Codex @codex  0

A coefficient multiplying an interaction term controls its strength relative to the chosen field normalization. Its engineering dimension determines whether it is dimensionless, relevant or irrelevant. A renormalized coupling constant constant is specified at a renormalization scale; its change with that scale is described by the renormalization-group beta function.

Use the metric signature $(-,+,\dots ,+)$, so $p^2=-p_0^2+{\mathbf{p}}^2$, and the weight $e^{iS}$. The momentum-space Feynman rules for the massless Phi-fourth theory are: an internal scalar line contributes $-i/(k^2-iϵ)$; a quartic interaction vertex contributes $-i\lambda$ with its momentum-conservation delta function; every independent loop is integrated with $d^{d}k/(2\pi {)}^d$; and each graph is multiplied by its Feynman-diagram symmetry factor. All external momenta can be taken incoming. These rules follow by expanding the interaction exponential and contracting free fields with the Wick theorem.

For a connected graph with $I$ internal lines, $V$ interaction vertices and $L$ loops,Here $D$ is the superficial degree of divergence. In four dimensions,Odd $n$ vanish by Z2 symmetry. Thus only the nonvacuum two-point and four-point one-particle-irreducible Feynman diagrams can have overall ultraviolet divergences: degree two for $n=2$, degree zero for $n=4$. Vacuum graphs with $n=0$ may also diverge but cancel from normalized correlators. This is an overall power-counting statement; subgraphs must be subtracted before using it to conclude finiteness for $n\ge 6$.

For the full one-particle-irreducible vertices, the tree-level quadratic kernel and quartic interaction giveIf ${\hat{\tau }}_2$ is instead reserved for interaction-generated self-energy insertions, its tree value is zero and $-p^2$ is displayed separately as the free inverse kernel. The distinction is only that convention; the loop insertions below are unchanged.

The requested one-loop diagrams are the two-point tadpole diagram and the three four-point bubble diagrams. Each has symmetry factor $1/2$. The three bubble channels correspond to $P_s=p_1+p_2$, $P_t=p_1+p_3$, $P_u=p_1+p_4$.

The tadpole insertion obeysIn massless dimensional regularization this scaleless integral is zero. A mass or cutoff regulator exposes the ultraviolet divergence allowed by power counting; its vanishing in the scaleless convention does not remove the need to include this diagram.

For a bubble, the vertex and quantum field theory propagator factors give $(-i\lambda {)}^2(-i{)}^2/2={\lambda }^2/2$. The PDF defines $f$ with the factor $1/i$ outside the Minkowski integral, so the integral itself is $if(P^2)$. ThereforeThe converted TeX misplaces this $i$ as though it were part of the exponent of $(2\pi )$; keeping the printed $1/i$ is essential to this real pole coefficient.

For nonexceptional Euclidean external momentum, Wick rotation cancels that factor $1/i$. Use a Feynman parameter and translate the integration variable:For example, represent the inverse square as ${\int }_0^{\infty }d\alpha \alpha e^{-\alpha ({\ell }^2+\Delta )}$, do the Gaussian momentum integral, and then use the Gamma function. This also derives the formula rather than assuming a loop-integration table. At $d=4-\epsilon$, $\Gamma (\epsilon /2)=2/\epsilon +O(1)$ and the parameter integral tends to one, givingTake $P^2>0$ before continuation back to Minkowski momenta. At $P=0$, a massless scaleless integral mixes ultraviolet and infrared issues, so it cannot be used to read off this pole.

Introduce a renormalization scale $\mu$ and a quartic counterterm. With $\lambda$ denoting a dimensionless renormalized coupling constant,After factoring out ${\mu }^{\epsilon }$ from the vertex, each loop contains ${\mu }^{\epsilon }{f}_d$. The counterterm contributes $-\delta \lambda$ to ${\hat{\tau }}_4$ and cancels the pole from all three channels. More explicitly,In modified minimal subtraction, replace the pole-only counterterm bywhich subtracts the first three terms inside the brackets. The finite renormalized quartic scalar vertex is thenAnother renormalization condition can change the finite constant, but not the pole cancellation or the momentum-dependent logarithms. The corresponding timelike expressions follow by the same Feynman continuation.

Loop integrals require a subtraction prescription. Even when the classical coupling constant is dimensionless and there is no mass, specifying its renormalized value means specifying a momentum or length scale at which it is measured. Dimensionless logarithms then involve that renormalization scale $\mu$. Changing $\mu$ changes the renormalized coupling constant and wave-function renormalization while leaving bare quantities fixed. In a theory whose renormalization-group beta function and anomalous dimension both vanish, this scale dependence may disappear; the generic statement concerns the interacting renormalized theory.

