Gaussian Fixed Point and Upper Critical Dimension

For a hundred years mean-field theory was the working tool of critical phenomena: van der Waals for fluids, Weiss for magnets, Landau for the general order-parameter picture. It predicts a universal set of critical exponents — $\beta=\tfrac12$ , $\gamma=1$ , $\nu=\tfrac12$ — that are the same in every dimension. And it is wrong: the exactly solved two-dimensional Ising model gives $\beta=\tfrac18$ , and careful three-dimensional experiments give $\beta\approx0.33$ . Why is mean-field theory sometimes exact, sometimes only qualitative, and sometimes badly off? The answer is a single number that lives at the simplest fixed point of the renormalization group. That fixed point is the Gaussian, or free-field, fixed point: keep only the quadratic part of the Landau–Ginzburg–Wilson Hamiltonian and set the quartic coupling $u$ to zero. There the field is a collection of independent Gaussian fluctuations, correlations are computable exactly, and the critical exponents are precisely the mean-field values. The deep content is not the fixed point but its stability. Power counting assigns the quartic coupling a scaling dimension $y_u=4-d$ . This number is positive below four dimensions (the coupling grows — mean-field breaks down), zero at four dimensions (marginal — logarithmic corrections), and negative above four dimensions (the coupling dies — mean-field is exact). The dimension where $y_u$ changes sign, $d_c=4$ , is the upper critical dimension. Above it the Gaussian fixed point is stable and mean-field exponents are exact; below it the fixed point turns unstable and the flow runs to a new interacting attractor, the Wilson–Fisher fixed point. This one sign change organizes the entire modern theory: it says exactly when fluctuations matter, exactly where mean-field theory earns its keep, and exactly why $d=4$ is the pivotal dimension of statistical field theory.

Key takeaways

The Gaussian (free-field) fixed point is the quadratic fixed point of the Landau–Ginzburg–Wilson action, with quartic coupling $u=0$ and free-field correlations; its critical exponents are exactly the mean-field (Landau) values.
Power counting at this fixed point gives the field dimension $[\phi]=(d-2)/2$ and the quartic coupling dimension $y_u=4-d$ — the single number that decides whether fluctuations matter.
The quartic coupling is irrelevant for $d>4$ (mean-field exact), marginal at $d=4$ (log corrections), and relevant for $d<4$ (mean-field fails, flow runs to Wilson–Fisher).
The dimension where $y_u$ changes sign, $d_c=4$ , is the upper critical dimension; the thermal coupling has $y_t=2$ for all $d$ , giving $\nu=1/y_t=\tfrac12$ at the Gaussian point.
Above $d_c$ mean-field exponents are exact, but the quartic coupling is a dangerous irrelevant variable: the free-energy amplitude is singular as $u\to0$ , so hyperscaling $\nu d=2-\alpha$ is violated and $\alpha=0$ rather than $2-d/2$ .
Key idea: power counting at the free-field fixed point gives the $\phi^4$ coupling scaling dimension $y_u=4-d$ ; the sign change at $d=4$ is the upper critical dimension separating the regime where mean-field exponents are exact ( $d>4$ ) from the regime governed by the Wilson–Fisher fixed point ( $d<4$ ).

The core insight

Power counting at the free-field fixed point gives the $\phi^4$ coupling scaling dimension $y_u=4-d$ ; the sign change at $d=4$ is the upper critical dimension $d_c$ , separating the regime where mean-field exponents are exact ( $d>4$ ) from the fluctuation-dominated regime controlled by the Wilson–Fisher fixed point ( $d<4$ ).

Where this lives today

The Gaussian fixed point and its upper critical dimension are the organizing principle behind the "triviality" of scalar field theory in four dimensions, a fact with direct consequences for particle physics: because the $\phi^4$ coupling is marginally irrelevant at $d=4$ , a pure single-scalar sector flows to a free theory in the continuum limit unless a cutoff is kept, which underlies lattice bounds on the Higgs mass. In condensed matter, the same counting explains why the fully connected (infinite-range) Ising model and high-dimensional simulations return exactly Landau exponents, and why four-dimensional lattice $\phi^4$ simulations show the tell-tale multiplicative logarithmic corrections. The concept also travels beyond $\phi^4$ : percolation and the Ising spin glass have upper critical dimension $d_c=6$ , self-avoiding walks and ordinary tricritical points sit at their own $d_c$ , and in every case the Gaussian-fixed-point power counting tells the practitioner which dimension marks the boundary of mean-field validity before a single loop is computed.

