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An additive subtraction can produce a source in the renormalization-group equation. If $G_R=G_B-a\log ({\Lambda }_0/\Lambda )$ and the bare correlator is invariant under $\mathcal{D}=\Lambda {\partial }_{\Lambda }+\beta (g){\partial }_g$, then $\mathcal{D}{G}_R=a$. A vanishing operator anomalous dimension does not remove this source.

For fixed external momenta, the massless Callan-Symanzik equation is $(\mu {\partial }_{\mu }+\beta (\lambda ){\partial }_{\lambda }+n{\gamma }_{\varphi })G_n=0$. Choose a running coupling satisfying $d\lambda /d\log \mu =\beta (\lambda )$ and $\lambda ({\mu }_0)={\lambda }_0$. The chain rule reduces the equation along this curve to $d{G}_n/d\log \mu =-n{\gamma }_{\varphi }{G}_n$, whose integrating factor gives the displayed solution. The reference correlation function at ${\mu }_0$ supplies boundary data; the renormalization group alone does not determine these data.

If $\beta =b{\lambda }^2$, then $\lambda (\mu )={\lambda }_0/[1-b{\lambda }_0\log (\mu /{\mu }_0)]$. If in addition the anomalous dimension is ${\gamma }_{\varphi }=c{\lambda }^2$, the integral in the exponent is $(c/b)(\lambda (\mu )-{\lambda }_0)$. These formulas use the stipulated truncated renormalization group functions and apply on a branch on which the running solution remains defined.

Renormalization can relate bare operators to a matrix of renormalized operators, ${\mathcal{O}}_{i,B}=Z_{ij}{\mathcal{O}}_{j,R}$. The corresponding anomalous dimension matrix acts on each insertion slot in the Callan-Symanzik equation.

A contact term is a distribution supported where insertion points coincide. Its Fourier transform is local in momentum, typically polynomial. Renormalizing an operator product can require additive contact counterterms even if each operator has zero anomalous dimension.

After integrating a free-field momentum shell, restore the cutoff with $x^{\prime }=x/b$ and ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$. The gradient coefficient is invariant, the mass changes as $r^{\prime }=b^{2}r$, and a uniform conjugate field changes as $h^{\prime }=b^{(D+2)/2}h$. This follows by counting measure, derivative and field factors in the quadratic Hamiltonian. A nonuniform source instead obeys $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, with the source on the right projected onto the retained Fourier modes. The field has engineering dimension $(D-2)/2$ and zero anomalous dimension.

A connected correlation function with critical anomalous dimension $\eta$ scales as $G(r)={\xi }^{-(D-2+\eta )}\mathcal{G}(r/\xi )$. If its large-distance shape is $s^{-(D-1)/2}{e}^{-s}$, substitution of $s=r/\xi$ produces the displayed amplitude. The extra ${\xi }^{-\eta }$ relative to the massive Gaussian field correlation tail is necessary for matching at $r$ of order $\xi$. Integrating over space gives magnetic susceptibility proportional to ${\xi }^{2-\eta }$; omitting that factor would instead make the long-distance contribution scale as ${\xi }^2$.

Critical exponents describe leading singular powers near a continuous phase transition. Using the reduced temperature $t=(T-T_c)/T_c$, the standard definitions includeThese are the order-parameter critical exponent, magnetic-susceptibility critical exponent, heat-capacity critical exponent and correlation-length critical exponent. The critical-isotherm exponent is defined by $M(0,H)\sim \mathrm{sgn}(H)|H{|}^{1/\delta }$. The anomalous dimension $\eta$ gives the critical long-distance connected correlation function, $G_c(x)\sim |x{|}^{-(d-2+\eta )}$, when an ordinary scale-invariant critical point exists.

In the Landau approximation, minimize the Landau free energy to obtain the equation of state, differentiate it with respect to the field conjugate to an order parameter to obtain magnetic susceptibility, and differentiate the minimized free energy with respect to temperature to obtain heat capacity. These operations determine $\beta ,\delta ,\gamma ,\alpha$, respectively. Expanding the Landau-Ginzburg theory about its uniform minimum gives the quadratic fluctuation kernel $K{q}^2+V^{\prime \prime }(M)$, whose Fourier transform determines the correlation length and the anomalous dimension. At the ordinary quartic critical point this gives ${\xi }^2=K/V^{\prime \prime }(M)\propto |t{|}^{-1}$ and $\eta =0$.

