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The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the Fluctuation-dissipation theorem.

Let $\delta (\mathbf{x})$ be a mean-zero density contrast. For a statistically homogeneous and isotropic field, its two-point correlation function is $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$, depending only on $r=|\mathbf{r}|$. For a galaxy point process the corresponding distinct-pair definition is $d{P}_{12}={\bar{n}}^2[1+\xi (r)]d{V}_{1}d{V}_2$, the excess pair probability relative to independent uniform positions. Self-pair shot noise is separated when using the density-field version.

Specify the Fourier convention byThe diagonal covariance follows from statistical homogeneity, and isotropy makes the matter power spectrum a function of $k$ alone. Substitute the inverse transforms into the correlation and integrate the delta function:Choose the polar axis parallel to $\mathbf{r}$. Angular integration gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceConversely, when the transform is defined, $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$. The stated convention fixes every $2\pi$ factor.

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.

Keep $Z=e^{iW}$, define the mean scalar field $\phi (x)=\delta W/\delta J(x)$, and locally invert the source-field relation. The quantum effective action is the Legendre transformIts value at zero source is expanded around ${\phi }_0=⟨\varphi ⟩$, which need not vanish. The kernels ${\Gamma }^{(n)}={\delta }^n\Gamma /\delta {\phi }^n{|}_{{\phi }_0}$ are the one-particle-irreducible vertices, with the Minkowski diagram vertex factor $i{\Gamma }^{(n)}$. A one-particle-irreducible Feynman diagram remains connected after any one internal line is cut. Amputation of external propagators removes the full two-point functions on its external legs. For higher-point connected graphs amputation alone need not remove internal one-particle-reducible Feynman diagrams; the derivatives of $\Gamma$ select precisely the irreducible kernels.

To derive the three-point relation, use condensed indices that include spacetime integration. Put $D_{ij}=W_{,ij}=i{G}_{2,ij}^c$ and $K={\Gamma }^{(2)}$. Differentiating ${\Gamma }_{,i}=-J_i$ gives $K_{ia}{D}_{aj}=-{\delta }_{ij}$, so $K=-D^{-1}$. A further derivative, using $\delta D_{ab}/\delta {\phi }_k=W_{,ab\ell }(D^{-1}{)}_{\ell k}$ and the derivative of an inverse, givesSince $W^{(3)}=-G_3^c$ and $D^{-1}=-i(G_2^c{)}^{-1}$, the three-point amputation in the connected-action convention isEquivalently, attach a full two-point correlation function to each leg of $i{\Gamma }^{(3)}$ to reconstruct $G_3^c$. The connected functions are evaluated at the same zero-source background. The displayed factors of $i$ follow from the declared Minkowski convention; Euclidean generating functionals redistribute these factors.

Choose the Fourier transform conventionStatistical homogeneity makes the Fourier covariance diagonal, and isotropy makes the matter power spectrum depend only on $k=|\mathbf{k}|$. The two-point correlation function $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$ follows by inserting the transform and using this covariance:Integrating the polar angle relative to $\mathbf{r}$ gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceThe inverse isotropic transform is $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$ whenever these transforms exist, with distributions or regularization otherwise.

For a spherical average of comoving radius $R$, the spherical top-hat window function isAt a fixed epoch let $P(k)=A{k}^n$. The scale-free smoothed density variance isThus the RMS fractional mass fluctuation scales as $R^{-(n+3)/2}$; the RMS mass fluctuation itself scales as $\bar{\rho }{R}^3{\sigma }_{\delta }\propto R^{(3-n)/2}$ at fixed epoch.

The comoving Poisson equation gives ${\Phi }_{\mathbf{k}}=-4\pi G{a}^2\bar{\rho }{\delta }_{\mathbf{k}}/k^2$. For the growing irrotational mode the linearized cosmological continuity equation gives ${\mathbf{v}}_{\mathbf{k}}=-iaHf\mathbf{k}{\delta }_{\mathbf{k}}/k^2$, where $f=d\log D_{+}/d\log a$. The averaged-potential and bulk-velocity variances are thereforeFor a single Cartesian component of the isotropic bulk velocity, divide its variance by three. Rescaling $kR$ givesThese are dimensional scaling laws, with an important convergence qualification. Since $W(y)\to 1$ at small $y$ and has large-$y$ envelope $O(y^{-2})$, the density average converges for $-3<n<1$, the absolute potential average for $1<n<5$, and the bulk velocity for $-1<n<3$. No one unbroken power law makes all three absolute averages finite on the entire range $0<k<\infty$. In particular, long-wavelength modes can dominate the potential. Physical infrared/ultraviolet cutoffs, potential differences, or fluctuation amplitudes in a band around $k\sim R^{-1}$ resolve the relevant divergences; with cutoffs the simple powers need not describe the full integrated variance. The scale-free potential and bulk-flow variance laws should be understood with these conditions, not as a claim that divergent RMS integrals are finite.

