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I have a question about renormalization schemes. Consider for example the self energy of the photon (i.e. renormalization of the electric charge in QED) and take two famous schemes: the Minimal Subtraction (MS) and the Momentum Subtraction (MO). In the MS case we choose the counter terms of the renormalized Lagrangian in order to just cancel the divergent part of the self energy loop:

$$

Z_e=1+\frac{e^2}{12\pi^2}\frac{1}{\epsilon},

$$

where [itex]\epsilon\to 0[/itex] when we restore the original 4-dimensional space-time.

This lead to a beta-function given by:

$$

\beta_{MS}(e)=\frac{e^3}{12\pi^2}.

$$

The MO scheme, on the other hand, requires the two-point function to be zero at a particular momentum [itex]-p^2=M^2[/itex]. This lead to a more complicated counter term and to a beta-function given by:

$$

\beta_{MO}(e)=\frac{e^3}{2\pi^2}\int_0^1dx x(1-x)\frac{M^2x(1-x)}{m^2+M^2x(1-x)}.

$$

My question is: these two beta-functions will lead to different behaviors of the running electric charge. How is that possibile? The running of the electric charge has been experimentally measured and the physical predictions of the theory should be independent on the renormalization scheme used. How do we reconcile these aspects?

Thank you

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# Minimal subtraction vs Momentum subtraction schemes

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