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For a stable massive Gaussian field theory, a finite blocking step yields $F(t,h)=F_{\mathrm{shell}}(t)+b^{-d}F(b^{2}t,b^{(d+2)/2}h)$ for free-energy density. The finite-step shell and measure terms are analytic backgrounds. Generic-dimension singular scaling is homogeneous after subtracting those backgrounds; resonances such as Gaussian free-energy logarithm at effective dimension two need additive logarithmic terms.

The Landau approximation evaluates a statistical field's partition function at a stable stationary configuration, omitting fluctuation-loop corrections to the free energy. Functional differentiation of that stationary value still gives a connected linear response: the inverse of the saddle's quadratic fluctuation operator. Freezing fields before differentiating would lose this response. A Gaussian field theory, in contrast, permits exact integration of its quadratic fluctuations.

For the unit-sphere nonlinear sigma model with action $\int d^{D}x|\nabla \sigma {|}^2/(2g)$, parameterize a slowly rotating background by $n_0=(\cos \theta ,\sin \theta ,0,\dots )$ and tangent vectors $e_1=(-\sin \theta ,\cos \theta ,0,\dots )$, $e_a$ for $a\ge 2$. Write $\sigma =n_0\sqrt{1-{\pi }^2}+{\sum }_{a=1}^{n-1}{e}_a{\pi }_a$. For constant slow $\nabla \theta$, the coefficient of $|\nabla \theta {|}^2$ in $|\nabla \sigma {|}^2$ is $1-{\sum }_{a=2}^{n-1}{\pi }_a^2$. The fast Gaussian field theory gives $⟨{\pi }_a^2⟩=g{I}_D$, where $I_D={\int }_{\Lambda /b<|p|<\Lambda }{d}^{D}p/[(2\pi {)}^D{p}^2]$. The omitted linear term integrates to zero, and terms cubic in the fast variables have zero one-loop average. Thus precisely $n-2$ tangent components reduce the phase stiffness $1/g$. Restoring the cutoff in $D=2+ϵ$ gives $dg/d\log b=-ϵg+(n-2)S_D{\Lambda }^{ϵ}{g}^2/(2\pi {)}^D+O(g^3)$. This weak-coupling expansion requires $n\ge 2$; the discrete target $S^0$ has no tangent expansion.

The uniform mode of a pure Gaussian field theory with negative squared mass has an unbounded quadratic energy. Its partition function diverges even with a finite-volume ultraviolet regulator, so it supplies no stable ordered branch. At positive squared mass the zero-source average is zero. Thus the scaling dimension of the field can yield a formal order-parameter critical exponent without establishing a negative-reduced-temperature magnetization law. A stabilizing quartic interaction is relevant below four dimensions and ordinarily changes the critical theory to the Wilson-Fisher fixed point; above four its coefficient is a dangerously irrelevant coupling for the ordered phase.

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.

The correlation length is the distance over which fluctuations remain appreciably correlated. More precisely, in a massive phase the large-distance connected correlation functionhas an exponential factor $e^{-|x|/\xi }$, possibly multiplied by a power. Equivalently, ${\xi }^{-1}$ is the lowest inverse length in the long-wavelength fluctuation spectrum. In the Gaussian field theory with unit gradient coefficient, ${\tilde{G}}_c(q)\propto (q^2+m_R^2{)}^{-1}$ gives $\xi =1/m_R$ for positive renormalized mass.

At an ordinary continuous phase transition, the restoring force for long-wavelength order-parameter fluctuations vanishes and there is no finite correlation scale. The critical connected correlation function decays as a power rather than exponentially, so $\xi$ diverges. The correlation-function susceptibility sum rule relates magnetic susceptibility to the integral of this connected correlation function; its divergence is consistent with correlations extending over increasingly large distances. Interactions change the powers, so the renormalized mass and the bare mass need not vanish at the same temperature.

Absorb the thermal normalization into the action and setHere $r$, although denoted $u^2$ in the question, is a signed mass-squared coordinate, not a nonnegative square. This matters because its interacting fixed-point value is negative. Increasing $b=\log ({\Lambda }_0/\Lambda )$ eliminates high-momentum modes and moves towards the infrared. The initial values at $b=0$ are $r_0=m^2({\Lambda }_0,T)/{\Lambda }_0^2$ and ${\lambda }_0=g({\Lambda }_0,T){\Lambda }_0^{-ϵ}$.

