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At wavelengths much longer than the lattice spacing, $u_{j+1}-u_j\simeq a{\partial }_{x}u$ gives $L=\int dx[\rho ({\partial }_{t}u{)}^2-K({\partial }_{x}u{)}^2]/2$. The field obeys a wave equation with sound speed $v=\sqrt{K/\rho }$. Its conjugate momentum density is $\Pi =\rho {\partial }_{t}u$ and canonical quantization gives $[u(x),\Pi (x^{\prime })]=i\hslash \delta (x-x^{\prime })$. The lattice remains the ultraviolet cutoff of this long-wave theory.

A continuum limit removes an ultraviolet cutoff while keeping specified physical observables finite. In renormalization-group language, the bare trajectory must approach an ultraviolet fixed point and the correlation length measured in cutoff units must diverge.

The cosmological constant problem is the unexplained smallness of the gravitationally observable vacuum energy compared with the contributions expected from quantum field theory. In units with $\hslash =c=1$, an ultraviolet cutoff ${\Lambda }_{\mathrm{UV}}$ generically gives contributions of order ${\Lambda }_{\mathrm{UV}}^4$, whereas the observed dark-energy density is of order $10^{-47}{\mathrm{GeV}}^4$. The Einstein equation depends on the sum of a bare cosmological constant and these contributions; subtracting a constant from a nongravitational Hamiltonian does not remove its gravitational effect. Exact global supersymmetry gives zero energy to a supersymmetric vacuum, but supersymmetry breaking generally reintroduces a much larger positive scale.

Cutoff regularization restricts loop momenta to $|\ell |<\Lambda$. The ultraviolet cutoff $\Lambda$ makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

A finite higher-point scalar vertex is a term in the quantum effective action with more external scalar fields than the original interaction, whose coefficient remains finite after renormalization. It can survive removal of an ultraviolet cutoff even when no corresponding independent counterterm is needed. Its coefficient is then predicted by the renormalized parameters and external momenta.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

For a Gaussian field theory with quadratic kernel proportional to $t+k^2$, let $t>0$ be the distance from the critical point in rescaled units. The relevant heat capacity correction contains the integralwith fixed ultraviolet cutoff $\Lambda$. Radial integration followed by $k=\sqrt{t}x$ giveswhere $C_d$ is the area of the unit sphere in $d$ dimensions.

For $0<d<4$, the rescaled integral has a finite nonzero limit, so $I_d\propto t^{d/2-2}$. At $d=4$, its large-$x$ integrand behaves as $1/x$, givingFor $d>4$, the leading term instead tends to $I_d(0,\Lambda )=C_d{\Lambda }^{d-4}/(d-4)$. It is dominated by the short-distance cutoff. A bare power law cannot describe the full leading integral in every dimension. ^[ref]

The divergent correction below four dimensions warns that the neglected interactions can become important. It is not an exact interacting critical exponent; the Ginzburg criterion tests the validity of the approximation.

For $H=\int [a{\varphi }^2/2+\kappa |\nabla \varphi {|}^2/2+\gamma ({\nabla }^2\varphi {)}^2/2+B{\varphi }^2|\nabla \varphi {|}^2]$ and an even positive trial kernel $J(q)$, define $S_0={\sum }_q^{+}1/J(q)$ and $S_2={\sum }_q^{+}{q}^2/J(q)$ using a positive-wavevector sum for a real field. Isserlis theorem gives $⟨H_4{⟩}_0=4B{S}_0{S}_2/V$. The Feynman-Bogoliubov inequality therefore has stationary kernelsThus both the mass and gradient coefficient shift. In the thermodynamic limit, their self-consistency equations areA coarse-graining ultraviolet cutoff $\Lambda$ is necessary for the first integral in three dimensions: its large-$k$ radial integrand tends to a constant. Physical solutions require $J(q)>0$.

An infrared regulator suppresses the small-momentum region responsible for an infrared divergence. Examples are a small positive mass or a finite volume. It must be distinguished from an ultraviolet cutoff: setting external momenta and masses to zero to extract an ultraviolet divergence can introduce a spurious infrared divergence. Keep a positive auxiliary mass until the divergent local coefficients have been separated, then take the desired limit in infrared-safe quantities.