The physical picture

Picture the order-parameter field near criticality as a bath of fluctuation modes, one for each wavevector $q$ . At the Gaussian fixed point these modes are independent Gaussian variables with a propagator $1/q^2$ at $T_c$ . Mean-field theory is exactly the assumption that these modes do not talk to one another — that the action is quadratic. The quartic coupling $u$ is the interaction that lets them scatter; whether it survives coarse-graining is the whole question. Whether $u$ matters is a competition between phase space and amplitude. Coarse-graining sums over short-wavelength modes; the more spatial dimensions there are, the more of those modes there are to self-average, and the more the nonlinearity washes out. Above four dimensions there is so much phase space that fluctuations are tame and $u$ is irrelevant — it flows to zero and the free-field description is exact. Below four dimensions the interaction accumulates faster than it can average away, and the free-field picture collapses. The borderline is exactly $d=4$ . The Ginzburg criterion reaches the same conclusion from a back-of-the-envelope estimate: compare the fluctuation of the order parameter inside one correlation volume $\xi^d$ to its mean-field value squared. The ratio scales as $|t|^{(d-4)/2}$ , which diverges as $t\to0$ for $d<4$ (fluctuations win) and vanishes for $d>4$ (mean-field is self-consistent). Both the RG power counting and the physical fluctuation estimate agree on $d_c=4$ .

Figure

The single number y_u = 4 − d and the sign flip at the upper critical dimension d_c = 4

Figure

RG flow near the Gaussian fixed point: stable above four dimensions, unstable below

The core equations

Every statement about the Gaussian fixed point starts from the coarse-grained Landau–Ginzburg–Wilson Hamiltonian, the effective energy for the smooth order-parameter field after microscopic degrees of freedom have been averaged out. It contains a gradient term that penalizes non-uniformity, a mass term whose coefficient is the reduced temperature, and a quartic self-interaction. The Gaussian fixed point is obtained by keeping only the first two.

\mathcal{H}[\phi]=\int \mathrm{d}^d x\left[\tfrac{1}{2}(\nabla\phi)^2+\tfrac{1}{2}\,t\,\phi^2+u\,\phi^4\right] \qquad (1)

where: $\phi(x)$ — coarse-grained order-parameter field (single component shown; an $n$ -component field replaces $\phi^2\to|\vec\phi|^2$ ); $t$ — reduced temperature, $t\propto(T-T_c)$ , the coefficient of the mass term; $u$ — quartic self-coupling whose relevance defines $d_c$ ; $d$ — spatial dimension. At the Gaussian fixed point one sets $u=0$ and studies whether that choice is stable under coarse-graining. In Fourier space the quadratic action is diagonal: each mode is an independent Gaussian variable governed by the inverse propagator $q^2+t$ . This is why "free field" and "mean-field" mean the same thing here — no mode feels any other, so fluctuations cannot correct one another.

\langle\tilde\phi(q)\,\tilde\phi(q')\rangle_0=\frac{(2\pi)^d\,\delta^d(q+q')}{q^2+t} \qquad (2)

where: $\tilde\phi(q)$ — Fourier component at wavevector $q$ ; $\langle\cdots\rangle_0$ — average in the Gaussian ensemble ( $u=0$ ); $q^2+t$ — inverse propagator, reducing to the massless form $1/q^2$ at criticality $t=0$ . The power-law decay $\langle\phi(x)\phi(0)\rangle\sim1/x^{d-2}$ that follows is exactly the $\eta=0$ statement: the field carries no anomalous dimension. The rescaling that leaves the gradient term invariant fixes the canonical (engineering) dimension of the field. This is the pivot: once $[\phi]$ is fixed, the scaling dimension of every coupling is pure counting.