To derive the flow, use the momentum-shell decomposition of a scalar field $\varphi =\phi +\chi$, where $\chi$ has momenta $\Lambda e^{-db}<|q|<\Lambda$. Its Gaussian shell covariance is $(q^2+m^2{)}^{-1}$. Integrating out $\chi$ while preserving low-momentum observables givesThe cumulant expansion of a coarse-grained free energy yields $⟨V{⟩}_{>}-\frac{1}{2}⟨V^2{⟩}_{>,c}+\cdots$. By Wick theorem,The last term is field independent, while the quadratic term gives the one-loop shell mass renormalization in scalar quartic theoryThis tadpole is independent of the external momentum, so no gradient renormalization occurs at this order.

For the quartic correction, the mixed vertex is $(g/4)\int {\phi }^2{\chi }^2$. Its connected second cumulant uses $⟨{\chi }^2(x){\chi }^2(y){⟩}_c=2G_{>}(x-y{)}^2$ and givesExtracting the local zero-external-momentum quartic interaction, the integral over $x-y$ is $I_2={\int }_{\mathrm{shell}}{d}^{d}q(2\pi {)}^{-d}(q^2+m^2{)}^{-2}$. Comparing its coefficient with $g{\phi }^4/4!$ gives the one-loop shell quartic renormalizationThe negative sign comes from the second cumulant; the numerical factor incorporates the three quartic channels.

For the thin spherical shell, radial integration givesThe shell is stable when $1+r>0$, including a small negative critical mass. Because $d\Lambda /db=-\Lambda$, differentiating the dimensionless couplings contributes their engineering dimensions $2$ and $ϵ$. Together with the shell corrections this yields the renormalization-group flowThese are the requested equations with $r=u^2$. They express independence of the long-distance theory from the arbitrarily chosen ultraviolet cutoff, to the retained perturbative order; higher interactions and gradient corrections enter at higher orders.

Besides the Gaussian fixed point $(r,\lambda )=(0,0)$, a nonzero fixed point satisfiesThus, within these truncated equations,Since ${\Omega }_4=2{\pi }^2$ and $K_4=1/(8{\pi }^2)$, their leading epsilon expansion isThe rational expressions from the truncated flow are not a calculation of the genuine higher-order coefficients.

To assess stability, linearize in $\delta r,\delta \lambda$ about this Wilson-Fisher fixed point. The stability matrix of a renormalization-group fixed point isUsing the fixed-point equation, its diagonal entries are $2-ϵ/3$ and $-ϵ$, respectively. Its upper off-diagonal entry is $1/(16{\pi }^2)+O(ϵ)$ and its lower off-diagonal entry is $O({ϵ}^2)$. Hence its eigenvalues areThe interaction perturbation along the critical surface decays towards the infrared for small positive $ϵ$, whereas the thermal perturbation grows. Therefore infrared attraction requires tuning to the critical surface. The source's unqualified wording cannot mean attraction in both coupling directions: the positive thermal eigenvalue is precisely what produces the divergence of the correlation length. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Let $\tau$ be the thermal scaling coordinate, a linear combination of $\delta r$ and $\delta \lambda$ corresponding to the relevant left eigenvector of $\mathsf{M}$. To leading order, $\tau =\delta r+[1/(32{\pi }^2)+O(ϵ)]\delta \lambda$ is possible. Its initial value is proportional to $T-T_c$ for a generic thermal path crossing the critical surface; the interaction also shifts the nonuniversal value of $T_c$. In the linear regime,Choose $b_{\ast }$ so that $|\tau (b_{\ast })|$ is of order one. At this scale the dimensionless correlation length is finite. Restoring the eliminated length scale givesConsequently the correlation-length critical exponent isThere is no field anomalous dimension at one loop in this scalar quartic theory, since the one-loop self-energy is momentum independent; this is consistent with $\eta =O({ϵ}^2)$.

Put $D=d-1$ for the membrane's internal dimension and assume $\sigma >0$. The small-slope expansion isThe zeroth-order term is the projected area and is independent of the height field. Discarding it changes only an overall normalization of the partition function. Thus the leading fluctuation energy and its statistical weight areOne obtains this Gaussian field theory by discretizing the height field on a microscopic mesh, integrating every height with its Boltzmann weight, and then writing the continuum measure. A microscopic cutoff is retained for quantities dominated by short wavelengths. Quadratic order here really means small slopes: a large rigid height shift is still free because the energy has no dependence on $h$ itself.