For $u>0$, $r=0$ gives $H=u{M}^3$ and hence $\delta =3$; the equilibrium free energy $-r^2/(4u)$ gives a finite heat capacity discontinuity, conventionally $\alpha =0$. Together with the explicit calculation below, the ordinary mean-field critical exponents areThese describe neglect of fluctuation corrections, rather than the exact exponents of every low-dimensional Landau-Ginzburg theory.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.

The identity-operator contribution to renormalization-group free energy $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to free energy, including entropy and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total partition function and its temperature derivatives. Iterating its recursion givesThus $N_p{C}_p$ is the original constant $N{C}_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.

To obtain the additive free-energy recursion under blocking, use energy-valued free energy per current lattice site,The blocking kernel normalization impliesSubstituting the recursion for $C^{\prime }$ cancels $C$, givingA second step gives $f_0(u_0)=b^{-2D}{f}_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction provesFor dimensionless free energy $-\log Z/N$, multiply $F,f_0,g,C$ throughout by ${\beta }_T$.

The singular free-energy density must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, thenThus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. Analytic subtraction of an inhomogeneous renormalization recursion explains this separation. A renormalization-group free-energy resonance can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.

Near a renormalization-group fixed point $u_{\ast }$, linearize the transformation:An eigenvector of $\mathsf{A}$ with eigenvalue $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i^{\prime }=b^{y_i}{t}_i$ to leading order. A relevant operator has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An irrelevant operator has $y_i<0$ and dies away; a marginal operator has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.

The coarse spins also require a multiplicative normalization. In continuum notation, writewhere ${\varphi }_{<}$ is the retained field and $\eta$ is its anomalous dimension. This restores the normalization of the critical long-distance connected correlation function. Since a uniform field conjugate to an order parameter enters through $\int d^{D}xH\varphi$, it has eigenvalue exponent $y_h=D-x_{\varphi }=(D+2-\eta )/2$.

Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular free-energy density obeysfor a large total scale factor $B=b^p$. Choosing $B=|t{|}^{-1/y_t}$ gives the scaling hypothesis for critical phenomenaTwo temperature derivatives give the heat-capacity critical exponent; one conjugate-field derivative gives the order-parameter critical exponent; two such derivatives give the magnetic-susceptibility critical exponent. Scaling the correlation length gives $\xi (t)=B\xi (B^{y_t}t)$. Consequently,At $t=0$, instead choose $B=|H{|}^{-1/y_h}$; differentiating gives $\delta =y_h/(D-y_h)$. These scaling relations for critical exponents assume that irrelevant variables can be set to their fixed-point values without singular consequences. A dangerously irrelevant coupling or logarithmic correction requires keeping the corresponding dependence.

For the Gaussian field theory, integrate out the Fourier modes in $\Lambda /b<|q|<\Lambda$ and retain ${\varphi }_{<}$. The modes are independent at zero external field, so this Gaussian thinning transformation changes the normalization by a Gaussian determinant and leaves a quadratic statistical Hamiltonian for the retained modes. Restore the cutoff withIn the integrated statistical Hamiltonian, the volume element contributes $b^D$, each gradient contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient ${\kappa }^{-1}$ of the gradient energy is unchanged, whileIn particular $y_t=2$, $y_h=(D+2)/2$ and $\eta =0$. The Gaussian critical exponents and the Gaussian order-parameter scaling index therefore give the requested indicesThe sign convention $+h\varphi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.

There are physical qualifications to these Gaussian indices. The strictly quadratic Gaussian field theory is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal Gaussian order-parameter scaling index. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a dangerously irrelevant coupling and gives the actual mean-field $\beta =1/2$, rather than the Gaussian hyperscaling value.

The heat capacity power can also be checked directly. Apart from constants, the zero-field Gaussian free-energy density isIf $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the heat capacity. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the Gaussian specific-heat infrared threshold, and it prevents confusing singular indices with the entire observable.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ to distinguish inverse temperature from an order-parameter critical exponent. For the Ising model, define the full two-point correlation function $G(r)=⟨{\sigma }_0{\sigma }_r⟩$ and its connected correlation function $G_c(r)=G(r)-M^2$ in a translation-invariant phase. Away from a thermodynamic critical point, the large-distance envelope decays exponentially, possibly multiplied by a power of distance:This defines the correlation length in physical distance units when the limit exists. At a continuous phase transition, $\xi \to \infty$ and the critical connected correlation function typically has algebraic decay $G_c(r)\sim |r{|}^{-(D-2+\eta )}$, where $\eta$ is the anomalous dimension. Below the transition the full $G(r)$ tends to $M^2$, so the decay definition must use $G_c$.

