Planck-scale modifications to electrodynamics characterized by a spacelike symmetry-breaking vector

In the study of Planck-scale (“quantum-gravity-induced”) violations of Lorentz symmetry, an important role was played by the deformed-electrodynamics model introduced by Myers and Pospelov. Its reliance on conventional effective quantum field theory, and its description of symmetry-violation effects simply in terms of a four-vector with a nonzero component only in the time direction, rendered it an ideal target for experimentalists and a natural concept-testing ground for many theorists. At this point however the experimental limits on the single Myers-Pospelov parameter, after improving steadily over these past few years, are “super-Planckian”; i.e. they take the model out of actual interest from a conventional quantum-gravity perspective. In light of this we here argue that it may be appropriate to move on to the next level of complexity, still with vectorial symmetry violation but adopting a generic four-vector. We also offer a preliminary characterization of the phenomenology of this more general framework, sufficient to expose a rather significant increase in complexity with respect to the original Myers-Pospelov setup. Most of these novel features are linked to the presence of spatial anisotropy, which is particularly pronounced when the symmetry-breaking vector is spacelike, and they are such that they reduce the bound-setting power of certain types of observations in astrophysics.

Figure 1

Behavior of the longitudinal component of the field (dotted line), and of the nonclassical part of the dispersion law (continuous line) as functions of $\frac{\theta }{\pi }$. For the behavior of the dispersive effects we simply show (up to an irrelevant overall factor introduced for visibility) the function $(n_0+|\overrightarrow{n}|\cos (\theta ){)}^3$, which indeed gives the dependence of these effects on the propagation direction, and similarly for the longitudinal component we show (up to another irrelevant factor introduced for visibility) $(n_0+|\overrightarrow{n}|\cos (\theta ){)}^2\sin (\theta )$, which indeed gives the dependence of the longitudinal component on the propagation direction. For panel (a) we took $|\overrightarrow{n}|=1$, $n_0=0.1$; for panel (b) we took $|\overrightarrow{n}|=1$, $n_0=0.5$; and for panel (c) we took $|\overrightarrow{n}|=1$, $n_0=0.9$.

Figure 2

In (a) we show the same case already shown in Fig. 1c ($|\overrightarrow{n}|=1$, $n_0=0.9$), but in a logarithmic plot which allows one to better appreciate the sizable partial blindness present between $\theta \simeq 0.8\pi$ to $\theta \simeq \pi$. [Notice that, this being a logarithmic plot, we characterize the dispersive effects by the absolute value of $(n_0+|\overrightarrow{n}|\cos (\theta ){)}^3$, whereas Fig. 1c showed also the behavior of the sign of $|(n_0+|\overrightarrow{n}|\cos (\theta ){)}^3|$.] We give in (b) an intuitive quantitative characterization (polar plot of one-half of the sky) of the portions of the sky that would typically have a certain blindness level, still for the case $|\overrightarrow{n}|=1$, $n_0=0.9$. The directions of “total blindness” form a circle in the sky, and for any given “level of blindness” one has an associated circular crown of corresponding thickness. Interestingly nearly the whole of the half of the sky shown in (b) has blindness amounting to a least suppression by a factor of 10, and the portion of the sky with blindness of ${10}^{-4}$ is evidently non-negligible. Note however that (b) was drawn assuming one is looking in the direction of $\overrightarrow{n}$, so essentially concerns $\pi /2\le \theta \le \pi$, which for $n_0>0$ is the portion of the sky where most “blindness” is found.