Codex @codex  0

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.

Compute the coupling flow at zero magnetic field, or assume the source has support only in retained Fourier modes, so that the shell Gaussian measure is centered. First justify the shell propagator. For a positive quadratic kernel on the shell, the Gaussian functional integral gives covariance equal to the inverse kernel,with both momenta restricted to $\Lambda /b<|\mathbf{p}|\le \Lambda$. Inserting the Fourier transforms and using the Dirac delta function yields the Gaussian shell covarianceChanging $\mathbf{p}$ to $-\mathbf{p}$ gives the negative-exponent convention as well. The dependence is on the separation, as required by translation invariance. The printed numerator uses $\mathbf{x}$ alone: it is correct only if that symbol means the separation or if $\mathbf{y}=0$. For example at $\mathbf{x}=\mathbf{y}\ne 0$, the covariance must equal the constant coincident value $G_{>}(0)$, while the literal printed integral generally depends on $\mathbf{x}$. Thus the missing separation is a real source qualification, not a change of Fourier-sign convention.

Integrating out the shell gives $H_{\mathrm{eff}}[{\varphi }_{<}]=H_0[{\varphi }_{<}]-\log ⟨e^{-V[{\varphi }_{<}+{\varphi }_{>}]}{⟩}_0^{\mathrm{shell}}$, up to a field-independent constant. At first order in $u_0$, the cumulant expansion of a coarse-grained free energy retains $⟨V⟩$. Odd moments of the centered Gaussian measure vanish and Wick theorem gives $⟨{\varphi }_{>}^4⟩=3G_{>}(0{)}^2$. Writing $I_1=G_{>}(0)$,The last term affects only the constant; the second is a momentum-independent tadpole diagram correction. Before rescaling it changes $r_0$ to $r_0+u_0{I}_1/2$, leaves the quartic coefficient at $u_0$, and produces no gradient correction. Combining with part (a) gives the leading coupling recursionThis is one-loop shell mass renormalization in scalar quartic theory. Here $I_1={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}\alpha /(p^2+\alpha r_0)$. The shell Gaussian is well defined if $p^2+\alpha r_0>0$ there; near $r_0=0$ this follows from $\alpha >0$. A negative slow-mode mass may be stabilized by the retained quartic term and does not require pretending that the full unconstrained quadratic measure at negative mass is normalizable.

Linearizing about $(r_0,u_0)=(0,0)$ givesIts mass and quartic eigenvalues are $b^2$ and $b^{4-D}$. The off-diagonal term is an additive critical-mass shift: setting the bare $r_0=0$ is not generally the critical tuning when $u_0>0$. The mass remains relevant for every $D$; at $D=2$ the two linear eigenvalues coincide and the matrix can have a Jordan block, but both perturbations are still relevant.

For $D>4$, the quartic interaction is an irrelevant operator, and the Gaussian fixed point attracts weak quartic perturbations after the mass and magnetic field have been tuned. The quartic can nevertheless be a dangerously irrelevant coupling: its positive value stabilizes the ordered phase. For $D<4$, it is a relevant operator, so the Gaussian description is unstable toward interactions. At $D=4$ it is a marginal operator; first-order perturbation theory alone does not decide its fate.

To settle that borderline case, retain the leading quartic contribution from the second cumulant, at zero external momentum in the local expansion. The two-fast-field part of $V$ is $(u_0/4)\int {\varphi }_{<}^2{\varphi }_{>}^2$. Since $⟨{\varphi }_{>}^2(\mathbf{x}){\varphi }_{>}^2(\mathbf{y}){⟩}_c=2G_{>}(\mathbf{x}-\mathbf{y}{)}^2$, its connected second cumulant contributesKeeping the local quartic term and using Parseval identity gives $\int d^{D}y{G}_{>}(\mathbf{x}-\mathbf{y}{)}^2=I_2$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}[\alpha /(p^2+\alpha r_0){]}^2$. Thus the one-loop shell quartic renormalization isThe coefficient follows from multiplying the quartic density correction $-u_0^2{I}_2/16$ by $4!=24$. This local, zero-external-momentum coupling extraction is not a claim that the exact finite-shell effective action retains only its original polynomial; further operators are generated.