The uniform height mode has zero energy. Fix its mean, impose a boundary condition, or divide out the integral over global translations to normalize the partition function. Height differences are independent of that choice. The approximation is the tension-only part of small-slope elastic membrane energy; there is no bending rigidity or confining potential in this part.

ChooseFor a real scalar field $\tilde{m}(-q)=\tilde{m}(q{)}^{\ast }$. The parallel derivative becomes $i{q}_{\parallel }$, and the transverse Laplacian becomes $-q_{\perp }^2$. The latter is squared in the energy, producing $q_{\perp }^4$, not $q_{\perp }^2$. ThusThe zero-momentum source is understood in a finite box or as a uniform-field distribution before taking the thermodynamic limit. With an orthonormal box transform the source is $-h\sqrt{V}{m}_0$. The positive coefficients and $t>0$ give a normalizable Gaussian field theory, with mean uniform field $\overline{m}=h/t$ and connected Fourier covariance $[K{q}_{\parallel }^2+L{q}_{\perp }^4+t{]}^{-1}$.

Use the connected correlation function $G(x)=⟨\varphi (x)\varphi (0)⟩-⟨\varphi {⟩}^2$, so that the constant ordered contribution is removed. Away from a thermodynamic critical point its long-distance decay defines the correlation length: $-{\lim }_{R\to \infty }{R}^{-1}\log |G(R)|={\xi }^{-1}$ when this limit exists. In a massive rotation-invariant phase the exponential decay is typically multiplied by an algebraic factor.

Expand the Landau-Ginzburg theory free energy about a stable uniform minimum $M$, writing $\varphi =M+\delta \varphi$. To quadratic order,integration of the Gaussian field theory gives the Fourier-space connected correlation function $\tilde{G}(q)=k_{B}T/(c+\kappa q^2)$. Thus the Ornstein--Zernike correlation function hasFor an ordinary quartic transition, $M=0$ above the transition and $M^2=-r/u$ below, so $c=r$ and $c=-2r$, respectively. The correlation length diverges as $|T-T_c{|}^{-1/2}$ on both sides, with different amplitudes. At criticality the mass vanishes and the Gaussian correlation function is proportional to $R^{-(D-2)}$ for $D>2$. Interactions can change this power to $R^{-(D-2+\eta )}$.

The mean-field approximation replaces each site's fluctuating surroundings by a self-consistent average. On a cubic lattice each spin has coordination number of a lattice $z=2D$, even though summing only over positive basis directions counts each bond once. Write ${\sigma }_n=M+\delta {\sigma }_n$ and drop products of deviations. Each Ising bond becomes ${\sigma }_n{\sigma }_m\simeq M{\sigma }_n+M{\sigma }_m-M^2$, so the approximate statistical Hamiltonian isThe constant corrects counting each bond twice. The resulting independent-spin partition function is $Z_{\mathrm{MF}}=e^{-{\beta }_{T}NzJ{M}^2/2}[2\cosh ({\beta }_{T}zJM){]}^N$, where ${\beta }_T=1/(k_{B}T)$. The one-spin probabilities giveEquating this average with the assumed $M$ givesHere $J>0$ is assumed; a ferromagnetic order parameter is not the appropriate choice for an antiferromagnetic coupling. Put $A=T_c/T$. For $A\le 1$, $\tanh (AM)<AM\le M$ at any $M>0$, so only $M=0$ is possible. For $A>1$, the function $\tanh (AM)-M$ has positive slope at zero, is strictly concave on $M>0$ and is negative at $M=1$. It therefore has exactly one positive zero and a spin-reversed negative zero. Expanding at the bifurcation givesThe transition is continuous, with order-parameter critical exponent $\beta =1/2$. Stability can also be checked from the Bragg-Williams free energy of the Ising model per Ising spin:Its derivative is $-zJM+k_{B}T\mathrm{arctanh}M$, giving the same self-consistency equation. Near zero,The quartic coefficient is positive at $T_c$, the stable minima approach zero continuously, and the minimized free energy has no first-derivative jump. Thus mean-field approximation predicts a second-order transition at $T_c=2DJ/k_B$. This is the mean-field Ising critical bifurcation.