Write ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$ and $G_{n,R}=Z_{\varphi }^{-n/2}{G}_{n,B}$. DefineBare correlation functions do not depend on the arbitrary subtraction scale. Differentiating their relation to renormalized correlation functions gives the Callan-Symanzik equationIt expresses the compensation between explicit scale dependence, running coupling and wave-function renormalization. In this convention the field scaling dimension is its engineering dimension plus ${\gamma }_{\varphi }$ at a fixed point; the critical-exponent notation $\eta$ often used for the anomalous dimension equals $2{\gamma }_{\varphi }$.

A renormalization-group fixed point satisfies $\beta (g_{\ast })=0$. It is ultraviolet attractive if the flow approaches it as $\mu \to \infty$, and infrared attractive if the flow approaches it as $\mu \to 0$. For a simple isolated zero, $\delta g\propto {\mu }^{{\beta }^{\prime }(g_{\ast })}$: a negative derivative gives ultraviolet attraction, a positive derivative infrared attraction. Marginal zeros, such as a cubic renormalization-group beta function at the origin, require looking at the first nonzero nonlinear term.

Put $z=p^2/{\mu }^2>0$, $t=\frac{1}{2}\log z$, and write the dimensionless quantum field theory propagator factor as $d(z,g)$. The two-point Callan-Symanzik equation becomesLet $\bar{g}(0)=g$ and $d\bar{g}/dt=\beta (\bar{g})$. The renormalization-group characteristic solution for a two-point function isWhere $\beta \ne 0$, the exponent is $2{\int }_g^{\bar{g}(t)}{\gamma }_{\varphi }(u)/\beta (u)du$. Thus the ultraviolet behavior depends on where the running coupling goes and on the anomalous dimension accumulated along the flow. A weak ultraviolet fixed point or asymptotically free limit permits perturbative evaluation; a flow to large values of the coupling constant does not.

For $\beta (g)=-b{g}^3$ and ${\gamma }_{\varphi }=c{g}^2$, $b>0$,ConsequentlyFor large positive $z$, the coupling constant tends to zero. If the normalization function $d(1,g)$ has a finite nonzero free-field limit $d(1,0)$, then $d(z,g)\sim d(1,0)[b{g}^2\log z{]}^{c/b}$. Equivalently, with $\Lambda =\mu \exp [-1/(2b{g}^2)]$,The positive exponent follows from the sign of $+n{\gamma }_{\varphi }$ in the printed RG equation; changing the field-renormalization anomalous dimension convention would change both signs together.

For the quintic renormalization-group beta function, set $B=-b>0$ and rename its positive quintic coefficient $A$ to avoid confusion with the later gauge-theory coupling constant. Then $\beta (g)=g^3(B-A{g}^2)$ has a nonzero fixed pointEvery nonzero positive running coupling in the perturbative basin approaches this ultraviolet fixed point, rather than approaching zero. The zero coupling constant is infrared attractive. If the same ${\gamma }_{\varphi }=c{g}^2$ is retained, the large-momentum factor is $d(z,g)\sim C{z}^{cB/A}$, assuming a regular nonzero matching factor at the fixed point. More generally the exponent is ${\gamma }_{\varphi }(g_{\ast })$. A fixed point at a large coupling constant inferred from a truncated renormalization-group beta function is only a formal extrapolation; perturbative control requires $B/A$ small.

For the gauge theory use the supplied convention $a=g^2/(16{\pi }^2)$ and evolution variable $L=\log {\mu }^2$. Asymptotic freedom requires ${\beta }_0>0$, or $n<11N/2$. To one-loop accuracy,The integration constant $\Lambda$ is an example of dimensional transmutation. With two loops this is the leading asymptotic expression, not an exact equality: if $\mathcal{L}=\log ({\mu }^2/{\Lambda }^2)$ is large,The expansion presumes fixed coefficients and a scale high enough to enter the ultraviolet regime.

Choose $n=11N/2-\delta$ with $0<\delta \ll N$, respecting integer flavor counts. The supplied coefficients becomeThus ${\beta }_1<0$ for sufficiently small $\delta /N$, and the two-loop flow has the Banks-Zaks fixed pointBoth $a_{\ast }$ and the loop-counting combination $N{a}_{\ast }$ are small. Its slope is ${\beta }^{\prime }(a_{\ast })=-{\beta }_0^2/{\beta }_1>0$, so it is infrared attractive. For even large $N$, taking $n=11N/2-1$ provides an explicit integer sequence with the desired parametric hierarchy. As a finite example, $N=3,n=16$ gives ${\beta }_0=1/3$, ${\beta }_1=-302/3$ and $a_{\ast }=1/302$. The fermion-count convention is the one encoded by the given coefficients.