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The parameters of a cutoff statistical field theory describe only the retained modes. Changing the ultraviolet cutoff changes which fluctuations have already been integrated out, so its effective mass, interaction coefficients and gradient normalization must change to preserve the same long-distance physics. They are not separately cutoff-independent observables. Because the source's weight is $e^{-H}$, its statistical Hamiltonian is dimensionless, with the physical inverse-temperature factor already absorbed.

For example split the scalar field into slow modes ${\varphi }_{<}$ with $|p|<\Lambda /b$ and shell modes ${\varphi }_{>}$ with $\Lambda /b<|p|<\Lambda$. Define the momentum-shell renormalization group step byThis integrates out short-wavelength fluctuations exactly if all generated terms are retained. The new Hamiltonian can be expanded in local symmetry-allowed operators when the retained external momenta are well below the shell scale. A cumulant expansion of a coarse-grained free energy gives perturbative coefficients. Restoring the cutoff with $x^{\prime }=x/b$ and, for a canonical gradient term, ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$, gives $m^{2\prime }=b^2{m}^2+\cdots$ and $g^{\prime }=b^{4-D}g+\cdots$. Interaction corrections also change the field normalization. Repeating the step produces the effective theory at successively longer distances.

To obtain the LG theory, assume a short-range scalar theory, slowly varying retained fields, analytic local couplings, positive gradient stiffness and stability, and a regime in which fluctuation corrections at the remaining scales are small. Keeping the leading gradient, quadratic and quartic operators gives a local Landau-Ginzburg theory functional. Its equilibrium in the Landau approximation is a uniform minimum, with a quadratic coefficient proportional to $T-T_c$ after the critical mass has been tuned. The RG explains why this is asymptotically consistent for the ordinary transition above four dimensions: the quartic interaction is irrelevant near the Gaussian fixed point, while it must still be retained to stabilize the ordered phase. Below four dimensions it cannot be dropped in the asymptotic critical region; an interacting Wilson-Fisher fixed point rather than the elementary saddle generally controls the transition. At four dimensions the interaction is marginal and produces logarithmic corrections. Tuning a quartic coefficient through zero requires a sextic stabilizing term and leads to tricriticality.

More concretely, in the Gaussian scaling regime let $u=g/6$ so the local quartic term is $u{\varphi }^4/4$. At a total blocking scale $\ell$, the leading couplings are $r^{\prime }={\ell }^{2}r$, $u^{\prime }={\ell }^{4-D}u$, $h^{\prime }={\ell }^{(D+2)/2}h$. Apply the saddle approximation to the blocked potential and multiply its minimum by ${\ell }^{-D}$ to convert back to original volume units. Rescaling the saddle field by $(|r^{\prime }|/u^{\prime }{)}^{1/2}$ givesAll blocking-scale factors cancel. This explicitly recovers mean-field scalar free-energy scaling with $r\propto t$. Although $u^{\prime }\to 0$ above four dimensions, the saddle free energy is proportional to $1/u^{\prime }$; setting it to zero before minimization would remove the ordered phase. This is the dangerously irrelevant coupling mechanism rather than homogeneous two-variable hyperscaling.

For the perturbative calculation make the source's kinetic convention explicit. Write $K={\alpha }^{-1}>0$. The canonical normalization of a scalar gradient term uses $\psi =\sqrt{K}\varphi$, so the canonical quadratic and quartic coefficients are $r/K$ and $g/K^2$. In what follows $m^2,g$ denote those canonical coefficients; then the reference propagator has denominator $p^2+M^2$. Without this normalization the propagator denominator is $K{p}^2+r$, and the unmodified printed integral would not apply. At one loop the quartic tadpole is momentum independent, so it produces no gradient renormalization at this order.

In the convention fixed by the displayed equation, the truncated two-point function is the one-particle-irreducible two-point vertex, not the connected two-point cumulant itself. If $W[J]=\log Z[J]$ and $\Gamma [\phi ]=\int J\phi -W[J]$ is the Legendre transform, then its second derivative is the inverse of $W^{(2)}=G$. For a translation-invariant background,Decompose the canonically normalized statistical Hamiltonian into a Gaussian part of mass $M^2$, a mass counterterm $\delta m^2=m^2(\Lambda ,T)-M^2$, and the quartic interaction $g{\varphi }^4/4!$. With the Euclidean sign convention in which a positive mass correction increases the inverse propagator, the self-energy expansion isHere $\Sigma$ contains loop corrections from proper two-point diagrams, excluding the separately displayed mass counterterm. The corresponding connected propagator begins $G=G_0-G_0(\delta m^2+\Sigma )G_0+\cdots$. This fixes the sign, which would be reversed if “self-energy” instead denoted the insertion added with a plus sign inside a Dyson series.