[\phi]=\frac{d-2}{2}\,,\qquad x'=x/b\;,\;\;\phi'(x')=b^{(d-2)/2}\,\phi(x) \qquad (3)

where: $b=e^{\ell}>1$ — the rescaling factor under coarse-graining; $[\phi]$ — canonical scaling dimension of the field, chosen so $\int\mathrm{d}^d x\,(\nabla\phi)^2$ is RG-invariant. Because there is no loop correction to this dimension at the Gaussian point, the anomalous dimension vanishes, $\eta=0$ . Counting powers of $b$ for the mass and quartic terms gives their RG eigenvalues. The thermal eigenvalue is $2$ in every dimension; the quartic eigenvalue is $4-d$ , and its sign is the whole story.

y_t=2\,,\qquad y_u=4-d\,,\qquad d_c=4 \iff y_u=0 \qquad (4)

where: $y_t$ — RG eigenvalue of the temperature coupling (always relevant, $y_t=2$ ); $y_u$ — RG eigenvalue of the quartic coupling; $d_c=4$ — upper critical dimension, the value of $d$ at which $y_u$ changes sign. Reach for these when you need to know whether an interaction survives coarse-graining: a positive eigenvalue means it grows and reshapes the critical behavior, a negative one means it dies and leaves mean-field intact. The thermal eigenvalue immediately fixes the correlation-length exponent, and the canonical field dimension fixes the rest at their Landau values.

\nu=\frac{1}{y_t}=\frac{1}{2}\,,\quad \eta=0\,,\quad \gamma=\nu(2-\eta)=1 \qquad (5)

where: $\nu$ — correlation-length exponent, $\xi\sim|t|^{-\nu}$ ; $\eta$ — anomalous dimension, zero at the Gaussian point; $\gamma$ — susceptibility exponent, $\chi\sim|t|^{-\gamma}$ . These are the mean-field exponents; they are exact for $d>4$ (where $u$ is irrelevant) and only the starting point of a correction series for $d<4$ . The Ginzburg criterion recovers $d_c=4$ from a self-consistency estimate rather than from RG power counting, which is why the two arguments together are convincing.

\frac{\langle\delta\phi^2\rangle_{\xi^d}}{\phi_{\rm mf}^2}\;\propto\;|t|^{(d-4)/2} \qquad (6)

where: $\langle\delta\phi^2\rangle_{\xi^d}$ — order-parameter fluctuation averaged over one correlation volume; $\phi_{\rm mf}\sim|t|^{1/2}$ — mean-field equilibrium order parameter. As $t\to0$ the ratio diverges for $d<4$ (fluctuations overwhelm the mean field) and vanishes for $d>4$ (mean field self-consistent); the borderline is $d=4$ .

Derivation AScaling dimensions and the upper critical dimension by power counting

The goal is to determine the RG eigenvalue of every coupling at the Gaussian fixed point purely by dimensional analysis, and to read off the upper critical dimension from where the quartic eigenvalue vanishes. At the Gaussian fixed point only the kinetic term survives, so every scaling dimension follows from requiring that one term be invariant under coarse-graining. The strategy is to fix the field dimension from the gradient term, then count powers of the rescaling factor for each operator; the exponent that appears is that operator's RG eigenvalue, and the operator is relevant, marginal, or irrelevant according to its sign.

Step 1. Fix the field dimension by demanding the gradient term be invariant: $\int\mathrm{d}^dx\,(\nabla\phi)^2$ picks up $b^{d}$ from the volume and $b^{-2}$ from the two derivatives, so the field must carry $b^{-(d-2)/2}$ , giving $[\phi]=(d-2)/2$ . Why: the gradient term is the only survivor at the Gaussian point, so it alone sets the field normalization, and the absence of a loop correction to it is precisely the statement $\eta=0$ .

