= Cubic two-point Wick-pairing classification
At second order in the <coupling constant (physics)> 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.
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