Editing Feynman diagram (section)

==== Spin 1: non-Abelian ghosts ====
To find the Feynman rules for non-Abelian gauge fields, the procedure that performs the gauge fixing must be carefully corrected to account for a change of variables in the path-integral.

The gauge fixing factor has an extra determinant from popping the delta function:

:<math> \delta\left(\partial_\mu A_\mu - f\right) e^{-\frac{f^2}{2}} \det M </math>

To find the form of the determinant, consider first a simple two-dimensional integral of a function {{mvar|f}} that depends only on {{mvar|r}}, not on the angle {{mvar|θ}}. Inserting an integral over {{mvar|θ}}:

:<math> \int f(r)\, dx\, dy = \int f(r) \int d\theta\, \delta(y) \left|\frac{dy}{d\theta}\right|\, dx\, dy </math>

The derivative-factor ensures that popping the delta function in {{mvar|θ}} removes the integral. Exchanging the order of integration,

:<math> \int f(r)\, dx\, dy = \int d\theta\, \int f(r) \delta(y) \left|\frac{dy}{d\theta}\right|\, dx\, dy </math>

but now the delta-function can be popped in {{mvar|y}},

:<math> \int f(r)\, dx\, dy = \int d\theta_0\, \int f(x) \left|\frac{dy}{d\theta}\right|\, dx\,. </math>

The integral over {{mvar|θ}} just gives an overall factor of 2{{pi}}, while the rate of change of {{mvar|y}} with a change in {{mvar|θ}} is just {{mvar|x}}, so this exercise reproduces the standard formula for polar integration of a radial function:

:<math> \int f(r)\, dx\, dy = 2\pi \int f(x) x\, dx </math>

In the path-integral for a nonabelian gauge field, the analogous manipulation is:

:<math> \int DA \int \delta\big(F(A)\big) \det\left(\frac{\partial F}{\partial G}\right)\, DG e^{iS} = \int DG \int \delta\big(F(A)\big)\det\left(\frac{\partial F}{ \partial G}\right) e^{iS} \,</math>

The factor in front is the volume of the gauge group, and it contributes a constant, which can be discarded. The remaining integral is over the gauge fixed action.

:<math> \int \det\left(\frac{\partial F}{ \partial G}\right)e^{iS_{GF}}\, DA \,</math>

To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case:

:<math> \partial_\mu A^\mu = f \,,</math>

Whose variation under an infinitesimal gauge transformation is given by:

:<math> \partial_\mu\, D_\mu \alpha \,,</math>

where {{mvar|α}} is the adjoint valued element of the Lie algebra at every point that performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action:

:<math> \det\left(\partial_\mu\, D_\mu\right) \,</math>

which can be rewritten as a Grassmann integral by introducing ghost fields:

:<math> \int e^{\bar\eta \partial_\mu\, D^\mu \eta}\, D\bar\eta\, D\eta \,</math>

The determinant is independent of {{mvar|f}}, so the path-integral over {{mvar|f}} can give the Feynman propagator (or a covariant propagator) by choosing the measure for {{mvar|f}} as in the abelian case. The full gauge fixed action is then the Yang Mills action in Feynman gauge with an additional ghost action:

:<math> S= \int \operatorname{Tr} \partial_\mu A_\nu \partial^\mu A^\nu + f^i_{jk} \partial^\nu A_i^\mu A^j_\mu A^k_\nu + f^i_{jr} f^r_{kl} A_i A_j A^k A^l + \operatorname{Tr} \partial_\mu \bar\eta \partial^\mu \eta + \bar\eta A_j \eta \,</math>

The diagrams are derived from this action. The propagator for the spin-1 fields has the usual Feynman form. There are vertices of degree 3 with momentum factors whose couplings are the structure constants, and vertices of degree 4 whose couplings are products of structure constants. There are additional ghost loops, which cancel out timelike and longitudinal states in {{mvar|A}} loops.

In the Abelian case, the determinant for covariant gauges does not depend on {{mvar|A}}, so the ghosts do not contribute to the connected diagrams.