For short-range interactions, large dimension gives many neighbours and suppresses the relative fluctuations of their average. More quantitatively, Gaussian field theory fluctuations averaged over a correlation volume have variance of order ${\xi }^{2-D}$, while the squared ordinary mean-field order parameter is of order $|t|\sim {\xi }^{-2}$. Their ratio is proportional to ${\xi }^{4-D}$. The Ginzburg criterion therefore makes neglect of critical fluctuations asymptotically consistent above four dimensions, inconsistent below four, and marginal at four, where logarithms occur. Large dimension is a route to validity; sufficiently long-range or infinite-range interactions can also produce mean-field behaviour. In low-dimensional short-range systems even the existence and location of a transition can differ from this approximation.

Split the Fourier modes into $|p|<\Lambda /b$ and the shell $\Lambda /b<|p|<\Lambda$. The source is uniform, so it couples only to the zero mode, which belongs to the retained part. The Gaussian field theory has no couplings between the shell and retained modes. Integrating out the shell therefore contributes only a source-independent determinant to the free energy, without changing the quadratic coefficients of the retained Hamiltonian.

Restore the cutoff by $x=b{x}^{\prime }$ and choose the field normalization ${\varphi }_{<}(b{x}^{\prime })=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. Counting the measure, derivative and field factors explicitly givesThus in the convention that preserves the kinetic normalization, the Gaussian momentum-shell scaling isThe field has engineering dimension $(D-2)/2$ and zero anomalous dimension. The measure Jacobian and shell determinant supply the additive constant described in Q2. A different choice of field rescaling can redistribute factors into $\alpha$, but must give the same physical transformed correlation length ${\xi }^{\prime }=\xi /b$.

Use the connected correlation function $G(r)=⟨{\sigma }_0{\sigma }_r⟩-⟨{\sigma }_0⟩⟨{\sigma }_r⟩$. Away from criticality its large-distance envelope decays exponentially on the correlation length $\xi$, often with an algebraic prefactor. At criticality $\xi$ diverges and a short-range scalar model instead has the power law $G(r)\sim r^{-(D-2+\eta )}$, with anomalous dimension $\eta$.

A blocking kernel $K_b({\sigma }^{\prime },\sigma )$ assigns coarse spins to each block of side $b$ lattice spacings, with ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. Define the coarse Hamiltonian byThe operator basis must be sufficiently complete to include interactions generated by blocking. Summing over coarse spins preserves the partition function, while the physical lattice spacing and site count change to $a^{\prime }=ba$, $N^{\prime }=b^{-D}N$. After $p$ steps they are $a_p=b^{p}a$ and $N_p=b^{-pD}N$. Exact blocking preserves the long-distance predictions; truncating the generated interaction space introduces an approximation.

Define the dimensionless free energy per site $F(u,C)=-N^{-1}\log Z(u,C,N)=f(u)+\beta C$. Preservation of $Z$ gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. For one step, removing the explicitly additive constant givesThe function $g$ records the additive free-energy density generated by eliminated degrees of freedom; it is usually regular near a finite blocking step's critical point. Iteration provesThis is the additive free-energy recursion under blocking. After removing regular backgrounds, its singular part obeys homogeneous scaling.

A renormalization-group fixed point satisfies $R_b(u^{\ast })=u^{\ast }$. Linearizing the map gives scaling coordinates $v_i^{\prime }=b^{{\lambda }_i}{v}_i$. A relevant operator has ${\lambda }_i>0$, an irrelevant operator has ${\lambda }_i<0$, and a marginal operator has ${\lambda }_i=0$ and needs nonlinear analysis. The critical surface consists of points flowing into the critical fixed point after all relevant directions are tuned. An outgoing or repulsive trajectory lies in the unstable manifold and describes perturbations that grow under coarse-graining.

With relevant thermal and magnetic coordinates $t,h$, the singular free-energy density obeys $F_s(t,h)=b^{-D}{F}_s(b^{{\lambda }_t}t,b^{{\lambda }_h}h)$. Choose $b=|t{|}^{-1/{\lambda }_t}$ to obtainHere ${\lambda }_t,{\lambda }_h$ are the positive scaling eigenvalues, and $f_{\pm }$ refer to the signs of $t$. Since $\xi (t,h)=b\xi (b^{{\lambda }_t}t,b^{{\lambda }_h}h)$, the correlation-length exponent is $\nu =1/{\lambda }_t$. Two temperature derivatives of $F_s(t,0)$ give $\alpha =2-D/{\lambda }_t$, establishing the hyperscaling relationThe magnetization exponent similarly is $\beta =(D-{\lambda }_h)/{\lambda }_t$. These expressions assume no additional dangerous scaling variables.

