We investigate the phenomenology of scalar diquarks with sub-TeV masses within the framework of the $E_6$ Supersymmetric Standard Model (E$_6$SSM) at the Large Hadron Collider (LHC). Focusing on the lightest of the six diquarks predicted by the model, we select some representative low masses for them in a parameter space region consistent with experimental constraints from direct searches for additional Higgs boson(s), Cold Dark Matter (CDM), and supersymmetry, as well as from flavor physics analyses. Using Monte Carlo ($MC$) simulations, we assess these benchmark points against the latest LHC results corresponding to an integrated luminosity of 140 fb$^{-1}$. We further evaluate the signal significance of the pair-production of these diquarks, when each of them decays into $tb$ pairs, at the $\sqrt{s}=13$ TeV LHC Run 3 with design integrated luminosity of 300 fb$^{-1}$, and also at the 3000 fb$^{-1}$ High-Luminosity LHC (HL-LHC). Our analysis yields a statistical significance exceeding $3\sigma$ at the HL-LHC for diquark masses up to 1 TeV, indicating promising prospects for their discovery.

Zero modes are an essential part of topological field theories, but they are frequently also an obstacle to the explicit evaluation of the associated path integrals. In order to address this issue in the case of Ray-Singer Torsion, which appears in various topological gauge theories, we introduce a massive variant of the Ray-Singer Torsion which involves determinants of the twisted Laplacian with mass but without zero modes. This has the advantage of allowing one to explicitly keep track of the zero mode dependence of the theory. We establish a number of general properties of this massive Ray-Singer Torsion. For product manifolds $M=N \times S^1$ and mapping tori one is able to interpret the mass term as a flat $\mathbb{R}_{+}$ connection and one can represent the massive Ray-Singer Torsion as the path integral of a Schwarz type topological gauge theory. Using path integral techniques, with a judicious choice of an algebraic gauge fixing condition and a change of variables which leaves one with a free action, we can evaluate the torsion in closed form. We discuss a number of applications, including an explicit calculation of the Ray-Singer Torsion on $S^1$ for $G=PSL(2,R)$ and a path integral derivation of a generalisation of a formula of Fried for the torsion of finite order mapping tori.