Columnar Recasting and High-Luminosity LHC Projections for a Light $L_\mu - L_\tau$ Gauge Boson in the Four-Muon Channel

Abstract

We present an independent recasting of light vector gauge boson ($Z'$) searches at the LHC within the leptophilic $U(1)_{L_\mu - L_\tau}$ framework. A simplified Standard Model extension is implemented in FeynRules, enforcing symmetric chiral leptonic currents while suppressing hadronic couplings. Signal generation for the four-muon ($4\mu$) final state is performed using MadGraph5_aMC@NLO, Pythia8, and a tuned Delphes3 detector emulation, covering the mass range 5-62 GeV under a reference coupling g=0.01. The analysis employs a vectorized columnar pipeline in Coffea with Awkward Array masks, enabling parallelized kinematic selections and the recovery of boosted muon topologies. Signal yields are statistically extracted via pyhf against digitized CMS Run 2 backgrounds (77.3 fb$^{-1}$). The resulting expected median limit profile tracks the absolute experimental sensitivity morphology within a factor of $\approx 2.5$, a discrepancy successfully quantified as a systematic consequence of fast-simulation detector limitations and the mass-window binned counting strategy relative to experimental unbinned multivariate profiling, thereby establishing a rigorous and conservative baseline sensitivity floor. Furthermore, we project the discovery potential for the future High-Luminosity LHC (HL-LHC) phase at $\mathcal{L}_{\text{int}} = 3000$ fb$^{-1}$ under the official CERN Scenario II recommendations. We demonstrate quantitatively that the HL-LHC will achieve exceptional expected local statistical discovery significances between 10.22$\sigma$ and 16.61$\sigma$ within the 5-28 GeV mass window, comfortably clearing the discovery threshold and establishing the multi-muon channel as a primary probe for lightweight vector portals in upcoming collider phases.