In my book of QCD, when talking about renormalization, the author mentions the difference between QED and QCD: In QED, the interaction paramenter $\alpha$ of renormalization factor $\mu$ and renormalization scale $M$ looks like $$\alpha(\mu)=\alpha_B(M)(1+$\beta_0\alpha_B(M)\ln\frac{\mu}{M}+\beta_0^2\alpha_B^2\ln^2\frac{\mu}{M}-...)=\frac{\alpha_B(\mu)}{1-\beta_0\alpha_B(\mu)\ln\frac{\mu}{M}}$$ $\alpha_B$ is bare parameter. This expansion follows from the expansions in number of loops in feynman diagrams.

Now in QCD, the author does not state the $\alpha$ expansion explicitly, but rather begins with the quantity $$R(Q)=\frac{\sigma(Q,ee\rightarrow\text{hadrons})}{\sigma(Q,ee\rightarrow\mu\mu)}=3(\sum e_i^2)(1-r(Q))$$ where the sum is said is from Quark parton model.
From the expansion of $r(q)$ into series(taylor?) he then proceeds to deduce the equation for beta function $$\frac{d\alpha(\mu, c_i)}{d\ln\mu}=-b\alpha(\mu, c_i)^2(1+c_0\alpha(\mu,c_i)+c_1\alpha^2(\mu,c_i)+c_2\alpha^3(\mu,c_i)+...)$$ where constants $c_i$ are said to be completely arbitrary. By this he is implying, that the expansion of the parameter $\alpha$ contains a different arbitrary constant in front of each order making it impossible to sum it, leading to infinite number of different beta function and interaction parameter freezing.

My question is, why is this a case? Why can't we just draw feynman diagrams for multiple loops for something, like a quark scattering, sum them up and get an expansion in just one constant $\beta_0$?

  • $\begingroup$ Apples versus oranges. The first is a theoretical calculation of a running coupling; the second a method for extracting it from experimental data. The running coupling of QCD was calculated just as in QED, with one surprising difference. The loop correction to the gluon propagator involves boson (gluon) as well as fermion (quark) loops, and the former have the opposite sign. $\endgroup$ – Bert Barrois Jul 3 '18 at 11:07
  • $\begingroup$ I know, but that is how the book is stylized. He claims that those numbers are different, which might be and that their exact value is not predicted by theory. But why wouldn't it be? Can't we just calculate those numbers by computing those multiloop diagrams? $\endgroup$ – user74200 Jul 3 '18 at 12:51
  • $\begingroup$ You didn’t say which book you are reading. Don’t get too skeptical. The RG calculation of the running coupling is supposedly accurate in the high-energy limit: ${{g}^{2}}\sim C/\log (p/\Lambda )$. If you believe the theory, the value of the coefficient C is exact, and the fractional error approaches zero. Only the leading term of the “beta function” is gauge-invariant, but non-leading terms don’t affect the weak-coupling limit. (I’d rather not know the ugly details.) At low energy, however, the calculations aren’t worth a hill of beans. $\endgroup$ – Bert Barrois Jul 4 '18 at 11:42

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