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An anomalous dimension is the interaction-generated difference between a full scaling dimension and its engineering dimension. For a scalar field, momentum-dependent self-energy corrections require wave-function renormalization and produce ${\Delta }_{\varphi }=(d-2+\eta )/2$.

The engineering dimension of a field, parameter or operator is its dimension under the rescaling that leaves the quadratic theory invariant. Interactions can change it into a full scaling dimension.

If ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$, differentiating a renormalized $n$-point function at fixed bare parameters gives $(\mu {\partial }_{\mu }+\beta {\partial }_g+n{\gamma }_{\varphi })G_{n,R}=0$. With this convention the field scaling dimension at a fixed point is its engineering dimension plus ${\gamma }_{\varphi }$. The critical exponent $\eta$ for scalar two-point decay is $2{\gamma }_{\varphi }$.

For $d\ge 2$, let $\omega$ be a Conformal Killing vector field, put $\sigma =\partial \cdot \omega /d$, and assign the massless scalar field scaling dimension $\Delta =(d-2)/2$. In a finite conformal transformation with metric scale ${\Omega }^2$, the transformation law is ${\varphi }^{\prime }(x^{\prime })=\Omega (x{)}^{-\Delta }\varphi (x)$. Its infinitesimal fixed-coordinate form isFor $\mathcal{L}=(\partial \varphi {)}^2/2$, contract the Conformal Killing equation with ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$. Direct differentiation then givesHere $2(1+\Delta )=d$. Flat conformal Killing integrability gives $\square \sigma =0$, with the Lorentzian version obtained by replacing the flat metric consistently. Thus the action variation is a boundary integral, vanishing for suitable boundary conditions. This proves conformal invariance with field dimension $(d-2)/2$, without using the field equation. The connected finite symmetry follows by integrating the infinitesimal generators. Equivalently its conformal currents follow from the improved stress-energy tensor of a free massless scalar.

In one dimension the metric conformality condition is vacuous. The same action-variation calculation contains $-{\varphi }^2{\omega }^{\prime \prime \prime }/4$, so the free action is invariant under the projective subgroup ${\omega }^{\prime \prime \prime }=0$, rather than arbitrary reparametrizations. The usual field-theory conformal-invariance claim above concerns $d\ge 2$.

In $d=2$, $\Delta =0$, so the field transforms simply as a scalar. There is no improvement term and no derivative of the conformal scale in the variation. All local independent reparametrizations of the two null coordinates preserve the action; after Euclidean continuation they are holomorphic and antiholomorphic maps. The conformal algebra is therefore infinite-dimensional, in contrast to $d>2$.

Now use Lorentzian coordinates with spatial period $\ell$. The canonical momentum is $\pi =\dot{\varphi }$, and canonical quantization imposes $[\varphi (t,x),\pi (t,y)]=i{\delta }_{\ell }(x-y)$, with the periodic delta distribution ${\delta }_{\ell }(s)={\ell }^{-1}{\sum }_{n\in \mathbb{Z}}{e}^{2\pi ins/\ell }$. Solving the wave equation and imposing reality gives a convenient oscillator normalization:Strict spatial periodicity excludes a winding term. The canonical commutation relation is equivalent towith ${\alpha }_n^{†}={\alpha }_{-n}$ and the corresponding tilded relation. Both oscillator families commute with $q,p$. To verify the normalization, the zero-mode commutator supplies $i/\ell$, while each oscillator family supplies $i(2\ell {)}^{-1}{\sum }_{n\ne 0}{e}^{2\pi in(x-y)/\ell }$. Their sum is exactly $i{\delta }_{\ell }(x-y)$. The positive-index modes are annihilation operators and the negative-index modes are creation operators, giving two independent bosonic Fock spaces together with the free-particle zero mode.

