $ \mu\tau_{\mathrm{h}} $ final states. The predicted background distributions correspond to the result of the global fit. The signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical, after the global fit. In the fit, $ a_{\tau} $ and $ d_{\tau} $ are fixed to their SM values. The ratio of the total predictions for an illustrative value of $ a_{\tau}= $ 0.008 to those with SM electromagnetic couplings is shown with a blue line in the lower panel of each plot.

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Figure 9-a:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 0, the lower panels showing the observed/expected ratio. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The minor inclusive diboson background contribution is drawn together with the DY background in the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, and $ \mu\tau_{\mathrm{h}} $ final states. The predicted background distributions correspond to the result of the global fit. The signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical, after the global fit. In the fit, $ a_{\tau} $ and $ d_{\tau} $ are fixed to their SM values. The ratio of the total predictions for an illustrative value of $ a_{\tau}= $ 0.008 to those with SM electromagnetic couplings is shown with a blue line in the lower panel of each plot.

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Figure 9-b:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 0, the lower panels showing the observed/expected ratio. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The minor inclusive diboson background contribution is drawn together with the DY background in the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, and $ \mu\tau_{\mathrm{h}} $ final states. The predicted background distributions correspond to the result of the global fit. The signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical, after the global fit. In the fit, $ a_{\tau} $ and $ d_{\tau} $ are fixed to their SM values. The ratio of the total predictions for an illustrative value of $ a_{\tau}= $ 0.008 to those with SM electromagnetic couplings is shown with a blue line in the lower panel of each plot.

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Figure 9-c:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 0, the lower panels showing the observed/expected ratio. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The minor inclusive diboson background contribution is drawn together with the DY background in the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, and $ \mu\tau_{\mathrm{h}} $ final states. The predicted background distributions correspond to the result of the global fit. The signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical, after the global fit. In the fit, $ a_{\tau} $ and $ d_{\tau} $ are fixed to their SM values. The ratio of the total predictions for an illustrative value of $ a_{\tau}= $ 0.008 to those with SM electromagnetic couplings is shown with a blue line in the lower panel of each plot.

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Figure 9-d:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 0, the lower panels showing the observed/expected ratio. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The minor inclusive diboson background contribution is drawn together with the DY background in the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, and $ \mu\tau_{\mathrm{h}} $ final states. The predicted background distributions correspond to the result of the global fit. The signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical, after the global fit. In the fit, $ a_{\tau} $ and $ d_{\tau} $ are fixed to their SM values. The ratio of the total predictions for an illustrative value of $ a_{\tau}= $ 0.008 to those with SM electromagnetic couplings is shown with a blue line in the lower panel of each plot.

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Figure 10:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 1. The description of the histograms is the same as in Fig. 9.

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Figure 10-a:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 1. The description of the histograms is the same as in Fig. 9.

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Figure 10-b:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 1. The description of the histograms is the same as in Fig. 9.

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Figure 10-c:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 1. The description of the histograms is the same as in Fig. 9.

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Figure 10-d:
Observed and predicted $ m_\text{vis} $ distributions in the $ \mathrm{e}\mu $ (upper left), $ \mathrm{e}\tau_{\mathrm{h}} $ (upper right), $ \mu\tau_{\mathrm{h}} $ (lower left), and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ (lower right) final states for events with $ N_\text{tracks}= $ 1. The description of the histograms is the same as in Fig. 9.

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Figure 11:
Observed and predicted $ N_\text{tracks} $ distributions for events passing the SR selection but with the relaxed requirement $ N_\text{tracks} < $ 10 and the additional requirement $ m_\text{vis} > $ 100 GeV, combining the $ \mathrm{e}\mu $, $ \mathrm{e}\tau_{\mathrm{h}} $, $ \mu\tau_{\mathrm{h}} $, and $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final states together. The acoplanarity requirement $ A < $ 0.015 is applied. The observed data and their associated Poissonian statistical uncertainty are shown with black markers with vertical error bars. The inclusive diboson background contribution is drawn together with that of the $ {\mathrm{t}\overline{\mathrm{t}}} $ process. The predicted distributions are adjusted to the result of the global fit performed with the $ m_\text{vis} $ distributions in the SRs, and the signal distribution is normalized to its best fit signal strength. The lower panel shows the difference between the observed events and the backgrounds, as well as the signal contribution. Systematic uncertainties are assumed to be uncorrelated between final states to draw the uncertainty band.

