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A limit of a theory or correlator as a mass parameter tends to zero. It can be singular because of infrared effects or changing degrees of freedom, so it must be justified before neglecting a running mass. The renormalization-group mass suppression criterion controls the mass-to-energy ratio but does not by itself prove existence of this limit.

Let ${\varphi }_B=Z_{\varphi }^{1/2}\varphi$ define the renormalized field in Phi-fourth theory. For an $n$-point correlation function, with no composite operator insertions, this impliesThe bare correlation function is independent of the arbitrary renormalization scale $\mu$ when the bare parameters and external momenta are fixed. DefineThe definitions of the running mass and wave-function renormalization functions specify their signs and the factor of two. Differentiating $Z_{\varphi }^{n/2}{G}_n$ using the chain rule yields the Callan–Symanzik equationHere ${\hat{\beta }}_{\lambda }$ denotes the renormalization-group beta function itself, not the function multiplied by an additional $\lambda$. The Callan-Symanzik equation concerns dependence on the arbitrary reference scale, with the displayed external momenta kept fixed.

For the minimal subtraction scheme in dimensional regularization, writeso ${\lambda }_B={\mu }^{ϵ}F$ and $m_B^2=m^{2}H$. The ${\mu }^{ϵ}$ factor gives the engineering contribution $-ϵ\lambda$ to the renormalization-group beta function. Put ${\hat{\beta }}_{\lambda }=-ϵ\lambda +{\beta }_{\lambda }$. Scale independence of the bare coupling then givesThe coefficient of ${ϵ}^0$ is $f_1-\lambda f_1^{\prime }+{\beta }_{\lambda }$. Its vanishing proves the first of the simple-pole formulas for minimal-subtraction renormalization group functions:The coefficients of ${ϵ}^{-k}$, $k\ge 1$, also vanish and impose $(\lambda {\partial }_{\lambda }-1)f_{k+1}={\beta }_{\lambda }{f}_k^{\prime }$. These relations explain how the higher poles are consistent with a finite renormalization-group beta function.

For the mass, differentiate the second bare-parameter relation:Its finite coefficient is ${\gamma }_{m^2}-\lambda b_1^{\prime }$. ThereforeThe higher poles impose $\lambda b_{k+1}^{\prime }={\gamma }_{m^2}{b}_k+{\beta }_{\lambda }{b}_k^{\prime }$. The formulas for the running mass and renormalization-group beta function thus follow directly from bare-parameter independence rather than from an identification with diagram coefficients by name.

At the specified loop order, $f_1=3{\lambda }^2/(16{\pi }^2)$ and $b_1=\lambda /(16{\pi }^2)$. There is no wave-function renormalization at this order. HenceThe regulated renormalization-group beta function is ${\hat{\beta }}_{\lambda }=-ϵ\lambda +3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$; after removing the regulator, ${\hat{\beta }}_{\lambda }={\beta }_{\lambda }$.

Now take $m^2=0$ and the four-dimensional limit. This massless surface is preserved since $\mu d{m}^2/d\mu =m^2{\gamma }_{m^2}$. Set $b=3/(16{\pi }^2)$, choose a reference scale ${\mu }_0$, and define the running coupling byDifferentiating $1/\lambda$ gives $d(1/\lambda )/d\log \mu =-b$, soThe zero-coupling solution is obtained by continuity. At one-loop accuracy ${\gamma }_{\varphi }=0$, and the chain rule turns the massless Callan-Symanzik equation intoThus the one-loop solution along a running-coupling trajectory isFor a prescribed endpoint coupling $\lambda$, substitute ${\lambda }_0=\lambda /[1+b\lambda \log (\mu /{\mu }_0)]$ on the right. The reference correlation function supplies arbitrary boundary data; the Callan-Symanzik equation determines their scale evolution, not their full momentum dependence. The running coupling formula is used on a branch on which its denominator is nonzero and the retained perturbative approximation is appropriate.

If one keeps $ϵ\ne 0$, the same characteristic calculation uses $d\lambda /d\log \mu =-ϵ\lambda +b{\lambda }^2$. Writing $r=\mu /{\mu }_0$ gives $\lambda (\mu )={\lambda }_0{r}^{-ϵ}/[1-(b{\lambda }_0/ϵ)(1-r^{-ϵ})]$, which tends to the displayed four-dimensional result as $ϵ\to 0$.

With a nonzero anomalous dimension, the characteristic equation instead reads $d{G}_n/d\log \mu =-n{\gamma }_{\varphi }(\lambda (\mu ))G_n$. Integration proves the characteristic solution of the massless Callan-Symanzik equation:For the stipulated ${\gamma }_{\varphi }=c{\lambda }^2$ and the same one-loop renormalization-group beta function, change variables using $d\log \nu =d\lambda /(b{\lambda }^2)$. The exponent's integral becomes $(c/b)(\lambda (\mu )-{\lambda }_0)$. Therefore the modified explicit solution isEquivalently its multiplicative factor is $\exp [-nc{\lambda }_0^2\log r/(1-b{\lambda }_0\log r)]$. This solves the equation with the stated truncated renormalization-group beta function and anomalous dimension; it does not introduce additional higher-order coefficients that were not specified.

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.

A renormalized mass changes with the reference scale according to its mass anomalous dimension $\delta (g)$. In a dimensionless correlator at increasing physical energy, its effective argument includes the canonical factor $e^{-t}$. This distinguishes the running mass itself from its ratio to the growing energy scale.

In the minimal subtraction scheme, let ${\lambda }_B={\mu }^{ϵ}F$, $F=\lambda +{\sum }_{k\ge 1}{f}_k(\lambda )/{ϵ}^k$, and $m_B^2=m^{2}H$, $H=1+{\sum }_{k\ge 1}{b}_k(\lambda )/{ϵ}^k$. Define the renormalization-group beta function at fixed bare parameters by $\hat{\beta }=\mu d\lambda /d\mu =-ϵ\lambda +\beta (\lambda )$, and the running mass function by ${\gamma }_{m^2}=\mu d\log m^2/d\mu$.

The bare coupling is scale independent, so $0=ϵF+\hat{\beta }{F}^{\prime }$. Its finite coefficient is $f_1+\beta -\lambda f_1^{\prime }$, giving the first formula. Similarly $0={\gamma }_{m^2}H+\hat{\beta }{H}^{\prime }$ has finite coefficient ${\gamma }_{m^2}-\lambda b_1^{\prime }$. The remaining poles cancel recursively: $(\lambda {\partial }_{\lambda }-1)f_{k+1}=\beta f_k^{\prime }$ and $\lambda b_{k+1}^{\prime }={\gamma }_{m^2}{b}_k+\beta b_k^{\prime }$. Thus the simple poles determine the finite renormalization group functions, while higher poles obey consistency conditions.