Step 2. Write the eigenvalue of a general operator $\partial^{N_\partial}\phi^{N_\phi}$ integrated over space as $y_{\mathcal{O}}=d-N_\partial-N_\phi\,[\phi]$ . Why: the volume element contributes $+d$ , each derivative $-1$ , and each field $-[\phi]$ ; the net power of $b$ multiplying the coupling is its RG eigenvalue.

Step 3. Apply to the mass term $t\,\phi^2$ ( $N_\partial=0$ , $N_\phi=2$ ): $y_t=d-2\cdot\tfrac{d-2}{2}=2$ . Why: the temperature coupling has no derivatives and two fields, and the $d$ -dependence cancels, leaving $y_t=2$ in every dimension — temperature is always relevant.

Step 4. Apply to the quartic term $u\,\phi^4$ ( $N_\partial=0$ , $N_\phi=4$ ): $y_u=d-4\cdot\tfrac{d-2}{2}=4-d$ . Why: four fields bring $-2(d-2)$ , and adding the volume's $+d$ leaves $4-d$ ; this is the number whose sign decides the fate of mean-field theory.

Step 5. Identify the marginal dimension by setting $y_u=0$ , giving $d_c=4$ . Why: at $d=4$ power counting is silent (the leading eigenvalue vanishes), so the coupling's fate is decided by one-loop corrections, producing the logarithmic corrections to scaling seen exactly at the upper critical dimension.

Step 6. Check that higher operators do not spoil the truncation: $y_{\phi^6}=d-6\cdot\tfrac{d-2}{2}=6-2d$ , negative for $d>3$ , so $\phi^6$ is strongly irrelevant near $d=4$ and can be dropped. Why: this justifies truncating the Landau–Ginzburg–Wilson Hamiltonian at fourth order — the sixth-order coupling has its own, lower upper critical dimension $d=3$ and never competes near $d=4$ . *Verified: verify_aq-sm-gaussian-fixed-point-upper-critical-dimension.py confirms $y_t=2$ , $y_u=4-d$ , $y_{\phi^6}=6-2d$ , and $d_c=4$ symbolically, and the sign of $y_u$ at $d=3,4,5$ .*

Go deeper · Derivation B · 2 minMean-field exponents from the Gaussian eigenvalues

Derivation BMean-field exponents from the Gaussian eigenvalues

The goal is to convert the two RG eigenvalues into the full set of critical exponents and confirm they reduce to the standard Landau values, fixing along the way a placeholder that the base export left in the anomalous-dimension relation. Two eigenvalues, the thermal $y_t$ and the magnetic $y_h$ (the eigenvalue of the field $h$ that couples linearly to $\phi$ ), determine every static exponent through the scaling and hyperscaling relations. The plan is to compute $y_h$ by the same power counting, then feed both eigenvalues through the exponent formulas, checking each against the Landau value at $d=4$ .

Step 1. Compute the magnetic eigenvalue: $h$ couples as $\int\mathrm{d}^dx\,h\phi$ , so $y_h=d-[\phi]=d-\tfrac{d-2}{2}=\tfrac{d+2}{2}$ . Why: the ordering field has one field and no derivatives, so its eigenvalue is $d$ minus a single field dimension.

Step 2. Read off the correlation length: $\nu=1/y_t=\tfrac12$ . Why: the correlation length diverges as $\xi\sim|t|^{-1/y_t}$ along the relevant thermal direction, and $y_t=2$ gives the Landau value directly.

Step 3. Fix the anomalous dimension from the eigenvalue relation $\eta=d+2-2y_h$ , which the base export wrote with a stray placeholder " $?$ " in the denominator. Substituting $y_h=(d+2)/2$ gives $\eta=d+2-(d+2)=0$ . Why: the exact eigenvalue-to- $\eta$ relation is $\eta=d+2-2y_h$ , not the malformed expression in the base; at the canonical field dimension it returns $\eta=0$ , consistent with the free-field propagator decaying as $1/x^{d-2}$ .

Step 4. Get the susceptibility from the Fisher relation $\gamma=\nu(2-\eta)=\tfrac12\cdot2=1$ . Why: with $\nu=\tfrac12$ and $\eta=0$ the Fisher relation returns the Landau value $\gamma=1$ .