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Figure 12:
Expected and observed negative log-likelihood as a function of $ a_{\tau} $ (left) and $ d_{\tau} $ (right), for the combination of all SRs in all data-taking periods.

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Figure 12-a:
Expected and observed negative log-likelihood as a function of $ a_{\tau} $ (left) and $ d_{\tau} $ (right), for the combination of all SRs in all data-taking periods.

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Figure 12-b:
Expected and observed negative log-likelihood as a function of $ a_{\tau} $ (left) and $ d_{\tau} $ (right), for the combination of all SRs in all data-taking periods.

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Figure 13:
Measurements of $ a_{\tau} $ (left) and $ d_{\tau} $ (right) performed in this analysis, compared with previous results from the OPAL, L3, DELPHI, ARGUS, Belle, ATLAS, and CMS experiments [25,26,24,28,27,9,10]. Confidence intervals at 68 and 95% CL are shown with thick black and thin green lines, respectively. The SM values of the $ \tau $ anomalous electromagnetic moments, $ a_{\tau}=$ 1.2$\times$10$^{-3} $ and $ d_{\tau}=-$7.3$\times$10$^{-38}$ e.cm, are indicated with the dashed blue lines.

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Figure 13-a:
Measurements of $ a_{\tau} $ (left) and $ d_{\tau} $ (right) performed in this analysis, compared with previous results from the OPAL, L3, DELPHI, ARGUS, Belle, ATLAS, and CMS experiments [25,26,24,28,27,9,10]. Confidence intervals at 68 and 95% CL are shown with thick black and thin green lines, respectively. The SM values of the $ \tau $ anomalous electromagnetic moments, $ a_{\tau}=$ 1.2$\times$10$^{-3} $ and $ d_{\tau}=-$7.3$\times$10$^{-38}$ e.cm, are indicated with the dashed blue lines.

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Figure 13-b:
Measurements of $ a_{\tau} $ (left) and $ d_{\tau} $ (right) performed in this analysis, compared with previous results from the OPAL, L3, DELPHI, ARGUS, Belle, ATLAS, and CMS experiments [25,26,24,28,27,9,10]. Confidence intervals at 68 and 95% CL are shown with thick black and thin green lines, respectively. The SM values of the $ \tau $ anomalous electromagnetic moments, $ a_{\tau}=$ 1.2$\times$10$^{-3} $ and $ d_{\tau}=-$7.3$\times$10$^{-38}$ e.cm, are indicated with the dashed blue lines.

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Figure 14:
Expected and observed 95% CL constraints on the real (left) and imaginary (right) parts of the Wilson coefficients $ C_{\tau B} $ and $ C_{\tau W} $ divided by $ \Lambda^2 $. The SM value is indicated with a cross. The blue shaded areas indicate excluded regions.

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Figure 14-a:
Expected and observed 95% CL constraints on the real (left) and imaginary (right) parts of the Wilson coefficients $ C_{\tau B} $ and $ C_{\tau W} $ divided by $ \Lambda^2 $. The SM value is indicated with a cross. The blue shaded areas indicate excluded regions.

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Figure 14-b:
Expected and observed 95% CL constraints on the real (left) and imaginary (right) parts of the Wilson coefficients $ C_{\tau B} $ and $ C_{\tau W} $ divided by $ \Lambda^2 $. The SM value is indicated with a cross. The blue shaded areas indicate excluded regions.
Tables

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Table 1:
Baseline selection criteria used in the different final states. The electrons, muons, and $ \tau_{\mathrm{h}} $ are required to be well identified and isolated. The $ p_{\mathrm{T}} $ and pseudorapidity ranges correspond to different sets of triggers, and different data-taking periods.

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Table 2:
Summary of the systematic uncertainties considered in the analysis. The sources of the uncertainties, the processes they affect, and their magnitudes are indicated.

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Table 3:
Observed and predicted event yields per final state in the signal-enriched phase space with $ m_\text{vis} > $ 100 GeV and $ N_\text{tracks}= $ 0. The signal and background yields are the result of the global fit including all sources of uncertainties.