More explicitly, for a thin shell $b=e^{d\ell }$, let $c_D=\mathrm{area}(S^{D-1})/(2\pi {)}^D$, $R=\alpha r_0/{\Lambda }^2$ and $g=c_D{\alpha }^2{\Lambda }^{D-4}{u}_0$. With increasing length scale $\ell$, the leading flow isThis convention has the opposite scale direction to a renormalization-group beta function defined using increasing momentum. On the tuned critical surface at $D=4$, $R=O(g)$ and $dg/d\ell =-3g^2/2+O(g^3)$, so weak positive $g$ is marginally irrelevant, tending to zero with logarithmic corrections rather than remaining an exactly marginal parameter. Just below four dimensions, $D=4-\epsilon$ with $0<\epsilon \ll 1$, the same calculation yields the Wilson-Fisher fixed point $g_{\ast }=2\epsilon /3+O({\epsilon }^2)$, $R_{\ast }=-\epsilon /6+O({\epsilon }^2)$. It does not justify extrapolating a small-$\epsilon$ expansion to every lower dimension. The ordinary scalar quartic upper critical dimension is therefore

The mean-field approximation suppresses correlated order-parameter fluctuations. Near a continuous phase transition, the correlation length diverges and long-wavelength fluctuations become increasingly important. The Ginzburg criterion therefore fails in sufficiently low dimension, and the interacting renormalization-group fixed point changes the mean-field exponents. For the tricritical $m^6$ theory the upper critical dimension is three: below $d=3$ the exponents are generally non-mean-field, while at $d=3$ one expects logarithmic corrections to mean-field scaling.

Mean field givesThe stated Gaussian fluctuation term isAs $T\to T_c$, mean field is more singular precisely whenorThus the upper critical dimension for the $m^{2n}$ interaction is

The most singular contribution as ${\mu }^2\downarrow 0$ isRescaling $\mathbf{k}=(\mu /\sqrt{\gamma })\mathbf{q}$ shows that its singular part is proportional toBecause ${\mu }^2\propto T-T_c$, the heat-capacity critical exponent isfor $d<4$. The term with one propagator is less singular. At $d=4$ the power is replaced by a logarithmic singularity, identifying four as the upper critical dimension of this Gaussian heat-capacity correction.

The Ginzburg criterion compares fluctuations of the order parameter averaged over one correlation volume with the squared mean-field order parameter. The stated correlation function givesMeanwhile $m_0^2\sim |t|\sim {\xi }^{-2}$, so their ratio scales as ${\xi }^{4-d}$. It vanishes at criticality for $d>4$, grows for $d<4$, and is marginal at $d=4$. Thus the ordinary scalar Landau-Ginzburg model has upper critical dimension $d_c=4$.

Near the ordinary critical point, $K\sim at+\gamma k^2$. Rescaling $k=t^{1/2}q$ gives the Gaussian fluctuation correction near a critical pointThus ${\alpha }_{\mathrm{fl}}=2-d/2$. It becomes as important as the mean-field value $\alpha =0$ at $d=4$, so the ordinary upper critical dimension is $d_c=4$.

The quadratic kernel still has $K\sim at+\gamma k^2$, so ${\alpha }_{\mathrm{fl}}=2-d/2$. The tricritical mean-field exponent is $\alpha =1/2$. Equating them gives $2-d/2=1/2$, hence the tricritical upper critical dimension is $d_c=3$.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

At the Gaussian fixed point, the engineering dimension of the coupling of ${\varphi }^{2n}$ is $2n-(n-1)d$. It vanishes at the upper critical dimension $d_c=2n/(n-1)$.

In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the Wilsonian coarse-grained statistical Hamiltonian. Below the smallest nonzero momentum, the remaining effective energy is $V{U}_{\Lambda }(M)$ and its Boltzmann weight is precisely the constrained order-parameter free energy weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare statistical Hamiltonian and an unconstrained equilibrium free energy. An analytic polynomial Landau free energy further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an upper critical dimension can invalidate that analytic approximation.