Gauge Couplings calculated from Multiple Point Criticality yield ${\alpha }^{-1}=136{.}_8\pm 9$: At Last the Elusive Case of $U(1)$

Abstract

We calculate the $U(1)$ continuum gauge coupling using the values of action parameters at the multiple point in the phase diagram of a lattice gauge theory. The multiple point is where a maximum number of phases convene. We obtain for the running inverse finestructure constant the values ${\alpha }_1^{-1}=56\pm 5$ and ${\alpha }_1^{-1}=99\pm 5$ at respectively the Planck scale and the $M_Z$ scale. The gauge group underlying the phase diagram in which we seek multiple point values of action parameters is what we call the Anti Grand Unified Theory (AGUT) gauge group $SM{G}^3$ which is the Cartesian product of 3 standard model groups (SMGs). There is one SMG factor for each of the $N_{gen}=3$ generations of quarks and leptons. In our model, this gauge group $SM{G}^3$ is the predecessor to the usual standard model group. The latter arises as the diagonal subgroup surviving the Planck scale breakdown of $SM{G}^3.Thisbreakdownleadstoaweakeningofthe$U(1)$couplingbya$N_{gen}$-relatedfactor.Themostimportantcorrectionobtainedfromusingmultiplepointparametervalues(inamulti-parameterphasediagraminsteadofthesinglecriticalparametervalueobtainedsayinthe1-dimensionalphasediagramofaWilsonaction)comesfromincludingtheinfluenceofhavingphasesconfinedsolelyw.r.t.discretesubgroups.Inparticular,whatmattersisthatthedegreeoffirst-ordernessistakenintoaccountinmakingthetransitionfromtheselatterphasesatthemultiplepointtothetotallyCoulomb-likephase.Combinedwiththeresultsofearlierworkonthenon-Abeliangaugecouplings,theresultspresentedhereleadtoourpredictionof$\alpha^{-1}=136._8\pm 9$ as the value for the fine-structure constant at low energies.

Comments: 4 figures, 85 pages, accepted for publication in Intl. J. Mod. Phys. A, this is a second replacement that should correct the lack of references in the original version