Let $S={\sum }_r{\sigma }_r$ and $F=-{\beta }_T^{-1}\log Z$ be the extensive Helmholtz free energy. Holding the field-independent constant $C$ and other physical couplings fixed,Because the Hamiltonian contains $-hS$, ${\partial }_h\log Z={\beta }_T⟨S⟩$ and ${\partial }_h^2\log Z={\beta }_T^2(⟨S^2⟩-⟨S{⟩}^2)$. Translation invariance then proves the correlation-function susceptibility sum rule:In the ordered zero-field phase these statements refer to a selected pure thermodynamic phase; averaging equally over opposite magnetizations introduces the separate spin-mixture contribution to zero-field susceptibility.

A normalized blocking kernel $K_b({\sigma }^{\prime },\sigma )\ge 0$ satisfies ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. For example, assign one block spin to each block of $b^D$ microscopic spins using majority sign, with equal probabilities for a tie. Define the blocked Hamiltonian, retaining the entire generated operator space, byThe field-independent contribution $N{g}_E$ belongs to the identity operator. With $N^{\prime }=N{b}^{-D}$ andsumming over ${\sigma }^{\prime }$ proves $Z(u,C,N)=Z(u^{\prime },C^{\prime },N^{\prime })$ exactly. Iterating gives $N_p=N{b}^{-pD}$ and the same partition function with $(u_p,C_p,N_p)$. Observables at large scales are reproduced by appropriately blocked observables and sources. Truncating the generated interactions makes this an approximation; dropping the additive constant already destroys the equality of free energies even when normalized spin expectations remain correct.

The extensive Helmholtz free energy is unchanged: $F(u_0,C_0,N)=F(u_p,C_p,N_p)$. In contrast, its value per site changes because the number of sites changes. Define the reduced zero-constant free-energy density $f_0(u)={\beta }_{T}F(u,0,N)/N$ in the thermodynamic limit. The preceding equality yieldsand iteration proves the additive free-energy recursion under blocking:Equivalently $C_p=b^{pD}[C_0+{\sum }_{j=0}^{p-1}{b}^{-jD}{g}_E(u_j)]$. Thus the inhomogeneous source records the identity-operator contribution to renormalization-group free energy from eliminated short-distance degrees of freedom.

To express this recursion for the singular part, write $f_0=a+f_s$, where $a$ is the regular analytic background. Then the same displayed recursion holds with $f_s$ andThis is the analytic subtraction of an inhomogeneous renormalization recursion. A purely analytic source can often be absorbed into $a$, after which it does not control the leading singular powers. It is not legitimate simply to discard an analytic source in the full free energy, where it can dominate the regular background. Moreover an analytic monomial $c{t}^m{h}^n$ contributesIf $m{l}_t+n{l}_h=D$, the sum is $p$ and hence generates a logarithm when $p\sim -\log |t|/(l_t\log b)$: a renormalization-group free-energy resonance. Such a term cannot be discarded when describing that logarithmic singularity. Neglect of the inhomogeneous term in the leading power-law scaling therefore assumes that its regular part has been subtracted, that any remaining source only changes finite scaling amplitudes, and that there are no relevant resonances or marginal logarithms. Omitting an irrelevant operator further requires absence of a dangerously irrelevant coupling, whose vanishing limit can make a scaling function singular.

A renormalization-group fixed point obeys $R_b{u}_{\ast }=u_{\ast }$. Linearization gives $\delta u^{\prime }=A\delta u$; in eigenvector coordinates $w_i^{\prime }=b^{l_i}{w}_i$. Positive $l_i$ label relevant operators, negative $l_i$ label irrelevant operators, and zero $l_i$ labels a marginal operator requiring nonlinear analysis. The critical surface is the stable manifold obtained by tuning all relevant scaling fields to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the observation length increases. Locally a typical zero-field section has $dw/d\ell =-\omega w$, $dt/d\ell =l_{t}t$, $\ell =\log b$, so the critical surface $t=0$ attracts along $w$, while trajectories with either sign of $t$ depart from it. A magnetic scaling field adds another relevant direction.