Use ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ for inverse temperature to distinguish it from the order-parameter exponent. The decaying two-point correlation function needed for the response identity is connected:Away from the critical point in a short-range Ising model pure phase, the correlation length controls the exponential long-distance falloff, $G(r)\sim A(r)e^{-|r|/\xi }$ with a possible algebraic prefactor. The full two-spin expectation below the transition instead tends to $M^2$. The spin-mixture contribution to zero-field susceptibility explains why a symmetric mixture is unsuitable for defining a finite connected correlation length in the ordered phase.

Let $F=-{\beta }_{\mathrm{th}}^{-1}\log Z$ be the total Helmholtz free energy. Take magnetization and susceptibility per site, soDifferentiating the Boltzmann weights at fixed temperature givesTranslation invariance then yields the correlation-function susceptibility sum ruleTotal magnetization and total susceptibility multiply these expressions by $N$; per physical volume they have an additional factor $a^{-D}$ relative to the per-site convention. These normalizations do not change the critical powers.

For the subsequent scaling requests, let $F_s$ denote singular free energy per site; the fixed extensive normalization is suppressed. The labels $+$ and $-$ denote $t>0$ and $t<0$, with distinct scaling functions and amplitudes on the two sides. Below the transition a one-sided derivative at $h=0$ selects a spontaneous-magnetization branch. Writing $x=h/|t{|}^{\Delta }$, differentiation of the scaling hypothesis for critical phenomena givesThe field-independent $I_{\pm }$ makes no contribution to these field derivatives, although it can change heat-capacity amplitudes. Assuming nonzero leading amplitudes and the indicated pure-phase limits, comparison with the exponent definitions givesThis is the Rushbrooke scaling relation. A temperature-independent critical isotherm requires the large-$|x|$ magnetization scaling function to grow as $\mathrm{sgn}(x)|x{|}^{\beta /\Delta }$: otherwise $M_s=|t{|}^{\beta }{\mathcal{M}}_{\pm }(x)$ would retain a temperature power as $t\to 0$ at fixed small $h$. Therefore $1/\delta =\beta /\Delta$. Since the preceding two equalities also give $\Delta =\beta +\gamma$, the Widom scaling relation follows:For the hyperscaling relation, additionally assume that one correlation volume supplies an order-one singular free-energy contribution and that there is a single isotropic correlation length, with no dangerously irrelevant coupling spoiling this argument. Then $F_s\sim {\xi }^{-D}\sim |t{|}^{D\nu }$. Two temperature derivatives give the singular heat capacity, soHyperscaling is an additional physical assumption, not a consequence of the two-variable equation alone, and generally fails above the upper critical dimension.

Finally the correlation scaling function $f_G(x)$ falls exponentially, up to algebraic factors, for $x\to \infty$ in a massive short-range pure phase. At $x\to 0$ it approaches a finite nonzero value in the scaling regime. Approximating the lattice sum at distances much larger than $a$, and denoting the area of the unit sphere by $S_{D-1}$, givesIf $\eta <2$ and the integral has a finite nonzero limiting value, microscopic distances contribute only a regular background and the leading divergence is ${\xi }^{2-\eta }$. Thus the Fisher scaling relation isThese statements concern leading singular powers; logarithmic corrections and cancellations of amplitudes require separate treatment.

Use ${\beta }_T=1/(k_{B}T)$ for inverse temperature, to distinguish it from the order-parameter critical exponent $\beta$. In a translation-invariant pure thermodynamic phase, the connected two-point correlation function isFor finite correlation length, at $|r|\gg \xi$ it decays exponentially, possibly with an algebraic prefactor; ${\xi }^{-1}$ is the asymptotic exponential decay rate. Below an ordered transition the pure phase is selected by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A zero-field symmetric mixture of the two ordered phases has a nondecaying connected term and does not describe the usual pure-phase magnetic susceptibility.

