Gravity, Lorentz Violation, and the Standard Model

TL;DR

The paper develops a comprehensive framework for gravity with Lorentz and CPT violation by embedding the SME in a general Riemann-Cartan spacetime. It constructs the leading-order gravitational and matter-sector terms, showing that explicit Lorentz violation clashes with RC geometry through Bianchi identities, whereas spontaneous Lorentz violation is compatible. The QED extension in curved spacetime is detailed, with explicit fermion and photon LV terms, and the full SME action—including gravity, matter, and gauge sectors—is presented in this background. The results illuminate how curvature and torsion modify LV coefficients, offer paths to experimental tests of Planck-scale physics at low energies, and point toward possible extensions beyond Riemann-Cartan geometry when LV is explicit.

Abstract

The role of the gravitational sector in the Lorentz- and CPT-violating Standard-Model Extension (SME) is studied. A framework is developed for addressing this topic in the context of Riemann-Cartan spacetimes, which include as limiting cases the usual Riemann and Minkowski geometries. The methodology is first illustrated in the context of the QED extension in a Riemann-Cartan background. The full SME in this background is then considered, and the leading-order terms in the SME action involving operators of mass dimension three and four are constructed. The incorporation of arbitrary Lorentz and CPT violation into general relativity and other theories of gravity based on Riemann-Cartan geometries is discussed. The dominant terms in the effective low-energy action for the gravitational sector are provided, thereby completing the formulation of the leading-order terms in the SME with gravity. Explicit Lorentz symmetry breaking is found to be incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this difficulty.