The beta function for QED is given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\beta=\frac{e^3}{16 \pi^2}*\frac{4}{3}*(Q_i)^2[/tex]

where [tex](Q_i)^2[/tex] represents the sum of the squares of the charges of all Dirac fields.

For one generation, for the charge squared you have (2/3)^2 for the up quark, (-1/3)^2 for the down quark, but this is all multiplied by 3 for the 3 colors of quarks, and then you have (-1)^2 for the electron and (0)^2 for its neutrino.

So all in all, 3[(2/3)^2+(-1/3)^2]+(-1)^2=8/3

However this gives a beta function that is not equal to the book value of:

[tex]\beta=\frac{e^3}{16 \pi^2}*\frac{20}{9}[/tex]

So is the book wrong?

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# Beta function QED

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