Let $\mathcal{F}=-k_{B}T\log Z$ denote extensive physical free energy and $S={\sum }_r{\sigma }_r$. Since the field contributes $-hS$ to the statistical Hamiltonian, differentiation givesExpanding the variance and using translation invariance proves the correlation-function susceptibility sum rule:The sum includes $r=0$. If the source is instead defined as the dimensionless quantity ${\beta }_{T}h$, the corresponding derivative omits the displayed inverse-temperature factor.

A real-space renormalization group step of length factor $b>1$ uses a local blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ satisfying ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. For example, a deterministic majority-spin rule assigns one coarse Ising spin to each block; tied blocks can be resolved with equal probabilities. Define the blocked statistical Hamiltonian byThe spacing becomes $a^{\prime }=ba$. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over ${\sigma }^{\prime }$ uses normalization to prove $Z(u^{\prime },C^{\prime },N^{\prime })=Z(u,C,N)$. Iteration gives $N_p=b^{-pD}N$ and $Z(u_p,C_p,N_p)=Z(u_0,C_0,N)$.

The coarse probability law is the exact marginal law of these block variables. Indeed, for any coarse observable $A$, its expected value is the microscopic expected value of ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )A({\sigma }^{\prime })$. A blocking rule that retains the long-wavelength order-parameter mode, with source and observable transformations tracked, therefore retains long-distance response and correlation functions. Their field normalization may change, while their physical length scales and critical singularities do not. Partition-function equality alone would not justify this assertion for an arbitrary kernel that discards all slow variables. Truncating the generated operator family is likewise an approximation.

For the free-energy recursion use dimensionless free energy per site,in the thermodynamic limit. Exact partition-function invariance gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. If one step sends $C$ to $C^{\prime }=b^{D}C+c(u)$, the coefficient $c(u)$ records the identity-operator contribution from eliminated fluctuations and normalization. ConsequentlyRepeated substitution derives the additive free-energy recursion under blocking:In physical energy units, divide this convention by ${\beta }_T$. The additive term is essential for absolute free energies even when it does not affect the leading critical powers.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $v_i$ so that $v_i^{\prime }=b^{y_i}{v}_i+O(v^2)$. A relevant operator has $y_i>0$, an irrelevant operator has $y_i<0$, and a marginal operator has $y_i=0$ with behaviour decided by nonlinear terms. Thus the eigenvalues of the discrete linear map are $b^{y_i}$; the sign criterion applies to the generator exponents $y_i$, not to the discrete multipliers. The critical surface is the stable manifold on which every relevant scaling field has been tuned away. Irrelevant coordinates flow to the renormalization-group fixed point along it. A repulsive renormalization-group trajectory leaves the renormalization-group fixed point along an unstable relevant direction, ultimately entering a noncritical phase or another scaling regime.

To separate the singular free-energy density from its regular background, suppose the eliminated-mode contribution $g$ is analytic near the renormalization-group fixed point. Look for an analytic background $a(u)$ satisfying $a(u)=b^{-D}a(R_{b}u)+g(u)$. Then $F_s=f-a$ satisfies $F_s(u)=b^{-D}{F}_s(R_{b}u)$. In linear thermal and field coordinates, an analytic monomial $g_{mn}{t}^m{h}^n$ has background coefficientprovided the denominator is nonzero. This explicitly explains when analytic subtraction of an inhomogeneous renormalization recursion works. The printed inhomogeneous equation may describe the free-energy function before this subtraction, or a singular part defined only up to analytic terms. The inhomogeneous sum need not be small compared with the full free energy: an analytic constant can dominate a vanishing singular contribution. It can be omitted in deriving leading singular powers after regular subtraction, provided no renormalization-group free-energy resonance, marginal logarithm or dangerously irrelevant coupling obstructs the homogeneous critical asymptotics. At a renormalization-group free-energy resonance $m{\lambda }_t+n{\lambda }_h=D$, additive logarithms can remain, so simply dropping the term would miss them.