Fix the uniform order-parameter phase and retain finite system size $L$ before evaluating a massless Gaussian functional integral. In $d\le 2$, its coincident and separated order-parameter phase covariances each acquire a divergent additive constant as $L\to \infty$. Their difference $2[G_L(0)-G_L(r)]$ instead has a finite fixed-separation limit for $d>0$. This is why physical XY model correlations are obtained from Gaussian phase averaging of angle differences. An ultraviolet cutoff also remains necessary for coincident-point quantities; infrared cancellation does not remove it. The logarithmic two-dimensional Green function follows from the $d\to 2$ power-law expression only after subtracting its constant pole.

For finitely many free physical degrees of freedom with nonnegative masses, assign positive weights to bosons and negative weights to fermions. With a common spherical ultraviolet cutoff, the unsubtracted vacuum energy density is finite precisely when the three displayed supertraces vanish. Expanding $\sqrt{p^2+m^2}=p+m^2/(2p)-m^4/(8p^3)+O(p^{-5})$ proves that they cancel the quartic, quadratic and logarithmic ultraviolet divergences. The remaining radial integral converges as ${\int }^{\infty }{p}^{-3}dp$. Equal-mass boson–fermion pairing is sufficient but stronger than the three conditions; a renormalized vacuum energy with a free counterterm need not satisfy them.

Compare the same total number of finite-box modes in the soliton and vacuum sectors before removing the ultraviolet cutoff. A localized bound state replaces a continuum mode, and its contribution must be retained. For the Sine-Gordon kink fluctuation operator and the odd phase branch $\delta (k)=2\arctan (1/k)$, the bound translation mode replaces the free $k=0$ oscillator. Thus the complete oscillator correction is $-m/2-(m/2)\int dk\delta (k)k/[2\pi \sqrt{k^2+1}]$ in this convention. The continuum integral alone misses the finite $-m/2$ contribution.

The self-energy bubble diagram at loop order one in six-dimensional cubic scalar field theory has Feynman-diagram symmetry factor two. With positive Euclidean scalar propagators, its amputated Green's function is the displayed integral. Its superficial degree of divergence is two. Expanding at large internal momentum and taking the angular average shows that its divergent part is a constant plus a multiple of $p^2$, precisely the mass counterterm and wave-function renormalization structures.

For a spherical ultraviolet cutoff $|k|\le \Lambda$ and $m^2>0$, its value at $p=0$ isThe divergent $p^2$ coefficient is $-g^2\log ({\Lambda }^2/m^2)/(768{\pi }^3)$. Finite terms depend on the regularization convention. Tadpole diagram insertions are excluded from the one-particle-irreducible Feynman diagram, and are removed from the connected two-point function by imposing a vanishing one-point function.

Put ${ϵ}_k={\hslash }^2{k}^2/(2m)>0$ for $\mathbf{k}\ne 0$ and $g=u\rho$, using $\rho =N/L^3$. Since the zero mode has zero kinetic energy, the exact kinetic energy in this trial state isTogether with part (d), the fixed-mean-number Bogoliubov variational energy isFor repulsive contact interaction, $u>0$, each ${ϵ}_k+g>g>0$. Using the double-angle hyperbolic-function identities,soThe stationary point therefore satisfiesThis is a minimum, not just a vanishing derivative. Define the positive Bogoliubov quasiparticle dispersionAt the displayed solution ${\theta }_k^{\ast }$, $\cosh 2{\theta }_k^{\ast }=({ϵ}_k+g)/E_k$ and $\sinh 2{\theta }_k^{\ast }=g/E_k$. HenceThe hyperbolic cosine has its unique minimum at zero, proving global minimization of each retained term. Treating equal-magnitude wavevectors together simply multiplies the same function by their multiplicity. At a fixed ultraviolet cutoff, the resulting energy change from $\theta =0$ isIndeed $E_k<{ϵ}_k+g$. More directly, for small positive $\theta$,The pair vacuum expectation value lowers the interaction energy at first order in $\theta$, whereas the nonzero occupation number and its kinetic energy cost start at second order. The two-mode squeezed vacuum admixes opposite-momentum pairs into the condensate, introducing spatial correlations that reduce the chance of two bosons coinciding. These correlations lower repulsive contact interaction enough to outweigh the added kinetic energy. A small quantum depletion of a Bose condensate can therefore lower the energy below the uncorrelated condensate value.