Under the stated homogeneous-scaling assumptions and after fixing irrelevant fields at their limiting values, two relevant scaling fields obey $t^{\prime }=b^{l_t}t$, $h^{\prime }=b^{l_h}h$. These fields are analytic coordinates near the renormalization-group fixed point, proportional to the physical reduced temperature and conjugate field to leading order. The singular free-energy density consequently obeysHere $F_s$ denotes the singular free energy per original site or unit volume, with regular dimensional factors absorbed; the extensive singular part is $N{F}_s$. Iterate and choose $b^p=|t{|}^{-1/l_t}$ to obtain the scaling hypothesis for critical phenomena:The sign labels the two sides of the transition: $f_{+}$ describes $t>0$, the disordered side, while $f_{-}$ describes $t<0$, the ordered side. The two functions need not coincide; $f_{-}$ has the zero-field one-sided derivative appropriate to spontaneous magnetization. The numbers $l_t,l_h$ are scaling eigenvalues in logarithmic form: the corresponding discrete RG multipliers are $b^{l_t},b^{l_h}$.

The correlation length obeys $\xi (t,h)=b\xi (b^{l_t}t,b^{l_h}h)$. The same scale choice gives $\nu =1/l_t$. Differentiating $F_s$ once and twice with respect to the conjugate field, and twice with respect to temperature, givesAt $t=0$, instead choose the scale $b^p=|h{|}^{-1/l_h}$. Then $F_s(0,h)\sim |h{|}^{D/l_h}$ and $M\sim \mathrm{sgn}(h)|h{|}^{D/l_h-1}$, so $\delta =l_h/(D-l_h)$. Thus differentiation and scale matching establishThe first is the Widom scaling relation and the second the hyperscaling relation. The hyperscaling relation requires precisely the regular homogeneous scaling used here. Above the ordinary upper critical dimension, the quartic coupling is dangerously irrelevant in the ordered phase and this derivation cannot discard it: the mean-field critical exponents $\nu =1/2$, $\alpha =0$ do not obey $\alpha =2-D\nu$ for $D>4$. At marginal dimensions logarithmic factors likewise require care beyond the displayed pure powers.

Changing the ultraviolet cutoff changes which fluctuations remain explicit. Integrating out a shell of Fourier modes shifts the coefficients of all symmetry-allowed operators and a field-independent constant. Hence the cutoff-dependent coefficients are chosen so that the remaining partition function and long-distance observables reproduce the same microscopic system. This is the Wilsonian coarse-grained statistical Hamiltonian; retaining only the printed operators is a local derivative expansion, not an exact closure.

In a massive symmetric phase, the quadratic long-wavelength inverse two-point function has the form ${\alpha }^{-1}{p}^2+m^2$. Its Landau scalar correlation length satisfies ${\xi }^{-2}=\alpha m^2$ in this approximation. For canonical normalization of a scalar gradient term, set $\varphi =\sqrt{\alpha }\psi$; then the mass and quartic coefficients become $r=\alpha m^2$ and $g_c={\alpha }^{2}g$. With this normalization the infrared mass is the inverse correlation length, $m_R={\xi }^{-1}$, within the massive quadratic or one-loop approximation. More generally the decay rate is a pole mass, whereas the zero-momentum inverse magnetic susceptibility is a curvature mass; an anomalous dimension or nontrivial momentum dependence can distinguish them. In the ordered phase one must first expand about a selected stable equilibrium magnetization, rather than identify a negative quadratic coefficient at the unstable origin with a squared inverse correlation length. Thus the intended mass identification includes a field normalization and a stable-phase qualification.

For momentum-shell renormalization group, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, retaining $|p|<\Lambda /b$ and eliminating $\Lambda /b<|p|<\Lambda$. DefineTaking a logarithm gives the cumulant expansion of a coarse-grained free energy. For a centered fast Gaussian measure with $G_{>}(0)=⟨{\varphi }_{>}^2⟩$, the first quartic cumulant isThe Wick contractions give a mass shift $\Delta r=g{G}_{>}(0)/2$ and an identity-operator contribution, while leaving the quartic term at this order. The second cumulant produces higher interactions and a quartic correction. After the length and field rescalings $x=b{x}^{\prime }$ and ${\varphi }_{<}(x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$, the gradient coefficient remains one, andThe one-loop shell mass renormalization in scalar quartic theory explicitly shows why the critical bare mass is shifted by fluctuations.