Suppose there are exactly two relevant scaling fields, thermal $t$ and magnetic $h$, with positive exponents ${\lambda }_t,{\lambda }_h$, and the remaining fields approach a regular limit. Physical reduced temperature and magnetic field agree with these coordinates to leading order, up to nonuniversal metric factors. The homogeneous scaling hypothesis for critical phenomena isas an asymptotic scaling equation. Choose $\ell =|t{|}^{-1/{\lambda }_t}$ to make the thermal argument $+1$ or $-1$. Defining $f_{\pm }(z)=F_s(\pm 1,z)$ givesThe two functions describe respectively $t>0$ and $t<0$: approaching the transition from disordered and ordered sides need not give the same function. In particular the ordered zero-field function has one-sided derivatives associated with spontaneous magnetization. Metric factors multiply this formula in different microscopic normalizations.

The correlation length in Bravais lattice units obeys $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$. At $h=0$, the same choice of $\ell$ gives $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s(t,0)$ give $2-\alpha =D/{\lambda }_t$. One and two field derivatives give, respectively, $\beta =(D-{\lambda }_h)/{\lambda }_t$ and $\gamma =(2{\lambda }_h-D)/{\lambda }_t$, assuming the relevant branch amplitudes are nonzero. At $t=0$, choose $\ell =|h{|}^{-1/{\lambda }_h}$; then $M\propto \mathrm{sgn}(h)|h{|}^{(D-{\lambda }_h)/{\lambda }_h}$, so $\delta ={\lambda }_h/(D-{\lambda }_h)$. HenceSimple algebra now gives the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the hyperscaling relation $\alpha =2-D\nu$. Also $\beta \delta ={\lambda }_h/{\lambda }_t=\beta +\gamma$, giving the Widom scaling relation. These statements refer to leading singular contributions with the homogeneous two-field scaling assumptions. Above the upper critical dimension, the quartic coupling is dangerously irrelevant because it is still needed to stabilize the ordered phase; ordinary mean-field critical exponents then satisfy Rushbrooke and Widom but generally not hyperscaling. Marginal logarithms also require qualifications at the upper critical dimension.

For the connected-correlation scaling form $G(r)=|r{|}^{-(D-2+\eta )}{f}_G(|r|/\xi )$, the function $f_G(x)$ tends to a finite nonzero value as $x\to 0$ and decays exponentially at large $x$, up to powers of $x$, in a massive short-range phase. The scaling ansatz alone does not fix that prefactor. If the usual massive Gaussian field correlation tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ applies, matching the powers gives $f_G(x)\sim B{x}^{(D-3)/2+\eta }{e}^{-x}$, with a correlation-length-dependent amplitude in $G$ fixed by matching the critical scaling form.

In physical coordinates the lattice sum has site-density factor $a^{-D}$. Its long-distance singular part therefore giveswhere ${\Omega }_{D-1}$ is the area of the unit sphere in $D$ dimensions. For $\eta <2$, finite $f_G(0)$, an integrable exponential tail and a nonzero integral, the last factor tends to a finite nonzero constant as $\xi \to \infty$. Microscopic distances contribute a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, proving the Fisher scaling relationEquivalently the field has scaling dimension $(D-2+\eta )/2$, so its conjugate field has ${\lambda }_h=(D+2-\eta )/2$, which gives the same identity from the exponent formulas above.

Normalize the vacuum path integral by $Z[0]$. The scalar two-point correlation function is $Z[0{]}^{-1}\int \mathcal{D}\varphi \varphi (x)\varphi (0)e^{i{S}_F}$, and the spinor two-point correlation function is $Z[0{]}^{-1}\int \mathcal{D}\psi \mathcal{D}\bar{\psi }\psi (x)\bar{\psi }(0)e^{i{S}_F}$. Vacuum boundary conditions make these time-ordered Feynman propagators. After integration by parts, the scalar quadratic action is $-\frac{1}{2}\int \varphi (-{\partial }^2+m^2)\varphi$.

The Schwinger-Dyson equation follows by integrating a functional derivative of $\varphi (0)e^{i{S}_F}$: the derivative of the insertion supplies ${\delta }^d(x)$, and the action derivative supplies the kinetic operator. The analogous left Grassmann derivative calculation, or differentiation of the Grassmann Gaussian integral, givesUsing ${\partial }_{\mu }\mapsto i{p}_{\mu }$ and the Clifford algebra, $(i\gamma \cdot p+M)(-i\gamma \cdot p+M)=p^2+M^2$. ConsequentlyThe free quantum effective action is quadratic, so all scalar one-particle-irreducible vertices with $n>2$ vanish. Its two-point vertex is the stated inverse kinetic form $-p^2-m^2$.

