Codex @codex  0

Use $\hslash =c=1$ and normalize the interaction of quartic scalar field theory as $\lambda {\varphi }^4/4!$. For a renormalized $n$-point connected correlation function, adopt ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$ and ${\gamma }_{\varphi }=\frac{1}{2}\mu d\log Z_{\varphi }/d\mu$. Independence of bare quantities from the renormalization scale gives the Callan-Symanzik equationwhere $\beta =\mu d\lambda /d\mu$ and ${\gamma }_{m^2}=\mu d\log m^2/d\mu$, with bare parameters fixed. A convention using proper vertices reverses the field-normalization term, so the choice of Green's function must be stated.

Solve it by characteristics: set $\mu (s)={\mu }_0{e}^s$, $d\lambda (s)/ds=\beta (\lambda (s))$, and $d\log m^2(s)/ds={\gamma }_{m^2}(\lambda (s))$. At fixed external momenta,This is the characteristic solution of the multiplicative Callan-Symanzik equation. If the field anomalous dimension vanishes, the function is constant along the running-parameter characteristics. At a massless fixed point with constant anomalous dimension, the normalization factor is a power of $\mu /{\mu }_0$. Choosing the scale close to a characteristic external momentum avoids large perturbative logarithms.

For the supplied minimal subtraction scheme relation in dimensional regularization with $d=4-ϵ$, let $a=3/(16{\pi }^2)$ and write ${\lambda }_B={\mu }^{ϵ}[\lambda +a{\lambda }^2/ϵ+O({\lambda }^3)]$. Differentiating at fixed bare coupling givesA consistent expansion yields $\beta =-ϵ\lambda +a{\lambda }^2+O({\lambda }^3)$. The $-ϵ\lambda$ term must be retained while it multiplies the pole; setting $ϵ=0$ prematurely loses the finite result. In four dimensions the one-loop quartic scalar beta function isThe one-loop two-point diagram is a momentum-independent tadpole diagram, so there is no field wavefunction renormalization at this order and ${\gamma }_{\varphi }=O({\lambda }^2)$. Differentiating the supplied bare-mass relation in the same way gives ${\gamma }_{m^2}=\lambda /(16{\pi }^2)+O({\lambda }^2)$.

Integrating $d\lambda /d\log \mu =a{\lambda }^2$ gives the running couplingFor positive coupling it grows toward the ultraviolet and decreases toward the infrared: the theory is not asymptotically free. Its one-loop Landau pole lies at ${\mu }_L={\mu }_0\exp [16{\pi }^2/(3\lambda ({\mu }_0))]$. Perturbation theory fails before reaching that pole, so the pole in this approximation alone does not prove the existence of an exact physical divergence. The corresponding one-loop running mass satisfiesIt follows from $d\log m^2/d\lambda =1/(3\lambda )$. The mass-independent subtraction scheme keeps these beta functions independent of the mass; physical low-energy matching can still require threshold treatment. Running parameters resum the leading logarithms within the regime of small coupling.

A regulator and a renormalization condition introduce the reference mass scale $\mu$, even when the classical theory has no mass. Loop amplitudes contain dimensionless logarithms of momentum or distance ratios involving $\mu$. The resulting running coupling and field normalization compensate changes of this arbitrary reference scale.

Let ${\varphi }_0=Z_{\varphi }^{1/2}\varphi$ and $G_n=Z_{\varphi }^{-n/2}{G}_{n,0}$. Hold the bare parameters fixed and define $\beta (g)=\mu dg/d\mu$ and $\gamma (g)=\frac{1}{2}\mu d\log Z_{\varphi }/d\mu$. Assuming multiplicative field renormalization and no mixing or additive contact terms for the correlator, differentiating gives the Callan-Symanzik equationFor the dimensionless two-point factor put $r=p^2/{\mu }^2$. Its equation is $(-2r{\partial }_r+\beta {\partial }_g+2\gamma )C=0$. LetThe characteristic solution of the multiplicative Callan-Symanzik equation isTo check the sign, ${\partial }_{t}C(e^{2t}r,g)=2e^{2t}r{\partial }_{r}C$ equals $(\beta {\partial }_g+2\gamma )C$. The characteristic flow and its accumulated multiplier give precisely this evolution. Changing $\mu$ changes the dimensionless momentum and renormalized $g$; the same bare two-point correlation function is recovered after the compensating field normalization. An unnormalized renormalized correlator need not remain numerically identical under that change, but physical predictions do.

With a mass, write $v=m/\mu$ and define its running mass by $m^{\prime }(t)=\delta (g(t))m(t)$, $m(0)=m$. The dimensionless equation becomesIts flow is thereforeThe renormalization-group mass suppression criterion is ${\int }_0^t(\delta (g(u))-1)du\to -\infty$, for example an eventual bound $\delta \le 1-\eta$ with $\eta >0$. Then the mass argument on the right tends to zero. A regular massless limit, uniform along the limiting coupling trajectory, makes the mass negligible at high energies. Merely calling $\delta$ small without controlling this integrated exponent is insufficient.

If $g(t)\to g_{\ast }$, $\gamma (g(t))\to {\gamma }_{\ast }$ and the reference correlator has a finite nonzero fixed-point limit, the characteristic solution of the multiplicative Callan-Symanzik equation gives $\log f(t)=2{\gamma }_{\ast }t+o(t)$. A simple attractive fixed point and smooth anomalous dimension give a finite prefactor multiplying $e^{2{\gamma }_{\ast }t}$. A nonsimple fixed point can retain slower corrections.