Assume short-range interactions, an analytic even local potential, a positive gradient coefficient, and a weak-coupling trajectory in the basin of the Gaussian fixed point. Above four dimensions the quartic interaction has negative engineering dimension and its dimensionless strength decreases under coarse graining; higher local powers and higher derivative terms also decrease. After absorbing analytic short-distance shifts into the coefficients, long-wavelength fluctuations are small compared with the ordered order parameter. The remaining coarse Landau free energy can therefore be evaluated at its stable saddle:The quartic stabilizer must still be retained for the ordered minimum even though it is dangerously irrelevant. Stationarity, $r_{R}M+g_R{M}^3/6=h$, gives the ordinary Landau-Ginzburg theory and its mean-field critical exponents. This derives the saddle approximation from decreasing fluctuation strength under the stated RG assumptions; it does not assume that every exact coarse-grained functional becomes a quartic polynomial. Below four dimensions the quartic coupling grows, long-wavelength loop corrections cease to be small, and the argument fails. At four dimensions the interaction is marginal and generates logarithmic corrections. The tuned sextic theory has the analogous boundary at three dimensions.

Use a consistent Fourier transform pair for the connected correlation function:The printed forward-transform expression integrates over $p$ while leaving $x$ free; that integration variable is an error. The above convention resolves it and produces the loop measure used below. At zero magnetization, $G_c(x)=⟨\varphi (0)\varphi (x)⟩$; otherwise the disconnected product of expectations must be subtracted.

In the inverse-propagator identity used here, the truncated two-point function means the full one-particle-irreducible two-point vertex, including its free quadratic part. This convention differs from using “truncated” for a connected cumulant. To derive the inverse relation, introduce $W[J]=\log Z[J]$, $\phi =\delta W/\delta J$ and the effective action $\Gamma [\phi ]=\int J\phi -W[J]$. Then $\delta \Gamma /\delta \phi =J$ and the chain rule gives ${\Gamma }^{(2)}{W}^{(2)}=1$. Since $W^{(2)}$ is the connected correlation function, translation invariance gives $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$.

Now use canonically normalized coefficients, denoting the cutoff mass by $m_b^2=m^2(\Lambda ,T)$ and the renormalized infrared mass by $m_R^2=m^2(0,T)$. Choose the free propagator ${\tilde{G}}_0(p)=(p^2+m_R^2{)}^{-1}$ and put $\delta m^2=m_b^2-m_R^2$ in the interaction as a quadratic counterterm. With the Euclidean self-energy convention in which an insertion changes the connected propagator by $-G_0\Sigma G_0$, the perturbative series beginsThe self-energy sums one-particle-irreducible loop insertions; the explicit mass counterterm is kept separate. Repeated insertions generate the geometric Dyson resummation, which explains why reducible chains occur in $G$ but not as independent terms in its inverse. Beyond one loop a momentum-dependent self-energy and wave-function renormalization must also be included.

At one loop the only quartic two-point graph is the tadpole diagram. At a $g{\varphi }^4/4!$ vertex, attaching the two labeled external fields uses $4\cdot 3=12$ choices and contracts the remaining two fields together. The factor is $12/4!=1/2$. Equivalently this is the same quadratic term in the shell cumulant expansion above. ThusIt is positive for $g>0$ and independent of external momentum, so there is no one-loop gradient renormalization. The condition $\tilde{\Gamma }(0)=m_R^2$ implies $\delta m^2+\Sigma (0)=0$, henceThe cutoff is necessary: the unregulated integral is ultraviolet divergent for $D\ge 2$. Using the renormalized mass internally is a self-consistent one-loop organization, not an exact resummation of all critical fluctuations. With nonunit $\alpha$, this formula instead applies to $r=\alpha m^2$, $g_c={\alpha }^{2}g$ after canonical normalization of a scalar gradient term.

