A search is presented for a third-generation leptoquark (LQ) coupled exclusively to a $\tau$ lepton and a b quark. The search is based on proton-proton collision data at a center-of-mass energy of 13 TeV recorded with the CMS detector, corresponding to an integrated luminosity of 138 fb$^{-1}$. Events with $\tau$ leptons and a varying number of jets originating from b quarks are considered, targeting the single and pair production of LQs, as well as nonresonant $t$-channel LQ exchange. An excess is observed in the data with respect to the background expectation in the combined analysis of all search regions. For a benchmark LQ mass of 2 TeV and an LQ-b-$\tau$ coupling strength of 2.5, the excess reaches a local significance of up to 2.8 standard deviations. Upper limits at the 95% confidence level are placed on the LQ production cross section in the LQ mass range 0.5-2.3 TeV, and up to 3 TeV for $t$-channel LQ exchange. Leptoquarks are excluded below masses of 1.22-1.88 TeV for different LQ models and varying coupling strengths up to 2.5. The study of nonresonant $\tau\tau$ production through $t$-channel LQ exchange allows lower limits on the LQ mass of up to 2.3 TeV to be obtained.
Product of acceptance and efficiency of a vector LQ signal as a function of LQ mass under the assumption of exclusive LQ couplings to b quarks and $\tau$ leptons. The acceptances and efficiencies are restricted to the sensitive region of $S_\mathrm{T}^\mathrm{MET} > 800\,\mathrm{GeV}$ and are computed with respect to all possible decay modes of two $\tau$ leptons.
Product of acceptance and efficiency of a vector LQ signal as a function of LQ mass under the assumption of exclusive LQ couplings to b quarks and $\tau$ leptons. The acceptances and efficiencies are restricted to the sensitive region of $\chi < 4$ and are computed with respect to all possible decay modes of two $\tau$ leptons.
Postfit distributions of $S_\mathrm{T}^\mathrm{MET}$ in the $\mathrm{e}\mu$ channel of the 0b category for the combined 2016-2018 data set after a simultaneous fit of the background and vector LQ signal to the data. The number of events in each bin are divided by the respective bin width. The last bin includes the overflow.
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