The Gauge Contribution

The Gauge Contribution

Next up is the contribution:

We see that part of the calculation involves ghost , and so is gives the same answer as above.

The only difference is the spin indices μ⌫ which give an extra factor of 4 after taking the trace. This means that

On rotational grounds, there is no term linear in F̄μ⌫ . This means that the first term comes from expanding out log gauge to quadratic order and focussing on the F̄μ⌫ terms,

Once again, we have a divergent integral to compute. This time we get

The sum then gives the contribution to the effective action, 

Here the 4/3 is the diagmagnetic contribution. In fact, it’s overkill since it neglects the gauge redundancy.

This is subtracted by including the contribution from the ghost fields.

Together, these give rise to a positive beta function. In contrast, the 8 term is the paramagnetic piece, and can be traced to the spin 1 nature of the gauge field.

This is where the overall minus sign comes from.

The coefficient of the kinetic terms is precisely the gauge coupling 1/g 2 . Combining both gauge and ghost contributions, and identifying the momentum k of the background field as the relevant scale μ, we have 

This is in agreement with the advertised result (2.58).

As explained previously, the overall minus sign here is important. Indeed, it was worth a Nobel prize.