Vector-like quarks (VLQs) that are partners of the heavy top and bottom quarks are predicted in many extensions of the Standard Model (SM). We explore the possibility that these states could explain not only the longstanding anomaly in the forward-backward asymmetry in $b$-quark production at LEP, $A_{rm FB}^b $, but also the more recent $sim 2sigma$ deviation of the cross section for the associated Higgs production with top quark pairs at the LHC, $sigma(ppto tbar t H)$. Introducing three illustrative models for VLQs with different representations under the SM gauge group, we show that the two anomalies can be resolved while satisfying all other theoretical and experimental constraints. In this case, the three different models predict VLQ states in the $1-2$ TeV mass range that can be soon probed at the LHC. In a second step, we discuss the sensitivity on the VLQ masses and couplings that could be obtained by means of a percent level accuracy in the measurement of ratios of partial Higgs decay widths, in particular $Gamma(H ! to! gammagamma)/Gamma(H ! to! ZZ^*)$ and $Gamma(H ! to ! bbar b)/Gamma(H ! to ! WW^*)$. We show that top and bottom VL partners with masses up to $sim 5$ TeV and exotic VLQs with masses in the $10$ TeV range can be probed at the high-luminosity LHC.
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