Define ${\alpha }_0={\tilde{\alpha }}_0=p/\sqrt{4\pi }$ and the normal ordering quadratic generatorsThese are the Fourier modes of the two chiral components of the stress-energy tensor. The oscillator operator commutators immediately give $[L_m,{\alpha }_n]=-n{\alpha }_{m+n}$. Applying this to both factors of $L_n$ gives $(m-n)L_{m+n}$, except for a scalar term when $m+n=0$, produced by reordering creators and annihilators. Its value is independent of the momentum sector. Evaluate it in the formal zero-momentum oscillator vacuum. For $m>0$,The two contractions of each pair explain the factor $1/2$. Hence the free-boson Virasoro central term yieldsThe tilded modes give a second commuting Virasoro algebra with $\bar{c}=1$. The adjoint condition $L_m^{†}=L_{-m}$ and the positive oscillator norms make this a unitary highest-weight Virasoro representation in each momentum sector. In a fixed momentum sector, $L_0=p^2/(8\pi )+{\sum }_{n>0}{\alpha }_{-n}{\alpha }_n$, so the oscillator ground state is a highest-weight state of weight $p^2/(8\pi )$ in either sector. For a noncompact field the zero mode is a free particle: definite momentum ground states are generalized states, rather than a normalizable zero-mode vacuum. The oscillator representation and central term remain well defined. With the standard cylinder vacuum-energy subtraction, the Hamiltonian isThe constant is the Casimir energy of a two-dimensional conformal field theory; omitting it amounts to a different additive Hamiltonian convention, not a change of the plane Virasoro algebra.

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.

Loop integrals require a subtraction prescription. Even when the classical coupling constant is dimensionless and there is no mass, specifying its renormalized value means specifying a momentum or length scale at which it is measured. Dimensionless logarithms then involve that renormalization scale $\mu$. Changing $\mu$ changes the renormalized coupling constant and wave-function renormalization while leaving bare quantities fixed. In a theory whose renormalization-group beta function and anomalous dimension both vanish, this scale dependence may disappear; the generic statement concerns the interacting renormalized theory.

Write ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$ and $G_{n,R}=Z_{\varphi }^{-n/2}{G}_{n,B}$. DefineBare correlation functions do not depend on the arbitrary subtraction scale. Differentiating their relation to renormalized correlation functions gives the Callan-Symanzik equationIt expresses the compensation between explicit scale dependence, running coupling and wave-function renormalization. In this convention the field scaling dimension is its engineering dimension plus ${\gamma }_{\varphi }$ at a fixed point; the critical-exponent notation $\eta$ often used for the anomalous dimension equals $2{\gamma }_{\varphi }$.

A renormalization-group fixed point satisfies $\beta (g_{\ast })=0$. It is ultraviolet attractive if the flow approaches it as $\mu \to \infty$, and infrared attractive if the flow approaches it as $\mu \to 0$. For a simple isolated zero, $\delta g\propto {\mu }^{{\beta }^{\prime }(g_{\ast })}$: a negative derivative gives ultraviolet attraction, a positive derivative infrared attraction. Marginal zeros, such as a cubic renormalization-group beta function at the origin, require looking at the first nonzero nonlinear term.

Put $z=p^2/{\mu }^2>0$, $t=\frac{1}{2}\log z$, and write the dimensionless quantum field theory propagator factor as $d(z,g)$. The two-point Callan-Symanzik equation becomesLet $\bar{g}(0)=g$ and $d\bar{g}/dt=\beta (\bar{g})$. The renormalization-group characteristic solution for a two-point function isWhere $\beta \ne 0$, the exponent is $2{\int }_g^{\bar{g}(t)}{\gamma }_{\varphi }(u)/\beta (u)du$. Thus the ultraviolet behavior depends on where the running coupling goes and on the anomalous dimension accumulated along the flow. A weak ultraviolet fixed point or asymptotically free limit permits perturbative evaluation; a flow to large values of the coupling constant does not.

For $\beta (g)=-b{g}^3$ and ${\gamma }_{\varphi }=c{g}^2$, $b>0$,ConsequentlyFor large positive $z$, the coupling constant tends to zero. If the normalization function $d(1,g)$ has a finite nonzero free-field limit $d(1,0)$, then $d(z,g)\sim d(1,0)[b{g}^2\log z{]}^{c/b}$. Equivalently, with $\Lambda =\mu \exp [-1/(2b{g}^2)]$,The positive exponent follows from the sign of $+n{\gamma }_{\varphi }$ in the printed RG equation; changing the field-renormalization anomalous dimension convention would change both signs together.

