Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the fermion sector

In the current paper the properties of a quantum field theory based on certain sets of Lorentz-violating coefficients in the nonminimal fermion sector of the Standard Model extension are analyzed. In particular, three families of coefficients are considered, where two of them are $CPT$ even and the third is $CPT$ odd. As a first step the modified fermion dispersion relations are obtained. Then the positive- and negative-energy solutions of the modified Dirac equation and the fermion propagator are derived. These are used to demonstrate the validity of the optical theorem at tree level, which provides a cross-check for the results obtained. Furthermore unitarity is examined and seems to be valid for the first set of $CPT$-even coefficients. However for the remaining sets certain issues with unitarity are found. The article demonstrates that the adapted quantum field theoretical methods at tree level work for the nonminimal, Lorentz-violating framework considered. Besides, the quantum field theory based on the first family of $CPT$-even coefficients is most likely well behaved at lowest order perturbation theory. The results are important for future phenomenological investigations carried out in the context of field theory, e.g., the computation of decay rates and cross sections at tree level.

Figure 1

Forward scattering amplitude of tree-level electron photon scattering that is related to the total cross section of electron photon scattering, if the optical theorem is valid. A modified electron is denoted by ${\tilde{\mathrm{e}}}^{-}$ and a photon by $\gamma$. The momenta are stated next to the particle symbols and the one-particle phase space is called $\mathrm{d}{\mathrm{\Pi }}_1$.

Figure 2

Plots of the integrand of Eq. (6.29) as a function of $p^4$ and $p$ with the choices $m_{\psi }=1$, $x^4=1$ for distinct values of the Lorentz-violating component coefficient; (a) for $c^{(6)0000}=0$ and (b) for $c^{(6)0000}=10^{-5}$.

Figure 3

Plots of the integrand of Eq. (6.51) as a function of $p^4$ and $p$ with the choices $m_{\psi }=1$, $x^4=1$ for distinct values of the Lorentz-violating component coefficient: (a) for $f^{(6)000}=0$ and (b) for $f^{(6)000}=10^{-5}$.