The one-loop proper diagram is the tadpole diagram. Attaching two external fields to the quartic vertex gives $4\cdot 3$ contractions, divided by $4!$, so its symmetry factor is $1/2$. With the loop momentum restricted by the cutoff,It is independent of external momentum. Impose the zero-momentum mass condition $M^2=\tilde{\Gamma }(0)=m^2(0,T)$. This gives $\delta m^2+(g/2)I_D(M^2;\Lambda )=0$, henceThis is the one-loop relation using a renormalized mass in the reference propagator, or the corresponding self-consistent tadpole approximation if solved without expanding in $g$. It is not an exact all-orders gap equation. Away from the critical infrared problem, replacing the loop mass by the bare one changes a strict perturbative result only at higher order.

Take $g>0$ smooth and nonzero for an ordinary stable quartic transition. To test the assumed linear thermal mass, work from the disordered side and put $R=m^2(0,T)>0$. For $D>2$, $I_D(0;\Lambda )$ is infrared finite. The critical bare mass is shifted, not generically zero: $m^2(\Lambda ,T_c)=-(g/2)I_D(0;\Lambda )$. Absorb the smooth temperature dependence of couplings into an analytic thermal tuning $\tau =A_t(T-T_c)+\cdots$, with $A_t>0$. Critical subtraction gives the one-loop critical-mass subtractionwhereThe infrared asymptotics of the critical-mass subtraction now distinguish the dimensions. For $D>4$, $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is finite, soFor $D=4$, $J_4(R)=(K_4/2)\log [({\Lambda }^2+R)/R]$ diverges logarithmically. For $2<D<4$, substitution $p=\sqrt{R}q$ givesso the correction to $\tau$ scales as $R^{(D-2)/2}$ and dominates the analytic linear term. Pure mean-field linear mass scaling is therefore consistent only aboveAt the boundary dimension logarithms modify the simple power law. Below it this calculation diagnoses the failure of the Gaussian expansion; the exponent obtained by treating the self-consistent one-loop equation as exact is not automatically the exponent of the interacting scalar theory. For $D\le 2$, the massless subtraction itself has an infrared divergence, so this perturbative argument cannot establish absence of a transition. In particular it does not rule out the two-dimensional Ising critical point.

The same upper dimension follows by engineering dimension counting: a canonical scalar field has dimension $(D-2)/2$, so $[g]=D-4(D-2)/2=4-D$. For a tricritical point tune the renormalized quadratic and quartic terms to zero and retain a positive sextic interaction $v{\varphi }^6$. Its engineering dimension isIt is marginal at $D_c=3$, irrelevant above three, and relevant below three. More generally the upper critical dimension of an even scalar interaction ${\varphi }^{2n}$ is $2n/(n-1)$.

A Ginzburg criterion check gives the same result: at tricritical mean-field scaling $m_0^2\sim |t{|}^{1/2}$ and $\xi \sim |t{|}^{-1/2}$, whereas fluctuations in a correlation volume scale as ${\xi }^{2-D}\sim |t{|}^{(D-2)/2}$. Their ratio to $m_0^2$ is $|t{|}^{(D-3)/2}$, which tends to zero only for $D>3$. ThusThe tricritical tuning concerns renormalized couplings: shell contractions of a sextic term can regenerate quadratic and quartic terms even when their bare coefficients vanish. At three dimensions the marginal sextic coupling produces logarithmic corrections rather than a strictly fluctuation-free mean-field limit.

In the convention implied by the printed expansion, the truncated two-point quantity is the inverse connected propagator, or one-particle-irreducible two-point vertex:It is the second functional derivative of the quantum effective action (or statistical Legendre effective action) about a translationally invariant zero-field equilibrium. The relation follows from the inverse Hessian relation for a connected two-point function. The subscript $c$ in $G$ already removes disconnected one-point products; the word “truncated” here is not a request to subtract that product a second time. Nor is a general amputated connected correlation function interchangeable with a one-particle-irreducible correlation function.

