# Beta function QED

by RedX
Tags: beta, function
 P: 969 The beta function for QED is given by: $$\beta=\frac{e^3}{16 \pi^2}*\frac{4}{3}*(Q_i)^2$$ where $$(Q_i)^2$$ 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: $$\beta=\frac{e^3}{16 \pi^2}*\frac{20}{9}$$ So is the book wrong?