Emergence of a universal limiting speed

We display several examples of how fields with different limiting velocities (the “speed of light”) at a high energy scale can nevertheless have a common limiting velocity at low energies due to the effects of interactions. We evaluate the interplay of the velocities through the self-energy diagrams and use the renormalization group to evolve the system to low energy. The differences normally vanish only logarithmically, so that an exponentially large energy trajectory is required in order to satisfy experimental constraints. However, we also display a model in which the running is a power-law type, which could be more phenomenologically useful. The largest velocity difference should be in the system with the weakest interaction, which suggests that the study of the speed of gravitational waves would be the most stringent test of this phenomenon.

Figure 1

The self-energy and vertex diagrams. Only self-energies will contribute to the running of the speeds, while the vertex is needed for the running of the coupling strength.

Figure 2

The smallest eigenvalue of $J$ (vertical) against the number of fermions $N_f$ (horizontal) taking $N_b=1$. The calculations are based on values of $z_{ab}^i$ between 0 and 1 generated randomly at each $N_f$. The plot shows that $\lambda \propto N_f$ for large $N_f$.

Figure 3

Numerical simulation of the running of the charge $e$, and ratio $\eta =1-c_f/c_g$. The horizontal axis is in units of $\log (\mu /M_f)$. We take $M_g=M_f=1$, ${\Gamma }_g={10}^{-3}$, ${\Gamma }_f=5$, ${\alpha }_f={10}^{-2}$, ${\mu }_{*}/M_f={10}^6$, and ${\alpha }_g=1$, 1.2, and 2 for the dashed, dot-dashed, and continuous lines, respectively. We also use the initial conditions $c_{f_{*}}=0.6$, $c_{g_{*}}=0.3$, and $e_{*}=0.5$. The running of $e$ is logarithmic, while the running of $\eta$ is power law. Very small values of $\eta$ are achieved in a very short interval of running as we increase ${\alpha }_g$, i.e. as we increase the number of gauge sectors. We also note that for ${\alpha }_g=2$ the value of $\eta$ is saturated by the error tolerance of the code. More powerful computations may give smaller values.

Figure 4

The allowed parameter space (number of species) for ${\mu }_{*}/{\mu }_{\mathrm{IR}}={10}^3$ (left panel) and ${10}^{16}$ (right panel). The horizontal and vertical axes are, respectively, $N_f$ and $N_g$. We take $e_{\mathrm{IR}}^2/4\pi \approx 1/129$, and then we solve for $e_{*}$ from the first formula in Eq. (63). In all cases we check that $e_{*}^2<4\pi$ so we can trust our perturbation theory even in the UV. The red region is the intersection between the yellow and green areas. The green (upper) region represents the parameters that satisfy $\eta \cong |\frac{c_g({\mu }_{\mathrm{IR}})-c_f({\mu }_{\mathrm{IR}})}{c_{g_{*}}-c_{f_{*}}}|\cong (\frac{e_{\mathrm{IR}}^2}{e_{*}^2}{)}^{2N_g/N_f}<{10}^{-14}$, while parameters in the yellow (lower) region ensure the validity of perturbation theory, i.e. $N_f{N}_g<16{\pi }^2/e_{*}^2$.