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Direct integration between the two renormalization scales gives

For incoming momenta, the Euclidean momentum-space Feynman rules are

The one-loop one-particle-irreducible diagrams with at most three external legs can be classified by their external species. There is a one-point $\chi$ tadpole diagram made from a $\varphi$ loop. There are two two-point bubble diagrams: the $\varphi$ self-energy has one internal $\varphi$ and one internal $\chi$, while the $\chi$ self-energy has two internal $\varphi$ lines and symmetry factor $1/2$. For three external legs, three vertices make a triangle: one triangle corrects the $\varphi \varphi \chi$ vertex and contains two internal $\varphi$ lines and one internal $\chi$ line; another has three external $\chi$ legs and a $\varphi$ loop, generating a ${\chi }^3$ interaction. The symmetry $\varphi \mapsto -\varphi$ forbids amplitudes with an odd number of external $\varphi$ legs. The classical $\varphi \varphi \chi$ vertex is the corresponding tree-level three-point diagram.

Adopt the self-energy conventionThis follows by summing the geometric series of exact propagators separated by amputated one-particle-irreducible two-point insertions. At one loop, after writing ${\lambda }_0={\mu }^{ϵ/2}\lambda +O({\lambda }^3)$,Introduce a Feynman parameter and shift the loop momentum. Withdimensional regularization givesUsing $\Gamma (-1+ϵ/2)=-2/ϵ+\gamma -1+O(ϵ)$ produceswhereand one convenient integral form of the finite part isChanging the definition of the dimensional-regularization scale only moves a finite constant between $C$ and the counterterm.

To make the two-point function finite, write ${\varphi }_0=Z_{\varphi }^{1/2}\varphi$, express $m_0^2$ in terms of a renormalized mass and a mass counterterm, and choose the pole parts of $\delta Z_{\varphi }$ and $\delta m^2$ to cancel $B{p}^2/ϵ$ and $A/ϵ$. In the minimal subtraction scheme no additional finite pieces are removed. The physical mass is the pole mass, so with the self-energy convention above it obeysThe explicit $\mu$ dependence of the finite self-energy cancels the running of $m(\mu )$, leaving $m_{\mathrm{phys}}$ independent of the renormalization scale.

The superficial degree of divergence counts the ultraviolet power before subdivergences and symmetry cancellations are considered. At $d=6$, a connected graph made from cubic vertices haswhere $E$ is its number of external legs. Hence one-, two-, and three-point functions can have quartic, quadratic, and logarithmic superficial divergences, whereas graphs with more external legs are superficially convergent.

Full renormalization also requires the $\chi$ mass and wave-function counterterms from its two-point function, a linear $\chi$ counterterm cancelling the tadpole, a coupling counterterm from the divergent $\varphi \varphi \chi$ triangle, and a ${\chi }^3$ counterterm from the three-$\chi$ triangle. A vacuum-energy counterterm removes divergent vacuum diagrams. These are precisely the local operators allowed by power counting in quantum field theory and the exact $\varphi \mapsto -\varphi$ symmetry.

Renormalization introduces an arbitrary renormalization scale $\mu$ even though the classical massless theory has no dimensionful parameter. Independence of the bare correlation function from that auxiliary scale gives the Callan-Symanzik equationHereis the beta function and $\gamma (g)$ is the field anomalous dimension, with its sign fixed by the displayed equation. The beta function determines the running coupling. Its zeros are renormalization-group fixed points, where the theory can become scale invariant. A positive beta function makes a positive coupling increase toward larger $\mu$, while a negative one makes it decrease.

For the propagator coefficient $C$, follow a characteristic with $t=\log (\mu /{\mu }_0)$ and $dg/dt=\beta (g)$. The $n=2$ equation becomesIntegration giveswith

For $\beta (g)=-b{g}^3$ with $b>0$, the only real fixed point is $g^{\ast }=0$. It is ultraviolet-attractive: the theory is asymptotically free. Integrating the running equation givesorfor the branch with the same sign as $g({\mu }_0)$. The perturbative expression has an infrared Landau pole atand therefore

Finally set ${\mu }_0=p$ and use $\gamma (g)=c{g}^2$. The characteristic factor becomesIn terms of the dynamically generated scale,

The bare coupling does not depend on the renormalization scale. Expressing it through the renormalized field and coupling gives, to the stated order,At one loop $\delta Z=0$. Put $a=3/(16{\pi }^2)$ and $\delta g=a{g}^2/ϵ$. If ${\beta }_g=\mu dg/d\mu$, differentiation and expansion through $g^2$ yieldSince $\lambda ={\mu }^{ϵ}g$,and in four dimensions

Introducing a Feynman parameter and subtracting at zero external momentum giveThe bare coupling ${\lambda }_0=\lambda +{\delta }_{\lambda }$ is independent of the renormalization scale. Differentiating its logarithmic counterterm at fixed ${\lambda }_0$ gives the leading beta function

The pole mass is the value of the invariant momentum at a pole of the full propagator. It is independent of the renormalization scheme, whereas a running mass parameter generally depends on the renormalization scale.

This prescription selects an arbitrary scale or scheme parameter by making a truncated perturbative observable stationary under its variation. The complete physical observable is independent of the renormalization scale, but truncation leaves residual dependence. Stationarity is a prescription, not a guarantee of uniquely optimal accuracy or of a stationary point's existence.

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

A running coupling is a renormalized coupling regarded as a function of the renormalization scale. Its scale derivative is its beta function.

The strong coupling constant parametrizes the strength of Quantum chromodynamics interactions. Its dependence on the renormalization scale is governed by the renormalization-group beta function. It is dimensionless in four dimensions.