Step 5. Get the magnetization and isotherm exponents from $\beta=(d-y_h)/y_t$ and $\delta=y_h/(d-y_h)$ ; at $d=4$ these give $\beta=\tfrac12$ and $\delta=3$ . Why: away from $d=4$ these are $d$ -dependent at the bare Gaussian point, but at the physical upper critical dimension they collapse to the Landau values, and for $d>4$ the dangerous-irrelevant-variable mechanism restores the $d$ -independent mean-field numbers.

Step 6. Get the specific-heat exponent from hyperscaling $\alpha=2-\nu d=2-d/2$ , which equals $0$ only at $d=4$ . Why: naive hyperscaling gives a $d$ -dependent $\alpha$ , but above four dimensions the true value is the $d$ -independent $\alpha=0$ — the signature that $u$ , although irrelevant, is dangerous and must be kept in the free-energy amplitude. *Verified: verify_aq-sm-gaussian-fixed-point-upper-critical-dimension.py confirms $y_h=(d+2)/2$ , the corrected relation $\eta=d+2-2y_h=0$ , and the Landau set $(\nu,\eta,\gamma,\beta,\delta,\alpha)=(\tfrac12,0,1,\tfrac12,3,0)$ at $d=4$ .*

Limiting cases

The entire behavior is dictated by the sign of $y_u=4-d$ , so the limiting cases are organized by dimension:

Case	Result	Meaning
$d>4$ (above the upper critical dimension)	$y_u=4-d<0$	Quartic coupling irrelevant; Gaussian fixed point stable; mean-field exponents exact (with dangerous-irrelevant amplitude corrections).
$d=4$ (upper critical dimension)	$y_u=0$	Quartic coupling marginal; mean-field exponents hold but with logarithmic corrections, e.g. $\chi\sim\|t\|^{-1}\|\ln\|t\|\|^{1/3}$ for $n=1$ .
$d<4$ (below the upper critical dimension)	$y_u=4-d>0$	Gaussian fixed point unstable; flow runs to the Wilson–Fisher fixed point; exponents depart from mean-field.
$d\le2$ , $n\ge2$ (lower critical dimension)	$d_\ell=2$	Continuous-symmetry order is destroyed by Goldstone fluctuations; the Gaussian/Landau picture of a finite- $T$ transition fails entirely.

Worked examples

Example 1 — The sign of $y_u$ across dimensions, and hyperscaling at $d=4$ . This computes the number that governs the whole theory and checks the specific-heat exponent at the borderline.

Step 1. Evaluate $y_u=4-d$ at three dimensions: at $d=3$ , $y_u=+1$ (relevant); at $d=4$ , $y_u=0$ (marginal); at $d=5$ , $y_u=-1$ (irrelevant). So mean-field theory is destabilized in three dimensions, marginal in four, and exact in five.

Step 2. Confirm the thermal eigenvalue is dimension-independent: $y_t=d-2\cdot\tfrac{d-2}{2}=d-(d-2)=2$ , giving $\nu=1/y_t=\tfrac12$ in every dimension.

Step 3. Test hyperscaling at $d=4$ : $\alpha=2-\nu d=2-\tfrac12\cdot4=0$ . The specific-heat exponent vanishes exactly at the upper critical dimension, consistent with the Landau value.

Step 4. Note the failure just above: at $d=5$ , naive hyperscaling gives $\alpha=2-\tfrac52=-\tfrac12$ , but the true mean-field value is $\alpha=0$ . The discrepancy is the dangerous-irrelevant-variable signal. The governing number is $y_u=4-d$ : relevant below four dimensions, marginal at four, irrelevant above.

Example 2 — Ginzburg criterion recovers $d_c=4$ . This shows the fluctuation estimate agrees with power counting.

Step 1. Take the mean-field order parameter squared, $\phi_{\rm mf}^2\sim|t|^{2\beta}=|t|$ with $\beta=\tfrac12$ .

