Violation of Fundamental Symmetries

Abstract

Gravity is the only known interaction which explicitly contains an energy scale, namely the Planck mass $M_\mathrm{Pl}$, and is thus non-renormalizable . General relativity is thus most likely an effective interaction which should be the classical low-energy limit of a more fundamental description at energies approaching the Planck scale, generally known as quantum gravity . Likely certain symmetries of ordinary quantum field theories such as the Lorentz symmetry and the discrete CPT symmetry are broken in quantum gravity and relic effects of such symmetry breaking will propagate down to low energies where they can be described within an effective field theory approach by non-renormalizable operators that are suppressed by some power of the Planck scale. In addition, renormalizable Lorentz-symmetry breaking operators can also be produced, for example, through loop diagrams. Finally, interactions induced by the exchange of light scalars or pseudo-scalars suppressed by a high energy scale can lead to fifth forces that violate the equivalence principle . In the present chapter we will describe possible phenomenological effects of such violations of fundamental symmetries and principles and how they can be constrained.

Problems

16.1

Dispersion Relations for Dimension Five Lorentz Symmetry Violations

Derive the dispersion relations Eq. (16.2) for the Standard electromagnetic Lagrangian Eq. (2.194) modified by the additional LIV terms Eq. (16.1). Hint: Derive the equations of motion for the modified Lagrangian and use plane waves as an ansatz to solve them.