Invalidation and Proof of the Mass Gap and Viability of The Standard Model on a Discrete Spacetime
Stéphane Maes
Abstract
<em>The Yang Mills Mass gap is often considered as phenomenologically solved, after all QCD does not halt, but mathematically, it remains an open problem, that must be theoretically proven, in order to axiomatically reconstruct Yang Mills, and prove its suitability. The mathematical problem has been elevated to one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of USD1,000,000 for its resolution. </em> <em> </em> <em>In past papers, we argued its resolution in a multi-fold universe with its discrete spacetime, relying on existing proofs in Lattice QCD (Quantum Chromodynamic). It works for discrete spacetime, but was not definitive in the continuous spacetime limit. In this paper, we revisit the standard model on lattices, in particular in the context of the Nielsen–Ninomiya theorem that identifies challenges for modeling both the Weak interaction and QCD on lattices, and thus the Standard Model (SM). </em> <em> </em> <em>Instead of aiming at solving what happens in the continuous limits, we rather treat the continuous case as a larger scale approximation of SM on a discrete spacetime. The paper studies Yang Mills fields and SM viability on discrete spacetimes besides continuous spacetime, and that includes the mass gap. Indeed, in previous papers we showed that General Relativity (GR) prescribes a discrete spacetime. Therefore, the mass gap and the Nielsen–Ninomiya theorem challenges must be addressed for SM in GR-based universes, which, a priori, includes our real universe.</em> <em> </em> <em>Doing so, the paper derives properties like asymptotic freedom, confinement and chiral symmetry breaking in non-abelian 4D Yang Mills theories. It also provides a new understanding of the link between confinement, and the chiral symmetry breaking, which typically occur around when confinement occurs, and explains the differences for abelian fields. These results are valid for discrete and continuous spacetime.</em> <em> </em> <em>In addition, the paper invalidates the mass gap conjecture and its Millennium prize formulation for 4D continuous spacetime by explaining that it can’t guarantee a mass gap. With reformulation, the mass gap Millennium Prize Problem is generally correct only in a 4D discrete spacetime, or in a multi-fold universe.</em> <em> </em> <em>SM<sub>G</sub> denotes the SM with gravity effects are not negligible at its scales. The present work gives new arguments for SM<sub>G</sub>: Yang Mills, QCD, and SM. The results also reinforce arguments for a (fundamental) particle desert above the electroweak scale, where the Ultimate Unification (UU) is encountered, and GUTs do not exist. The mass gap is generically proven for 2D random walk multi-fold universes, which characterize all of Physics.</em> <em> </em>