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A bare coupling is an interaction coefficient in a regulated action. It is related to a renormalized coupling through counterterms and any required powers of the renormalization scale. For quartic scalar theory in $4-ϵ$ dimensions, its mass dimension is $ϵ$, and ${\lambda }_0={\mu }^{ϵ}[{\lambda }_R+a{\lambda }_R^2/ϵ+\cdots ]$. At fixed regulator, changing $\mu$ while leaving the theory unchanged keeps ${\lambda }_0$ fixed; the corresponding change of ${\lambda }_R$ defines its renormalization-group beta function.

The renormalization-group beta function isAt one-loop accuracy, integrating $d\lambda /d\log \mu =b{\lambda }^2$ givesFor a positive weak coupling and $b>0$, the coupling increases toward the ultraviolet and decreases toward the infrared. Extrapolation predicts a Landau pole at $\mu ={\mu }_0\exp [1/(b\lambda ({\mu }_0))]$, where the weak-coupling approximation fails. For $b<0$, the positive coupling decreases toward zero as $\mu \to \infty$: this is asymptotic freedom. Toward the infrared it instead grows, so perturbation theory eventually fails. The zero coupling is a fixed point. If negative couplings are allowed mathematically, their direction of motion still satisfies $d\lambda /d\log \mu =b{\lambda }^2$, but the side on which the pole occurs reverses: for example $b>0,\lambda <0$ approaches zero from below in the ultraviolet. Negative quartic coupling does not give a stable scalar potential.

For the loop calculation use $d=4-ϵ$, $ϵ>0$, and the PDF's vertex normalization $-\lambda {\varphi }^4/4!$. A Feynman parameter combines the denominators. Shift the Euclidean loop momentum to $\ell =k+xp$, and put $M_x^2=m^2+x(1-x)p^2$. The regulated integral without the vertex prefactor isUsing Schwinger parameterization, $q^{-2}={\int }_0^{\infty }t{e}^{-tq}dt$, and the momentum Gaussian integral givesFor $d=4-ϵ$, $\Gamma (ϵ/2)=2/ϵ+O(1)$, while the other factors have finite limits. Thus the massive scalar bubble pole in four dimensions isThis pole is independent of external momentum and mass and hence is canceled by a local quartic counterterm. There are three channels, corresponding to the three pairings of four external momenta. Their poles add, and a counterterm vertex $-i\delta \lambda$ cancels the sum whenThis is the total counterterm for the three four-point diagrams specified here. The minimal subtraction scheme removes their poles only. The one-loop tadpole is momentum independent and requires a mass counterterm, but does not produce one-loop wave-function renormalization; that is why it does not alter this one-loop coupling calculation.

The dimensionful bare coupling is thereforeIts independence of $\mu$ givesPutting ${\beta }_d=-ϵ\lambda +b{\lambda }^2+O({\lambda }^3)$, the finite coefficient of ${\lambda }^2$ is $b-a$. ThusThis is the one-loop quartic scalar beta function, positive for nonzero weak real coupling. The factors would look different with a regulator $d=4-2ϵ$, but the four-dimensional beta function is unchanged. For massless theory take nonexceptional external momentum, or an infrared regulator, so an infrared singularity is not confused with the ultraviolet pole.

Set $ϵ=4-n$ and $a=3/(16{\pi }^2)=-C$. The kinetic term gives the scalar mass dimension $(n-2)/2$, so the quartic bare coupling has mass dimension $4-n=ϵ$. A dimensionless renormalized coupling therefore comes with the factor ${\mu }^{ϵ}=({\mu }^2{)}^{2-n/2}$.

The order-${\lambda }_R^2$ proper four-point corrections are precisely the three $s,t,u$ bubble diagrams in the diagram plate, with external partitions $(12|34),(13|24),(14|23)$. Each carries a symmetry factor one half. Its two-propagator integral has logarithmic ultraviolet divergence in four dimensions, represented in dimensional regularization by a simple pole at $n=4$. That pole's residue is independent of external momenta, so a local quartic counterterm cancels it. The three channels add, rather than describing three separate couplings. The two-point tadpole has no momentum dependence, so it does not supply a one-loop field-normalization correction to the four-point coupling. The first momentum-dependent field correction contributes to this relation only at higher order.

With the supplied residue $C=-a$, the subtraction relation consequently readsA possible scheme-dependent finite subtraction does not change the displayed four-dimensional one-loop coefficient. Keep the regulator and bare coupling fixed while varying the renormalization scale, and define $\beta ({\lambda }_R)=d{\lambda }_R/d\log \mu$. Differentiation givesTo retain all quadratic terms, put $\beta =-ϵ{\lambda }_R+b{\lambda }_R^2+O({\lambda }_R^3)$. The engineering term times the pole factor contributes $-2a{\lambda }_R^2$, and the remaining quadratic coefficient is $a-2a+b$. It vanishes only when $b=a$. ThusDropping the product of the engineering term and pole factor would give the wrong sign for the loop term.

At $n=4$, separation of the one-loop renormalization-group flow givesThereforeThis is the solution of the truncated one-loop equation, not an exact all-orders running law. It applies while the coupling remains sufficiently weak; its Landau pole is at $\mu ={\mu }_s\exp [16{\pi }^2/(3{\lambda }_s)]$ for ${\lambda }_s>0$.

A renormalization-group fixed point has $\beta ({\lambda }^{\ast })=0$. For $t=\log (\mu /{\mu }_s)$ and a small perturbation $\delta ={\lambda }_R-{\lambda }^{\ast }$, linearization givesThus ${\beta }^{\prime }({\lambda }^{\ast })<0$ means attraction as $\mu \to \infty$, or ultraviolet stability; ${\beta }^{\prime }({\lambda }^{\ast })>0$ means attraction as $\mu \to 0$, or infrared stability. If the derivative vanishes, higher terms decide marginal stability. These statements concern the specified coupling direction; other couplings or the mass can be relevant even when this direction is stable.

For $ϵ>0$, the quadratic renormalization-group beta function has zeros $0$ and $ϵ/a$. At the nonzero zero,ConsequentlyThis is the weak-coupling Wilson-Fisher fixed point, perturbatively controlled near four dimensions. Reaching its full critical limit also requires tuning the mass, the thermal relevant direction at the Wilson-Fisher fixed point.

At $n=4$, this nonzero fixed point merges with the Gaussian fixed point. Although the derivative at zero vanishes, $\beta =a{\lambda }_R^2>0$ for a positive weak coupling, so the running solution tends to zero logarithmically toward the infrared. The four-dimensional critical long-distance limit is Gaussian: the quartic interaction is marginally irrelevant, rather than flowing to a nonzero weakly interacting infrared fixed point. This is the Gaussian infrared limit of positive four-dimensional quartic coupling. It is not ultraviolet asymptotic freedom; the positive coupling grows in the ultraviolet. Nor does it say a massive theory has no interactions at finite energy: the critical or massless scaling regime is the one described by this limiting flow. Extrapolating the one-loop law to arbitrarily high cutoff at fixed positive low-energy coupling encounters its Landau pole; avoiding that pole as the cutoff is removed forces the perturbative coupling toward zero. This motivates the usual triviality conclusion, but the one-loop calculation alone is not a nonperturbative theorem excluding all possible strongly coupled ultraviolet completions.