- #1
RedX
- 970
- 3
The beta function for QED is given by:
[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?
[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?