Using the renormalization group approach, coupling constants are "running". If we apply this to the fine structure (coupling) constant, we do know that, e.g., at energies around the Z mass, $$\alpha \approx 1/128$$ instead of 1/137. We know that $$\alpha =e^2/ \hbar c$$. Therefore, if alpha is running with energy, at least one of the 3 "constants" there (electric charge, the Planck's constant or the speed of light is varying with energy). I find hard to see (due to gauge invariance) why e should vary, but the remaining options are not much better. Making c vary with energy drives us to varying speed of light theories, and I believe that energy variations of the speed of light are well bound from different experiments. A varying Planck constant? I can not see a physical meaning of it! Therefore, my question is:
HOW PEOPLE can not find "disturbing" the issue of a "running coupling constant" like alpha? And related to this: Is there some experiment to search for energy variations of the Planck constant beyond those with respect to the speed of light? An about a varying electric charge with energy? I find it difficult due to gauge invariance! So, how can people live with " a varying fine structure constant withoug being "puzzled" too much?