Step 2. Estimate the fluctuation inside a correlation volume: integrating the propagator gives $\langle\delta\phi^2\rangle_{\xi^d}\sim\xi^{-(d-2)}$ and, with $\xi\sim|t|^{-1/2}$ , this is $|t|^{(d-2)/2}$ .

Step 3. Form the ratio: $\langle\delta\phi^2\rangle/\phi_{\rm mf}^2\sim|t|^{(d-2)/2-1}=|t|^{(d-4)/2}$ .

Step 4. Take $t\to0$ : the ratio diverges for $d<4$ and vanishes for $d>4$ , with the borderline at $d=4$ . The fluctuation estimate reproduces the RG result: the upper critical dimension is $d_c=4$ .

Origins · 1 mingaussian fixed point and how this was discovered

gaussian fixed point

free-field fixed point of the RG · The free-field or "Gaussian" fixed point of the renormalization group is the fixed point at which the coarse-grained action is purely quadratic and every fluctuation mode is an independent Gaussian variable. It is the anchor of the modern theory of criticality: its stability analysis, through the scaling dimension

y_u=4-d

of the quartic coupling, defines the upper critical dimension

d_c=4

and demarcates exactly where mean-field theory is exact. Formalized in the momentum-shell language by Kenneth Wilson in 1971, it is the point from which the Wilson–Fisher fixed point branches for

d<4

, and the reference against which all anomalous dimensions are measured.

History

1937

— Landau formulates the general mean-field (order-parameter) theory of continuous transitions, predicting dimension-independent exponents

1944

— Onsager's exact 2D Ising solution gives β = 1/8, contradicting mean-field β = 1/2 and proving the Landau exponents are not universal

1960

— Ginzburg gives the fluctuation criterion locating where mean-field theory fails, implicitly identifying four dimensions

1966

— Kadanoff's block-spin picture supplies the scaling intuition behind fixed points

1971

— Wilson formulates the momentum-shell RG, making the Gaussian fixed point and its stability precise

1972

— Wilson and Fisher show the Gaussian point is unstable for d < 4 and locate the interacting fixed point that replaces it

Going further

References. Kardar, Statistical Physics of Fields, Ch. 4–5 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 5–7 · Cardy, Scaling and Renormalization in Statistical Physics, Ch. 3–5 · Pathria & Beale, Statistical Mechanics (3rd ed.), Ch. 14 · Wilson & Kogut, Phys. Rep. 12 (1974) 75 · Amit & Martín-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena.

Provenance. All scaling dimensions follow from canonical power counting at the Gaussian fixed point: a verifier recomputes $[\phi]=(d-2)/2$ from invariance of $\int\mathrm{d}^dx\,(\nabla\phi)^2$ , then $y_t=d-2[\phi]=2$ and $y_u=d-4[\phi]=4-d$ , giving $d_c=4$ . Mean-field exponents $(\nu,\eta,\gamma,\beta,\delta,\alpha)=(\tfrac12,0,1,\tfrac12,3,0)$ at $d=4$ are standard (Pathria & Beale Ch. 12–14; Kardar Ch. 4–5). The Ginzburg exponent $(d-4)/2$ and the one-loop coefficient $(n+8)$ are standard (Wilson–Kogut; Amit). The logarithmic-correction exponent $1/3$ for $n=1$ at $d=4$ is the textbook value (Goldenfeld). Base export correction: the anomalous-dimension relation was carried as a malformed expression with a placeholder denominator; the correct eigenvalue relation $\eta=d+2-2y_h$ (giving $\eta=0$ ) is used and verified here.

Builds on: Momentum-Shell Renormalization Group · Fixed Points of the RG · Linearization About a Fixed Point and Eigenvalue Spectrum · Relevant, Irrelevant, and Marginal Operators · Ginzburg Criterion. Leads to: Wilson–Fisher Fixed Point · The Epsilon-Expansion · Critical Exponents from the RG.

Is the Gaussian fixed point the same thing as mean-field theory?