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Table 4:
Selection criteria to define the fiducial cross section. Events where the two $ \tau $ leptons decay both to electrons or to muons, with neutrinos, are considered to be outside the fiducial region. All requirements are applied using generator-level quantities, as detailed in the text.
Summary
The photon-fusion production of a pair of $ \tau $ leptons, $ {\gamma\gamma\to\tau\tau} $, has been observed for the first time in proton-proton collisions, with a significance of 5.3 standard deviations. The $ \tau $ leptons are reconstructed in their leptonic and hadronic decay modes. The signal has been identified by requiring low track activity around the di-$ \tau $ vertex and low azimuthal acoplanarity between the $ \tau $ candidates. Data in a control region with two muons were used to determine corrections for the simulations to accurately model the track multiplicity and to predict the signal contribution in the final state of two $ \tau $ leptons. The signal strength, fiducial cross section, and constraints on the anomalous electromagnetic moments of the $ \tau $ lepton have been extracted using the di-$ \tau $ invariant mass distributions in four di-$ \tau $ final states. The measured fiducial cross section of $ {\gamma\gamma\to\tau\tau} $ is $ \sigma^\text{fid}_\text{obs}= $ 12.4$ ^{+3.8}_{-3.1} $ fb. The anomalous $ \tau $ magnetic moment is determined to be $ a_{\tau}= $ 0.0009$_{-0.0031}^{+0.0032} $, whereas the electric dipole moment of the $ \tau $ lepton is constrained to $ |d_{\tau}| < $ 2.9$\times $10$^{-17}$ e.cm at 95% confidence level. They are both in good agreement with the predictions of the standard model of particle physics, and the measurements do not show any evidence for the presence of new physics that would modify the electromagnetic moments of the $ \tau $ lepton. This is the most stringent limit on the $ \tau $ lepton magnetic moment to date, improving on the previous best constraints by nearly an order of magnitude.
Additional Figures

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Additional Figure 1:
Event weights applied to simulations in the 2016 pre-VFP data-taking period as a function of the dilepton vertex position along the $ z $ axis and the pileup track multiplicity in a 0.1 cm-wide window around the dilepton vertex, where VFP stands for preamplifier feedback bias corrections due to inefficiencies in the strip modules of the tracker during the 2016 data-taking period.

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Additional Figure 2:
Event weights applied to simulations in the 2016 post-VFP data-taking period as a function of the dilepton vertex position along the $ z $ axis and the pileup track multiplicity in a 0.1 cm-wide window around the dilepton vertex, where VFP stands for preamplifier feedback bias corrections due to inefficiencies in the strip modules of the tracker during the 2016 data-taking period.

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Additional Figure 3:
Event weights applied to simulations in the 2017 data-taking period as a function of the dilepton vertex position along the $ z $ axis and the pileup track multiplicity in a 0.1 cm-wide window around the dilepton vertex.

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Additional Figure 4:
Event weights applied to simulations in the 2018 data-taking period as a function of the dilepton vertex position along the $ z $ axis and the pileup track multiplicity in a 0.1 cm-wide window around the dilepton vertex.

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Additional Figure 5:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 6:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 7:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 8:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 9:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 10:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 11:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 12:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 13:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 14:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the high-$ m_{\mathrm{T}} $ CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 15:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 16:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm+\pi^{0} $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

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Additional Figure 17:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 18:
Multiplicative $ N_\text{tracks} $-dependent corrections to the $ \tau_{\mathrm{h}} $ misidentification factors in the $ \mu\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 19:
Multiplicative $ N_\text{tracks} $-dependent corrections to the leading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 20:
Multiplicative $ N_\text{tracks} $-dependent corrections to the leading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm+\pi^{0} $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 21:
Multiplicative $ N_\text{tracks} $-dependent corrections to the leading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 22:
Multiplicative $ N_\text{tracks} $-dependent corrections to the leading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 23:
Multiplicative $ N_\text{tracks} $-dependent corrections to the subleading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 24:
Multiplicative $ N_\text{tracks} $-dependent corrections to the subleading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm+\pi^{0} $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 25:
Multiplicative $ N_\text{tracks} $-dependent corrections to the subleading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm $ decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 26:
Multiplicative $ N_\text{tracks} $-dependent corrections to the subleading $ \tau_{\mathrm{h}} $ misidentification factors in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state in the same-sign CR, for the $ h^\pm h^\mp h^\pm+\pi^{0} $(s) decay mode. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 27:
OS-to-SS scale factors used to estimate the mis-ID background in the $ \mathrm{e}\mu $ final state. They are measured in an $ \mathrm{e}\mu $ CR with inverted muon isolation.

png pdf
Additional Figure 28:
Correction to the OS-to-SS scale factors in the $ \mathrm{e}\mu $ final state to account for the inversion of the muon isolation. They are measured as the ratio of the OS-to-SS scale factors measured in these two CRs: a CR with inverted electron isolation and nominal muon isolation, and a region with inverted electron and muon isolations.

