Quartic interaction (${\varphi }^4$ theory)

Phi-fourth theory has a scalar interaction $-\lambda {\varphi }^4/4!$. In four dimensions its coupling is classically marginal, and its one-loop four-point function receives bubble corrections in the three Mandelstam channels.

For a real scalar field with interaction $-\lambda {\varphi }^4/4!$, the first Dyson series term is $-i\lambda \int {\varphi }^4{d}^{4}x/4!$. Four labeled external legs can attach to the four fields in $4!$ ways, or in $\left(\genfrac{}{}{0ex}{}{4}{2}\right)2!2!=24$ ways after separating incoming and outgoing legs. The factorial cancels and spacetime integration supplies the four-momentum conservation delta function. This gives the connected tree scattering amplitude $-\lambda$. The separate identical final-state symmetry factor belongs in the event phase-space integral and is not another vertex factor.

For a connected six-external-leg Feynman diagram made from quartic vertices, $4V=6+2I$ and $L=I-V+1$ give $V=L+2$, $I=2L+1$. A tree therefore has two vertices and one internal line; partitioning six labelled external legs into two triples gives ten tree channels. A one-loop triangle has three vertices, three internal lines and two external legs at each vertex. With external labels fixed, that triangle has symmetry factor one. Other one-loop topologies also contribute to the full amplitude.

With interaction $-{\mu }^{ϵ}\lambda {\varphi }^4/4!$ and $d=4-ϵ$, the two-point tadpole diagram has symmetry factor $1/2$. The Euclidean massive loop integral gives its pole through $I_{d,1}=-m^2/(8{\pi }^2ϵ)+O(1)$. The three four-point bubble diagrams each have symmetry factor $1/2$, and the massive scalar bubble pole in four dimensions gives the quartic counterterm. The tadpole is independent of external momentum, so no one-loop wave-function renormalization is required. A vacuum-energy counterterm is additionally needed if vacuum diagrams are retained.

With interaction $-{\lambda }_0{\varphi }^4/4!$, the tadpole diagram has symmetry factor one half. After Wick rotation and a rotationally invariant Euclidean cutoff, its self-energy is $({\lambda }_0/2){\int }_{|k|<\Lambda }{d}^{4}k(2\pi {)}^{-4}(k^2+m_0^2{)}^{-1}$. The sphere area $2{\pi }^2$ and radial integral ${\int }_0^{\Lambda }{k}^{3}dk/(k^2+m_0^2)=\frac{1}{2}[{\Lambda }^2-m_0^2\log (1+{\Lambda }^2/m_0^2)]$ give the displayed shift. It is independent of external momentum, so no one-loop momentum-dependent wave-function renormalization arises from this graph. Different regulator schemes can treat power divergences differently while defining the same chosen pole mass.

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.

With interaction $\lambda {\varphi }^4/4!$ for one real scalar, the three one-loop four-point bubble channels give the four-dimensional renormalization-group beta function $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$. Its positive sign for $\lambda >0$ excludes asymptotic freedom and produces a perturbative Landau pole on extrapolation.

For a positive weak quartic scalar field theory coupling in four dimensions, the one-loop renormalization-group beta function is $\beta =a{\lambda }^2$, $a=3/(16{\pi }^2)>0$. Integration gives the displayed inverse-coupling law, so $\lambda$ tends to zero logarithmically as $\mu \to 0$ and grows toward the ultraviolet. The interacting Wilson-Fisher fixed point of $4-ϵ$ dimensions merges with the Gaussian fixed point at $ϵ=0$. This describes the quartic direction in the massless or critically tuned infrared regime, not disappearance of every finite-energy interaction in a massive theory. The ultraviolet Landau pole signals breakdown of that perturbative extrapolation; one loop alone does not prove a nonperturbative triviality theorem.

At quadratic order, $\mu d\lambda /d\mu =-ϵ\lambda +3{\lambda }^2/(16{\pi }^2)$ for $ϵ>0$. Solving the linear equation for $1/\lambda$ gives the displayed expression. If $0<\lambda ({\mu }_0)<{\lambda }_{\ast }$, the ultraviolet limit is zero and the infrared limit is ${\lambda }_{\ast }$. Initial data above ${\lambda }_{\ast }$ reach a Landau pole at a finite ultraviolet scale. The interacting value is the Wilson-Fisher fixed point along the quartic coupling direction; the full scalar theory also has a thermal relevant direction at the Wilson-Fisher fixed point, so a critical infrared limit requires mass tuning.