Unification of Gravity and Standard Model: Weyl-Dirac-Born-Infeld action

Abstract

We construct a unified (quantum) description, by the gauge principle, of gravity and Standard Model (SM), that generalises the Dirac-Born-Infeld action to the SM and Weyl geometry, hereafter called Weyl-Dirac-Born-Infeld action (WDBI). The theory is formulated in $d =4-2\epsilon$ dimensions. The WDBI action is a general gauge theory of SM and Weyl group (of dilatations and Poincar\'e symmetry), in the Weyl gauge covariant (metric!) formulation of Weyl geometry. The theory is SM and Weyl gauge invariant in $d=4-2\epsilon$ dimensions and there is no Weyl anomaly. The WDBI action has the unique elegant feature, not present in other gauge theories or even in string theory, that it is mathematically well-defined in $d=4-2\epsilon$ dimensions with no need to introduce in the action a UV regulator scale or field. This action actually {\it predicts} that gravity, through (Weyl covariant) space-time curvature $\hat R$, acts as UV regulator of both SM and gravity in $d=4$. A series expansion of the WDBI action (in dimensionless couplings) recovers in the leading order a Weyl gauge invariant version of SM and the Weyl (gauge theory of) quadratic gravity. The SM and Einstein-Hilbert gravity are recovered in the Stueckelberg broken phase of Weyl gauge symmetry, which restores Riemannian geometry below Planck scale. Sub-leading orders are suppressed by powers of (dimensionless) gravitational coupling ($\xi$) of Weyl quadratic gravity.