To test a linear thermal mass, approach the ordinary transition from the symmetric side and set $R=m_R^2>0$. For $D>2$ the massless tadpole is infrared finite. Subtract its critical value, absorb regular temperature dependence of the coupling into a nonzero thermal coefficient $A$, and write $m_b^2(T)-m_b^2(T_C)=A(T-T_C)+\cdots$. The one-loop critical-mass subtraction giveswhereFor $D>4$, $J_D(0)=c_D{\Lambda }^{D-4}/(D-4)$ is finite, so the bracket tends to a finite constant and $R\propto T-T_C$ is self-consistent. At $D=4$,and the growing logarithm obstructs an asymptotically constant linear coefficient at nonzero coupling. For $2<D<4$, put $k=\sqrt{R}q$ to obtainThe divergent bracket then invalidates the assumed linear thermal mass. The radial critical-mass subtraction in dimensions three to five gives explicit checks of all three behaviors. For $D\le 2$, the massless tadpole subtraction itself is infrared divergent; that failure of the Gaussian expansion does not prove absence of an interacting transition. Therefore the ordinary upper critical dimension is $D_C=4$, with the unmodified Landau-Ginzburg theory asymptotically justified by this test only for $D>4$. Exponents inferred by solving the self-consistent one-loop equation below four dimensions would not be exact interacting exponents.

For a tricritical point the renormalized quartic coefficient must be tuned to zero, and a positive sextic interaction stabilizes the order parameter. The gradient term gives the scalar field engineering dimension $(D-2)/2$. A coefficient of ${\varphi }^6$ therefore has dimension $D-6(D-2)/2=6-2D$, which becomes marginal at $D=3$. This gives $D_C^{\mathrm{tri}}=3$. A direct Ginzburg criterion reaches the same result: long-wavelength variance in a correlation volume iswhile the tricritical saddle has $M^2\sim |r{|}^{1/2}$. Their ratio scales as $|r{|}^{(D-3)/2}$, vanishing only above three dimensions. At three dimensions it is marginal; below three it grows. In contrast the ordinary saddle $M^2\sim |r|/g$ gives a ratio $g|r{|}^{(D-4)/2}$. Both calculations assume the thermal mass is measured relative to its shifted critical value. A sextic interaction generates quartic terms under coarse graining, so tricriticality requires tuning the renormalized quartic scaling field as well as the mass.

The partition function sums the Boltzmann weights of all statistical configurations. For discrete states $s$ of energy $E_s$, $Z={\sum }_s{e}^{-{\beta }_{\mathrm{th}}{E}_s}$; for a regulated scalar field it is a functional integral, $Z=\int \mathcal{D}\varphi e^{-{\beta }_{\mathrm{th}}H[\varphi ]}$. It normalizes probabilities and generates thermodynamic averages. Its Helmholtz free energy isA quantum version uses $Z=\mathrm{Tr}{e}^{-{\beta }_{\mathrm{th}}\hat{H}}$. In the following identification, $F(M)$ is a constrained free-energy density, so its units match those of a Hamiltonian density rather than a total extensive energy.

Fix a microscopic cutoff ${\Lambda }_0$. Split the field into retained modes ${\varphi }_{<}$ with momenta below $\Lambda$ and eliminated modes ${\varphi }_{>}$ between $\Lambda$ and ${\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian exactly byIts coefficients and additive constant depend on $\Lambda$ so that integrating the retained modes gives precisely the original $Z$. With sources and observables carried along, the same construction preserves long-wavelength responses. Thus changing this intermediate ultraviolet cutoff is a change of description, not a change of the physical system. It is advantageous because one can integrate out fluctuations irrelevant to the observable length scale, organize a local gradient expansion, and use renormalization-group flow toward simpler universal fixed-point descriptions. Exact invariance retains all generated operators; truncating them is an approximation.

The quadratic gradient coefficient generally becomes $K_{\Lambda }$, not a constant. Field renormalization sets a chosen kinetic normalization, for example ${\varphi }_R=\sqrt{K_{\Lambda }}\varphi$ when this makes the gradient coefficient one in the energy convention. Sources and observables must be transformed inversely so that their physical meaning is preserved. Under rescaled coordinates $x^{\prime }=x/b$, the critical field scaling isIts anomalous part is the anomalous dimension; it gives $G(r)\sim r^{-2x_{\varphi }}$ and the conjugate-field exponent $y_h=D-x_{\varphi }$. Different conventions for the wave-function factor require corresponding changes in its logarithmic derivative. A fixed physical magnetization must not be confused with a cutoff-dependent dimensionless renormalized field.

