LHC Constraints on a $(B-L)_3$ Gauge Boson

In this paper, we explore the constraints that the LHC can place on a massive gauge boson $X$ that predominantly couples to the third generation of fermions. Such a gauge boson arises in scenarios where the $B-L$ of the third generation is gauged. We focus on the mass range $10 \leq m_X \lesssim 2\,m_W$, where current constraints are lacking, and develop a dedicated search strategy. For this mass range, we show that $b \bar b \tau^+ \tau^-$, where at least one of the $\tau$s decay leptonically is the optimal channel to look for the $X$ at the LHC. The QCD production of $b$ quarks, combined with the cleanliness of the leptons coming from the decay of the $\tau$ allow us to detect $X$ gauge boson with couplings of $g_X \sim(0.005-0.01)$, for $m_X < 50 \ \text{GeV}$, and a coupling of $O(0.1)$ for heavier $X$ gauge boson with $100 \ \text{fb}^{-1}$ of integrated luminosity. This is about a factor of 2-10 improvement over previous constraints coming from the decay of $\Upsilon \to \tau^+ \tau^-$. Extrapolating to the full HL-LHC luminosity of $3000 \ \text{fb}^{-1}$, the bounds on $g_X$ can be enhanced by another factor of $\sqrt{2}$ for $m_X < 50 \ \text{GeV}$.