Wick contraction

A Wick contraction replaces a selected pair of free fields by their two-point function. Summing pairings computes Gaussian correlation functions and supplies the propagator lines and combinatorial factors of a Feynman diagram.

For two identical even-degree $n$ scalar Feynman vertices, choose $r$ half-edges on each and pair them across the vertices. The remaining $n-r$ half-edges pair locally. This gives the displayed number of Wick contractions, with even $r$ and $(-1)!!=1$. The coefficient in the unnormalized partition function is $N_{n,r}/[2!(n!{)}^2]$. The $r=0$ contribution is disconnected and is removed from the logarithm, but remains in the partition function itself. This rule produces the quartic and sextic oscillator vacuum graph coefficients without drawing-dependent symmetry-factor guesses.

At second order in the coupling constant of a real phi cubed theory, the two-point numerator has eight fields, hence $7!!=105$ Wick contractions. Of these, six give a free external line times a vacuum theta graph, nine give a free line times a vacuum dumbbell, 36 give a two-line bubble, 36 give a connected tadpole attachment, and 18 give two separate one-point tadpoles. Dividing by $2!(3!{)}^2=72$ gives weights $1/12,1/8,1/2,1/2,1/4$. The normalized vacuum generating functional cancels the first two contributions, while the last remains in the full two-point correlation function as the product of one-point functions. Thus cancellation of vacuum bubbles does not, by itself, take the connected part.