What constant varies in the fine structure constant? 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 $$\alpha =K_Ce^2/ \hbar c$$, or using units with $K_C=1$, 
$$\alpha=\dfrac{e^2}{\hbar c}$$ 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?
 A: First of all is the definition of coupling constant:

In physics, a coupling constant, usually denoted g, is a number that determines the strength of the force exerted in an interaction. Usually, the Lagrangian or the Hamiltonian of a system describing an interaction can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. 

It is called constant because of historical reasons. It is a number that enters in the expansion to a mathematical series of a calculation of cross sections using Feynman diagrams, when we assume the interaction to be small to allow for a converging series expansion. It was neat that in electromagnetic diagrams it was a real constant, alpha.
There is no reason though to expect that the nice expression for alpha is also the exact expression needed in the final form of the Feynman series expansion of the cross section.
It was realized that the coefficients of the series had a calculable  energy dependence.  Look for the format in the link, page 9.
Thus alpha is not changed. The running is on the coefficient of expansion, quite legitimate, versus the energy for which the expansion is made. Physicists who have taken a course on this have no problem with the definition.
A: This is different from the possible time-variation of the low-energy fine-structure constant, but the same considerations apply when you try to attribute any such variation in $\alpha$ to a variation in $e$, $\hbar$, or $c$. You can't. See Duff, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093 . I also don't see why it would be disturbing. It represents the fact that a charge is "dressed" with a cloud of virtual particles.