Two qualifications delimit this result. First, the paper does not state the sign of $u$. More generally a finite stable minimum exists only when ${ϵ}_k+g>|g|$ for every retained nonzero mode; repulsion ensures this. For example, with $g=-{ϵ}_k$ in some mode, $f_k(\theta )={ϵ}_k\sinh (2\theta )/2$ tends to $-\infty$ as $\theta \to -\infty$, so an unconditional minimum claim for arbitrary $u$ is false. Also the formula minimizes the truncated Bogoliubov theory energy, not the exact quartic Hamiltonian including its discarded terms, and it must remain in the regime of small quantum depletion of a Bose condensate.

Second, a three-dimensional unregulated contact interaction cannot give a finite absolute beyond-leading-order ground-state energy by this bare sum. For large $k$, $E_k-{ϵ}_k-g=-g^2/(2{ϵ}_k)+O({ϵ}_k^{-2})$, whose three-dimensional sum has an ultraviolet divergence. A finite ultraviolet cutoff, or renormalization of a three-dimensional dilute-gas contact coupling matched to a physical scattering length, is needed. This does not alter the modewise minimization or the finite-cutoff lowering mechanism just derived.

Fix the uniform angular mode, for example by fixing the mean phase in a large box, and introduce a microscopic ultraviolet cutoff $a$. The quadratic Goldstone-mode effective free energy becomes, under a Fourier transform,To evaluate the Gaussian functional integral, add a source $j$. Completing the square in every nonzero Fourier transform mode givesTaking two source derivatives yields the correlation functionThe cutoff and the treatment of the uniform mode regulate this expression. At separations much larger than $a$, its nonconstant part is the Green function satisfying $-\bar{K}{\nabla }^{2}G={\delta }^{(d)}(x)$. Rotation invariance and the radial Laplacian give $(r^{d-1}{G}^{\prime }{)}^{\prime }=0$ away from the origin. Integrating the Green function equation over a small sphere, whose unit-sphere area is $S_d=2{\pi }^{d/2}/\Gamma (d/2)$, fixes the flux:Thus for $d\ne 2$ the required long-distance correlation function isAt $d=2$, the finite separation-dependent limit isThe constant absorbs the divergent constant term in the $d\to 2$ limit. In low dimension the individual $G(r)$ has no regulator-independent infinite-volume value; its differences do. This is why the phase-difference variance, rather than an absolute phase variance, is the useful observable.

The dot product of the two magnetizations is ${\bar{m}}^2\cos [\theta (x)-\theta (0)]$. For the Gaussian spin wave field, the Gaussian phase averaging identity $⟨e^{iX}⟩=e^{-⟨X^2⟩/2}$ for a centered Gaussian variable givesKeeping this exponential is essential: expanding the cosine to quadratic order would fail when phase differences grow with separation. Equivalently, the phase-difference variance isFor $d>2$, the integral defining the absolute phase variance is finite at small momentum after fixing the uniform mode; its microscopic ultraviolet cutoff still matters. Its nonconstant Green function decays as $r^{2-d}$, so $D(r)$ tends to a finite constant $D_{\infty }$ andIn two dimensions, the logarithmic Green function gives $D(r)=(\pi \bar{K}{)}^{-1}\log (r/a)+O(1)$. ConsequentlyFor $1\le d<2$, $D(r)$ grows as $2r^{2-d}/[(2-d)S_d\bar{K}]$, up to bounded microscopic terms, and the correlation function again vanishes. In particular, in one dimension $S_1=2$ gives $D(r)=r/\bar{K}$ and exponential decay ${\bar{m}}^2{e}^{-r/(2\bar{K})}$. Therefore there is no conventional long-range order for $d\le 2$ at positive temperature in this short-range continuous-symmetry model. In two dimensions, algebraic decay can still support quasi-long-range order.

These infrared spin-wave correlations of the XY model explain the mechanism behind the Mermin-Wagner theorem. They are a test of the low-temperature spin wave approximation, not a claim that every $d>2$ XY model is ordered at every temperature. Large local fluctuations and defects destroy the ordered phase at sufficiently high temperature. Also, the assumed small angle is a local smooth-field approximation; phase differences at arbitrarily large separation need not remain small.

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 $\eta =O({ϵ}^2)$.

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.