png pdf
Additional Figure 29:
Multiplicative $ N_\text{tracks} $-dependent correction to the OS-to-SS scale factors used to estimate the jet mis-ID background in the $ \mathrm{e}\mu $ final state. The cyan shaded area corresponds to the fit uncertainty. This correction was measured for the 2018 data-taking period and corrections in the other data-taking periods are similar. The cyan shaded area corresponds to the fit uncertainty.

png pdf
Additional Figure 30:
Observed and predicted $ N_\text{tracks} $ distributions in the $ \mathrm{e}\mu $ final state for events passing the SR selection with the additional requirement $ m_\text{vis} < $ 100 GeV. The inclusive diboson background contribution is drawn together with the $ {\mathrm{t}\overline{\mathrm{t}}} $ process. The predicted distributions are adjusted to the result of the global fit performed with the $ m_\text{vis} $ distributions in the SRs, and the signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical.

png pdf
Additional Figure 31:
Observed and predicted $ N_\text{tracks} $ distributions in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state for events passing the SR selection with the additional requirement $ m_\text{vis} < $ 100 GeV. The inclusive diboson background contribution is drawn together with the $ {\mathrm{t}\overline{\mathrm{t}}} $ process. The predicted distributions are adjusted to the result of the global fit performed with the $ m_\text{vis} $ distributions in the SRs, and the signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical.

png pdf
Additional Figure 32:
Observed and predicted $ N_\text{tracks} $ distributions in the $ \mu\tau_{\mathrm{h}} $ final state for events passing the SR selection with the additional requirement $ m_\text{vis} < $ 100 GeV. The inclusive diboson background contribution is drawn together with the $ {\mathrm{t}\overline{\mathrm{t}}} $ process. The predicted distributions are adjusted to the result of the global fit performed with the $ m_\text{vis} $ distributions in the SRs, and the signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical.

png pdf
Additional Figure 33:
Observed and predicted $ N_\text{tracks} $ distributions for events in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state passing the SR selection with the additional requirement $ m_\text{vis} < $ 100 GeV. The predicted distributions are adjusted to the result of the global fit performed with the $ m_\text{vis} $ distributions in the SRs, and the signal distribution is normalized to its best fit signal strength. The uncertainty band accounts for all sources of background and signal uncertainty, systematic as well as statistical.

png pdf
Additional Figure 34:
Observed negative log-likelihood scans as a function of the signal strength $ \mu $, assuming SM values for $ a_{\tau} $ and $ d_{\tau} $, for the combination of all SRs in all data-taking periods. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 35:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in all data-taking periods. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 36:
Observed negative log-likelihood scans as a function of $ d_{\tau} $, for the combination of all SRs in all data-taking periods. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 37:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in the $ \mathrm{e}\mu $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 38:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values. The double-minimum structure corresponds to an excess of observed events that can be described by non-zero value of $ \deltaa_{\tau} $, where BSM effects only moderately depend on the sign of $ \deltaa_{\tau} $.

png pdf
Additional Figure 39:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in the $ \mu\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 40:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 41:
Observed negative log-likelihood scans as a function of $ d_{\tau} $, for the combination of all SRs in the $ \mathrm{e}\mu $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 42:
Observed negative log-likelihood scans as a function of $ a_{\tau} $, for the combination of all SRs in the $ \mathrm{e}\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values. The double-minimum structure corresponds to an excess of observed events that can be described by BSM values of $ d_{\tau} $, where BSM effects do not depend on the sign of $ d_{\tau} $.

png pdf
Additional Figure 43:
Observed negative log-likelihood scans as a function of $ d_{\tau} $, for the combination of all SRs in the $ \mu\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 44:
Observed negative log-likelihood scans as a function of $ d_{\tau} $, for the combination of all SRs in the $ \tau_{\mathrm{h}}\tau_{\mathrm{h}} $ final state. The scan with statistical uncertainty in data only is shown with the dashed blue line while the scan including all uncertainties is shown with the solid black line. For the fit with statistical uncertainty only, the nuisance parameters are frozen to their best-fit values.

png pdf
Additional Figure 45:
Acoplanarity distribution in data and in simulation before correction, in the 2018 data-taking period. The background prediction is normalized to match the data yield and only the statistical uncertainty is shown.
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Compact Muon Solenoid
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