Phi cubed theory (${\varphi }^3$ theory)

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.

The two-vertex cubic bubble in $d=6-ϵ$ dimensions has the displayed amputated pole. A Feynman parameter gives mass $m^2+x(1-x)k^2$, and $\Gamma (-1+ϵ/2)$ supplies the pole. The full propagator receives $D_0^{2}I$, while its inverse receives $-I$. This sign distinguishes a two-point insertion from an inverse-propagator correction.

With additive counterterms $\frac{1}{2}\delta Z(\partial \varphi {)}^2+\frac{1}{2}\delta m^2{\varphi }^2$, their Euclidean propagator insertion is $-(\delta Z{k}^2+\delta m^2)$. These minimal subtraction scheme coefficients cancel the cubic bubble insertion. A bare mass shift also subtracts $m^2\delta Z$ after wavefunction renormalization; it is not the same coefficient as the additive mass counterterm.

A canonically normalized real scalar field in six dimensions has mass dimension two. The cubic coupling is classically a marginal coupling, whereas a local quartic coupling has mass dimension minus two. This theory is perturbatively renormalizable by power counting in quantum field theory, but a real cubic Euclidean potential is unbounded below. Its expansion about a massive Gaussian theory is therefore formal perturbation theory, not a convergent positive functional integral at nonzero real $g$.

The self-energy bubble diagram at loop order one in six-dimensional cubic scalar field theory has Feynman-diagram symmetry factor two. With positive Euclidean scalar propagators, its amputated Green's function is the displayed integral. Its superficial degree of divergence is two. Expanding at large internal momentum and taking the angular average shows that its divergent part is a constant plus a multiple of $p^2$, precisely the mass counterterm and wave-function renormalization structures.

For a spherical ultraviolet cutoff $|k|\le \Lambda$ and $m^2>0$, its value at $p=0$ isThe divergent $p^2$ coefficient is $-g^2\log ({\Lambda }^2/m^2)/(768{\pi }^3)$. Finite terms depend on the regularization convention. Tadpole diagram insertions are excluded from the one-particle-irreducible Feynman diagram, and are removed from the connected two-point function by imposing a vanishing one-point function.

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.

In a six-dimensional cubic scalar field theory, three labelled box Feynman diagrams give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4{\varphi }^4/4!$, their contribution is $\delta g_4=-3g^4{I}_4$. To check its sign and multiplicity, expand the one-loop scalar effective action as $\frac{1}{2}\mathrm{Tr}\log (D+g\varphi )$, where $D=-{\partial }^2+m^2$. Its fourth-order term is $-g^4\mathrm{Tr}(D^{-1}\varphi {)}^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.

For $m>0$ and $0<\Lambda <{\Lambda }_0$, defineThe radial formula uses the area ${\pi }^3$ of the unit five-sphere. Its mass dimension is $6-2r$. As ${\Lambda }_0\to \infty$, $I_1$ diverges quartically, $I_2$ quadratically, $I_3$ logarithmically, and $I_r$ is ultraviolet finite for $r\ge 4$ at fixed $\Lambda ,m$. This follows directly by comparison with $q^{5-2r}$.

The four-point tree-level Feynman diagram has two cubic vertices joined by one propagator. The three ways to divide four labeled external legs into vertex pairs give the $s$, $t$, and $u$ channels. For interaction $+\lambda {\varphi }^3/3!$, the amplitude is $\mathcal{M}=-{\lambda }^2[(s-m^2{)}^{-1}+(t-m^2{)}^{-1}+(u-m^2{)}^{-1}]$, with the Feynman i-epsilon prescription understood. All channels interfere in its modulus squared.

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.