Use the zero-momentum renormalized mass $R=m^2(0,T)$ in the free propagator ${\tilde{G}}_0(p)=(p^2+R{)}^{-1}$, and split the quadratic coupling into $R$ plus a mass counterterm $\delta m^2=m^2(\Lambda ,T)-R$. The self-energy $\Sigma (p)$ is the sum of loop 1PI insertions, excluding the separately displayed counterterm. Summing repeated insertions by Dyson resummation givesThe sign convention is that a positive tadpole shifts the inverse propagator upwards. At first order, the connected correlation function correction is $-G_0^2\Sigma$, consistent with this inverse-propagator convention. A different split between the reference mass and counterterm produces the same renormalized result.

For the positive quartic interaction $g{\varphi }^4/4!$, the one-loop tadpole diagram has no external-momentum dependence. Its symmetry factor is $1/2$: assigning the two external legs to four vertex fields gives $4\cdot 3$ contractions, and division by $4!$ gives $1/2$. With the dimensionless statistical-action convention,The physical zero-momentum condition $\tilde{\Gamma }(0)=R$ sets $\delta m^2+{\Sigma }_1(0)=0$, hence the one-loop mass relation isWriting the internal line with $R$ is a renormalized or self-consistent one-loop convention. Away from critical infrared singularities it differs from a bare-mass insertion only at higher perturbative order. This equation does not by itself provide exact critical exponents once loop corrections become large.

For $D>2$, subtract the critical-temperature condition $0=m^2(\Lambda ,T_C)+(g/2)I_D(0)$. Take $g$ and the regular coefficients at their critical values, absorbing smooth changes into a coefficient $A>0$. Sincethe one-loop critical-mass subtraction becomesLet $K_D$ be the area of the unit $(D-1)$-sphere divided by $(2\pi {)}^D$. Radial integration gives $J_D(R)=K_D{\int }_0^{\Lambda }{k}^{D-3}(k^2+R{)}^{-1}dk$.

For $D>4$, $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is infrared finite, so it merely renormalises the coefficient and $R\propto T-T_C$ is consistent. For $2<D<4$, setting $k=\sqrt{R}x$ yieldswith a finite positive dimensionless integral. The correction is singular relative to the term $R$, invalidating the finite-coefficient linear-mass assumption. At $D=4$,so the boundary is logarithmically marginal. For $D\le 2$, even the subtraction using $I_D(0)$ needs an infrared regulator; it cannot be used to restore a finite linear critical expansion. Thus the ordinary upper critical dimension isA fixed-coupling self-consistent one-loop formula is not the full marginal renormalization group analysis, but its logarithm already shows why an uncorrected linear power law is not generic at $D=4$.

At a tricritical point, both the quadratic and quartic scaling directions must be tuned; the leading stabilising interaction is sextic. With canonical scalar-field engineering dimension $(D-2)/2$, the sextic coupling has eigenvalue $D-6(D-2)/2=6-2D$. It becomes marginal at $D=3$, givingLower even couplings generated by coarse-graining must remain tuned. This is why using an untuned quartic tadpole to diagnose a tricritical point would give the wrong boundary. The tricritical sextic beta function supplies marginal logarithmic corrections at three dimensions.

For a general multicritical even Landau potential, assume the lower stabilising even terms have been tuned away and the first remaining one is $A_{2n}{M}^{2n}$, with $n\ge 2$ and $A_{2n}>0$. Minimising the potential on its ordered branch givesIts curvature at the minimum is $4(n-1)a_2|t|$. For a finite positive gradient stiffness $c$, the longitudinal correlation length therefore scales as $\xi \propto |t{|}^{-1/2}$. The Ginzburg criterion compares the order-parameter fluctuation averaged over a correlation volume with this squared mean-field value. Keeping momenta of order ${\xi }^{-1}$ or less,Consequently the multicritical Ginzburg ratio behaves asOnly for a positive exponent do these relative fluctuations vanish on approaching the critical point. Thus the general upper critical dimension isEquivalently the interaction eigenvalue $y_{2n}=2n-(n-1)D$ vanishes there. The cases $n=2$ and $n=3$ reproduce 4 and 3 respectively.

For $D<D_C$, the ratio diverges and the mean-field assumptions lose self-consistency arbitrarily close to the transition. At $D=D_C$ the marginal Ginzburg criterion is scale-independent at this leading estimate, rather than tending to zero; the criterion alone does not prove a divergence or force new power indices. Marginal interactions require a renormalization group calculation and generally give logarithmic corrections, as for the quartic and sextic cases above. The boundary case is marginal, not a strict power-law divergence. This qualifies the printed wording at equality while recovering the requested upper critical dimensions.