For the Gaussian field theory, decompose the field into low and high Fourier modes, integrate out $\Lambda /b<|p|<\Lambda$, and restore the cutoff using $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. The gradient coefficient remains fixed, the mass coefficient transforms as $m^{\prime 2}=b^2{m}^2$, and the uniform source as $h^{\prime }=b^{(D+2)/2}h$. Hence ${\lambda }_t=2$, ${\lambda }_h=(D+2)/2$, and the Gaussian momentum-shell scaling givesThe sign convention $+h\varphi$ reverses the magnetization's sign but not these exponents. For $2<D<4$, these are Gaussian scaling predictions, not the exponents of a generic interacting scalar critical point: the quartic interaction is relevant there. The free quadratic Hamiltonian also has no stable ordered phase at negative $m^2$, so the quoted $\beta$ is its formal order-parameter scaling index, not a spontaneously magnetized state of the unstabilized Gaussian integral. The relevance distinction is also explained in Tong's renormalization-group lecture notes.

Take a positive gradient coefficient ${\kappa }^{-1}$, a positive squared mass, a finite-volume regulator and an ultraviolet cutoff. Use the dimensionless statistical action convention $Z=\int \mathcal{D}\varphi e^{-H}$: inverse temperature is absorbed into the coefficients. If $H$ denotes dimensional energy instead, the exponent below gains a factor ${\beta }_{\mathrm{th}}$ and the covariance gains ${\beta }_{\mathrm{th}}^{-1}$. These conventions cannot be mixed.

Choose the Fourier transform conventionIntegration of the quadratic gradient and source terms givesComplete the square with ${\tilde{\varphi }}_{\mathrm{cl}}(p)=\tilde{\Delta }(p{)}^{-1}\tilde{J}(p)$. Translation of the regulated real Gaussian variables leaves their measure unchanged, so the Gaussian integral givesHere $Z_G=Z_0[0]$ is the source-free Gaussian partition function. For $n$ independent regulated real coordinates and a positive quadratic matrix $A$, it is $(2\pi {)}^{n/2}(\det A{)}^{-1/2}$ under Lebesgue normalization. The continuum determinant needs the stated regulator and a measure convention. Reality pairs the nonzero Fourier modes, so they must not be counted as two independent real fields.

A connected correlation function subtracts the product of the means. The logarithm $W[J]=\log Z_0[J]$ generates its cumulants: differentiating $W$ twice with respect to the source, using the measure $d^{D}p/(2\pi {)}^D$ in the functional derivative convention, yieldsThe source changes the mean but not the covariance; the stated zero-source limit therefore follows immediately. Momentum conservation uses $p+q$, appropriate for a real field without complex conjugation on either displayed factor.

For the continuum massive Gaussian field correlation tail, set $\mu =\sqrt{\kappa }m>0$ and $r=|x|$. Fourier inversion givesThe large-argument Modified Bessel function of the second kind has $K_{\rho }(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$. Hence $G_0(r)$ is proportional to $r^{-(D-1)/2}{e}^{-\mu r}$ at large distance andAt zero mass the exponential length diverges. For $D>2$ the critical continuum tail is algebraic, $G_0(r)\propto r^{2-D}$; in lower dimensions infrared regulation needs additional care.

For a uniform source $h$, perform a momentum-shell renormalization group step: separate slow modes $|p|<\Lambda /b$ and fast modes $\Lambda /b<|p|<\Lambda$. In a Gaussian field theory they decouple; the uniform source couples only to the retained zero mode. Integrating fast modes contributes only a field-independent determinant to the free energy. Restore the cutoff by $x^{\prime }=x/b$, $p^{\prime }=bp$ andThe factors from the measure, gradient and two fields cancel in the gradient term. The mass term gains $b^2$, and the linear source gains $b^{(D+2)/2}$. ThereforeHere $m$ is the positive mass; for signed thermal coordinates it is the squared-mass relation that applies. The Gaussian fixed point has thermal scaling eigenvalue ${\lambda }_t=2$, source eigenvalue ${\lambda }_h=(D+2)/2$ and zero anomalous dimension.