To justify the zero-cutoff identification of constrained free energy, work first in a finite periodic volume $V$. Define the physical spatial mean $M=V^{-1}\int \varphi (x)d^{D}x$ and constrain that mean in the microscopic integral:Retain the constant mode $M$ while eliminating every nonzero mode. When $\Lambda$ is below the smallest nonzero box momentum, the remaining energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$. Consequently $e^{-{\beta }_{\mathrm{th}}V{U}_{\Lambda }(M)}=Z_M$ with a consistently fixed, possibly field-independent measure normalization. This provesFor a uniform field, $U_{\Lambda }(M)$ is the Hamiltonian density $\mathcal{H}(\Lambda ,M)$ in the question. The statement assumes that the physical zero mode is retained, all other modes are integrated exactly, field-rescaling factors are undone, and density and additive-normalization conventions agree. Taking the cutoff limit in the finite box before its thermodynamic limit makes this derivation unambiguous. With an applied uniform source, $Z(h)=\int dM{e}^{-{\beta }_{\mathrm{th}}V[F_c(M)-hM]}$ up to the same measure convention; its large-volume saddle minimizes $F_c(M)-hM$.

The Landau approximation further assumes that this potential can be approximated by a regular local polynomial on the homogeneous branches, and that the remaining fluctuations are negligible for the observables sought. It is not an assertion that an exact zero-cutoff free energy is the bare polynomial. In a coexistence region the exact constrained density can be convexified by domains, while a nonconvex Landau branch potential describes local phases and their metastability.

For the breakdown criterion use a short-range scalar theory with positive quadratic gradient stiffness. At the Gaussian fixed point, invariance of $\int (\nabla \varphi {)}^2{d}^{D}x$ gives field dimension $(D-2)/2$. A coupling $g_{2n}$ multiplying ${\varphi }^{2n}$ therefore has engineering scaling exponentIt is irrelevant only above its upper critical dimension. For the ordinary critical point the leading stabilizing interaction is quartic, while at a tricritical point the renormalized quartic term is tuned away and the leading interaction is sextic. ThereforeOne can also see the failure through the Ginzburg criterion. Long-wavelength fluctuations averaged over a correlation volume obey $⟨(\delta M{)}^2{⟩}_{\xi }\sim {\xi }^{2-D}$, up to temperature and stiffness factors. The quartic saddle has $M^2\sim {\xi }^{-2}$, giving a fluctuation/mean-square ratio proportional to ${\xi }^{4-D}$. The tricritical sextic saddle has $M^2\sim {\xi }^{-1}$, giving a ratio proportional to ${\xi }^{3-D}$. Both ratios grow without bound below their respective critical dimension, invalidating the fluctuation-free Landau exponents sufficiently close to a transition.

At equality the ratios are marginal, not power-law divergent. The loop expansion resolves this boundary: the quartic vertex correction contains $\int d^{D}q/(q^2+{\xi }^{-2}{)}^2$, logarithmic at $D=4$; the sextic vertex correction contains a two-loop three-propagator integral whose overall degree is $2D-6$, logarithmic at $D=3$. Marginal running therefore generally adds logarithmic corrections at these dimensions. Below them, where the corresponding transition exists, interactions control the long-distance critical behavior. Above them, leading thermodynamic powers are mean-field, although stabilizing dangerously irrelevant couplings explain why naive hyperscaling need not hold.

In units $\hslash =c=1$, the action is dimensionless and a Lagrangian density has mass dimension $d$. The kinetic term therefore gives $2+2[\varphi ]=d$. Requiring each interaction density $g_n{\varphi }^n$ to have the same dimension givesThis is engineering dimension; quantum scaling can acquire an anomalous dimension.

For a connected Feynman diagram with no derivative interactions, each independent loop integration contributes $d$ powers of momentum, and each internal scalar propagator contributes $-2$. Thus its superficial degree of divergence is $D=dL-2I$. Let $V={\sum }_n{V}_n$. A connected graph has $L=I-V+1$ independent cycles, and counting the half-edges incident on vertices gives $2I+E={\sum }_{n}n{V}_n$. Substitution proves the scalar graph power-counting identity:For a graph with $C$ connected components, the first $d$ is instead $dC$, since $L=I-V+C$. The printed expression therefore uses the usual connected-graph convention. This is the overall degree under simultaneous rescaling of all loop momenta. Subgraphs can still have ultraviolet divergences when the whole graph has $D<0$, so their counterterms must be subtracted before declaring an integral finite.