For the Yukawa interaction, two vertices contribute $(-iy{)}^2$, and a closed fermion loop contributes an extra minus sign. Tracing the two spinor numerators gives $4[M^2-k\cdot (k-p)]$, since the one-gamma traces vanish. Removing the overall $i$ from the amplitude gives the displayed loop integral. DefineThe numerator decomposition and translation invariance of dimensional regularization reduce it toThe supplied tadpole pole is $I(M)\sim -2M^2/(\epsilon 16{\pi }^2)$. A Feynman parameter combines the two bubble denominators. Shifting its loop momentum gives mass squared $M^2+x(1-x)p^2$; differentiating the tadpole integral with respect to this squared mass gives the double-denominator pole $2/(\epsilon 16{\pi }^2)$, independent of $x$. Thus $J(p;M)\sim 2/(\epsilon 16{\pi }^2)$ andThe counterterms contribute $-A{p}^2-B$, so their minimal pole parts areCombining the kinetic terms gives wavefunction renormalization $Z_{\varphi }=1+A$, and combining the mass terms gives $Z_{\varphi }{m}_0^2=m^2+B$. ThereforeThese are the Yukawa scalar self-energy pole coefficients; finite parts depend on the chosen renormalization condition.

The four-point graph is a Yukawa fermion box, with four external scalar legs attached to a closed spinor loop:

Each high-momentum dirac propagator is $O(k^{-1})$. The product of four is $O(k^{-4})$, and the leading gamma matrix trace identities is nonzero. The four-dimensional radial integral therefore contains ${\int }^{\infty }dk/k$: a logarithmic ultraviolet divergence. This local four-scalar divergence cannot be absorbed by scalar mass or field normalization. Add $-\lambda {\varphi }^4/4!$, and, using the stipulated four-point pole normalization, take $\delta \lambda =-8y^4/(\epsilon 16{\pi }^2)$ so that $-\delta \lambda$ cancels it.

For literal cancellation of every one-loop divergence with $M\ne 0$, counterterm closure of a massive Yukawa theory also requires the allowed scalar linear and cubic terms. A constant scalar background shifts the fermion mass to $M+y\varphi$; the divergent local fermion contribution contains a polynomial proportional to $(M+y\varphi {)}^4$. Its linear and cubic terms are not forbidden by a symmetry when the fermion mass is nonzero. A closed renormalizable family therefore haswith field, mass and coupling redefinitions for both scalar and spinor fields. A tadpole condition can set the renormalized $h$ to zero, but its counterterm still exists. The vacuum constant is needed if vacuum energy is retained. If an exact discrete chiral symmetry is imposed with $M=0$, the scalar potential can be even and the odd terms are forbidden; the essential new interaction is then the quartic one.

A regulator and a renormalization condition introduce the reference mass scale $\mu$, even when the classical theory has no mass. Loop amplitudes contain dimensionless logarithms of momentum or distance ratios involving $\mu$. The resulting running coupling and field normalization compensate changes of this arbitrary reference scale.

Let ${\varphi }_0=Z_{\varphi }^{1/2}\varphi$ and $G_n=Z_{\varphi }^{-n/2}{G}_{n,0}$. Hold the bare parameters fixed and define $\beta (g)=\mu dg/d\mu$ and $\gamma (g)=\frac{1}{2}\mu d\log Z_{\varphi }/d\mu$. Assuming multiplicative field renormalization and no mixing or additive contact terms for the correlator, differentiating gives the Callan-Symanzik equationFor the dimensionless two-point factor put $r=p^2/{\mu }^2$. Its equation is $(-2r{\partial }_r+\beta {\partial }_g+2\gamma )C=0$. LetThe characteristic solution of the multiplicative Callan-Symanzik equation isTo check the sign, ${\partial }_{t}C(e^{2t}r,g)=2e^{2t}r{\partial }_{r}C$ equals $(\beta {\partial }_g+2\gamma )C$. The characteristic flow and its accumulated multiplier give precisely this evolution. Changing $\mu$ changes the dimensionless momentum and renormalized $g$; the same bare two-point correlation function is recovered after the compensating field normalization. An unnormalized renormalized correlator need not remain numerically identical under that change, but physical predictions do.