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For a real phi cubed theory with labelled external points, the connected three-point function at third order in the coupling constant has seven labelled one-loop Feynman diagrams: one triangle, three external-branch bubble insertions, and three tadpole-ended chains. The symmetry factors are respectively $1,2,2$, with graph weight $1/S$. To prove completeness, use $3V=2I+3$ and $L=I-V+1$: $V=3$ gives $I=3$ and $L=1$. The unique internal cycle is then a triangle, a pair of parallel edges, or a self-loop; the degree-three condition fixes the external attachments. Only the triangle remains one-particle irreducible after amputation of external propagators. A tadpole-free prescription removes the three chains but must be stated separately.

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.

A connected five-point tree in phi cubed theory has three cubic vertices and two internal propagators. For labeled external legs there are fifteen diagrams: choose the leg attached to the middle vertex and partition the other four legs into two unordered pairs.

In phi cubed theory, a two-point bubble diagram has two cubic vertices, two internal scalar propagators and symmetry factor $1/2$. Its connected amputated insertion is $+{\mu }^{ϵ}{g}^2{B}_d(p)/2$. The inverse scalar propagator, or quadratic one-particle-irreducible vertex, therefore receives the negative insertion:Equivalently, the connected two-point correction is $G_0(p{)}^2{\mu }^{ϵ}{g}^2{B}_d(p)/2$, and expansion of its inverse changes the sign. The same sign follows from the second term of $\frac{1}{2}\mathrm{Tr}\log (K+{\mu }^{ϵ/2}g\varphi )$.

Using the six-dimensional residue and ${\int }_0^{1}x(1-x)dx=1/6$, the divergent inverse-propagator correction isIts momentum term requires wave-function renormalization, unlike the massive quartic tadpole. The minimal-subtraction two-point counterterms in cubic scalar theory areThese are additive coefficients in a Lagrangian expressed in terms of the renormalized field. If ${\varphi }_0=Z^{1/2}\varphi$, $Z=1+\delta Z$, and the bare mass is separately written $m_0^2=m^2+\Delta m^2$, then $\delta m^2=m^2\delta Z+\Delta m^2$ at this order, so $\Delta m^2=-5g^2{m}^2/[6(4\pi {)}^3ϵ]$. Distinguishing these conventions prevents a spurious disagreement between mass counterterms. The local Minkowski counterterms are $\delta Z(\partial \varphi {)}^2/2-\delta m^2{\varphi }^2/2$. A complete cubic theory also has one-point and three-point subtractions; they are not the specified two-point bubble poles.

In phi cubed theory, each vertex has three legs. A connected four-point tree-level Feynman diagram therefore needs two cubic vertices and one internal propagator. There are exactly three ways to partition the labeled external legs between these vertices:

       s channel               t channel              u channel

 p \          / p'         p ---o--- p'            p ---o--- q'
    o--------o                  |                       |
 q /          \ q'         q ---o--- q'            q ---o--- p'

 internal: p+q             internal: p-p'          internal: p-q'

Solid lines denote the real scalar field propagator. The labels specify which external pair meets at each vertex, so the last two graphs are different even though their unlabeled shapes agree. This is the four-point tree amplitude in phi cubed theory:There is no four-point contact vertex in the given cubic interaction.

The momentum-space Feynman rules for Phi cubed theory areEach vertex carries $(2\pi {)}^4{\delta }^{(4)}$ of momentum conservation, each independent loop momentum is integrated with $d^4\ell /(2\pi {)}^4$, and a graph is divided by its Feynman-diagram symmetry factor. External propagators are omitted from an amputated scattering amplitude.

Phi cubed theory has interaction $-g{\varphi }^3/3!$, giving a cubic vertex $-ig$. In four spacetime dimensions the scalar field has mass dimension one and $g$ has mass dimension one.