Almost, but with a sharpened meaning. Mean-field theory neglects fluctuation feedback, which is exactly the statement that the action is quadratic — that is the Gaussian fixed point. The RG upgrade is that mean-field is not just an approximation but the exact long-distance description whenever the quartic coupling is irrelevant, i.e. for $d>4$ . Below four dimensions the Gaussian fixed point still exists as a solution of the RG equations, but it is unstable, so it no longer controls the physics — the Wilson–Fisher fixed point does.

Why is the anomalous dimension exactly zero at this fixed point?

At the Gaussian fixed point the action is purely quadratic, so the only term that can fix the field's normalization under rescaling is the gradient term $(\nabla\phi)^2$ . Requiring it to be RG-invariant forces the canonical value $[\phi]=(d-2)/2$ , and there is no loop to correct it. Since $\eta$ measures precisely the correction to the field's scaling dimension away from canonical — equivalently, the two-point function decays as $1/x^{d-2+\eta}$ — a canonical dimension means $\eta=0$ . A nonzero $\eta$ requires interactions, which first appear at the Wilson–Fisher point (and there only at two loops).

Why is

d=4

special rather than

d=3

d=2

Because the quartic operator $\phi^4$ has scaling dimension $y_u=4-d$ , which changes sign precisely at $d=4$ . The number 4 is not an accident of Ising physics; it is set by the operator you are testing. A $\phi^6$ interaction has $y=6-2d$ and its own upper critical dimension $d=3$ ; percolation and spin glasses, controlled by different operators, have $d_c=6$ . The upper critical dimension is a property of the interaction, computed by counting the fields and derivatives it contains against the canonical field dimension.

What actually happens right at

d=4

The quartic coupling is marginal — power counting says it neither grows nor shrinks — so its fate is decided by the one-loop term in its flow, $\mathrm{d}u/\mathrm{d}\ell\sim-u^2$ . This drives $u$ slowly to zero as $u(\ell)\sim1/\ell$ , a logarithmic rather than power-law approach to the Gaussian fixed point. The physical fingerprint is multiplicative logarithmic corrections to mean-field scaling: for the $n=1$ class the susceptibility behaves as $\chi\sim|t|^{-1}|\ln|t||^{1/3}$ . Four-dimensional lattice field-theory simulations confirm these log corrections.

u

is irrelevant for

d>4

, why do we keep talking about it?

Because "irrelevant" is a statement about the flow, not about every observable. The exponents above four dimensions are indeed the $u$ -independent mean-field values. But the free-energy amplitude is singular as $u\to0$ : the scaling form carries a factor like $u^{-(d-4)/2}$ . An irrelevant variable that appears singularly in an observable is called dangerous, and $u$ is the textbook example. Keeping its singular dependence is what restores the correct $d$ -independent $\alpha=0$ and explains why naive hyperscaling ( $\nu d=2-\alpha$ ) fails above four dimensions.

How does this fixed point connect to the Wilson–Fisher fixed point?

They are two solutions of the same RG flow equations. For $d>4$ the Gaussian point ( $u^*=0$ ) is stable and the only game in town. Exactly at $d=4$ they coincide. For $d<4$ the Gaussian point becomes unstable in the $u$ direction (its eigenvalue turns positive), and a second fixed point — Wilson–Fisher — splits off at $u^*\propto\varepsilon=4-d>0$ , taking over as the stable attractor. The two fixed points exchange stability at $d=4$ , a transcritical bifurcation in the space of couplings. The Gaussian point supplies the starting point and the small parameter for the epsilon-expansion that reaches the Wilson–Fisher point.

Does the lower critical dimension come out of this same analysis?

Only partly. The upper critical dimension $d_c=4$ is a Gaussian-fixed-point statement: it is where fluctuations begin to overwhelm mean-field theory. The lower critical dimension — below which no ordered phase survives at $T>0$ — is a different question about whether long-range order can exist at all, and it depends on the symmetry: $d_\ell=1$ for the discrete Ising case, $d_\ell=2$ for continuous $O(n\ge2)$ symmetry (destroyed by Goldstone modes, the Mermin–Wagner statement). The Gaussian power counting locates $d_c$ ; the lower critical dimension needs a separate spin-wave or domain-wall argument.