For the quintic renormalization-group beta function, set $B=-b>0$ and rename its positive quintic coefficient $A$ to avoid confusion with the later gauge-theory coupling constant. Then $\beta (g)=g^3(B-A{g}^2)$ has a nonzero fixed pointEvery nonzero positive running coupling in the perturbative basin approaches this ultraviolet fixed point, rather than approaching zero. The zero coupling constant is infrared attractive. If the same ${\gamma }_{\varphi }=c{g}^2$ is retained, the large-momentum factor is $d(z,g)\sim C{z}^{cB/A}$, assuming a regular nonzero matching factor at the fixed point. More generally the exponent is ${\gamma }_{\varphi }(g_{\ast })$. A fixed point at a large coupling constant inferred from a truncated renormalization-group beta function is only a formal extrapolation; perturbative control requires $B/A$ small.

For the gauge theory use the supplied convention $a=g^2/(16{\pi }^2)$ and evolution variable $L=\log {\mu }^2$. Asymptotic freedom requires ${\beta }_0>0$, or $n<11N/2$. To one-loop accuracy,The integration constant $\Lambda$ is an example of dimensional transmutation. With two loops this is the leading asymptotic expression, not an exact equality: if $\mathcal{L}=\log ({\mu }^2/{\Lambda }^2)$ is large,The expansion presumes fixed coefficients and a scale high enough to enter the ultraviolet regime.

Choose $n=11N/2-\delta$ with $0<\delta \ll N$, respecting integer flavor counts. The supplied coefficients becomeThus ${\beta }_1<0$ for sufficiently small $\delta /N$, and the two-loop flow has the Banks-Zaks fixed pointBoth $a_{\ast }$ and the loop-counting combination $N{a}_{\ast }$ are small. Its slope is ${\beta }^{\prime }(a_{\ast })=-{\beta }_0^2/{\beta }_1>0$, so it is infrared attractive. For even large $N$, taking $n=11N/2-1$ provides an explicit integer sequence with the desired parametric hierarchy. As a finite example, $N=3,n=16$ gives ${\beta }_0=1/3$, ${\beta }_1=-302/3$ and $a_{\ast }=1/302$. The fermion-count convention is the one encoded by the given coefficients.

At a critical point, a local field operator $\mathcal{O}$ has scaling dimension ${\Delta }_{\mathcal{O}}$ if $\mathcal{O}(x)\mapsto b^{-{\Delta }_{\mathcal{O}}}\mathcal{O}(x/b)$ under coarse-graining by $b$. The coupling $u$ in $\int d^{d}xu\mathcal{O}$ has renormalization-group eigenvalueA perturbation is a relevant operator when $y_u>0$, an irrelevant operator when $y_u<0$, and a marginal operator when $y_u=0$; nonlinear terms in its renormalization-group beta function decide the fate of a marginal coupling. Irrelevant microscopic interactions decay under coarse-graining, so many distinct systems approach the same fixed point and share one universality class.

Near criticality the singular free energy density is of order one per correlation volume:The heat capacity contains two derivatives with respect to temperature, soConsequently the hyperscaling relation is

A scalar field of scaling dimension ${\Delta }_{\varphi }$ has a scale-invariant two-point correlation function proportional to $|x{|}^{-2{\Delta }_{\varphi }}$. Comparison with the stated form givesThe difference $\eta /2$ from the engineering value is the field's anomalous dimension, generated by fluctuations and interactions at a non-Gaussian renormalization-group fixed point.

A free scalar has scaling dimension one. Setting the meson dimension to one givesThe expected interacting supersymmetric conformal window is therefore

At the Gaussian fixed point, the engineering dimension of the field isThe operator ${\varphi }^n({\nabla }^2\varphi {)}^m$ contains $n+m$ fields and $2m$ derivatives. Since its integral must be dimensionless,It is marginal when this vanishes, namelywhen $n+m\ne 2$. Since $n,m$ are positive, the exceptional case is $n=m=1$: $\varphi {\nabla }^2\varphi$ has a dimensionless coupling in every $d$ and differs from the kinetic term by integration by parts. If the displayed formula gives no positive $d$, there is no positive spatial dimension in which that operator is naively marginal.

At an interacting fixed point, field and composite operators acquire anomalous dimensions, and operators with the same symmetries can mix under renormalization. Their full scaling dimensions can therefore differ from these naive engineering dimensions.

By the state–operator correspondence, a CFT operator of scaling dimension $\Delta$ creates a cylinder state of energy $E=\Delta /R$. The identity gives the vacuum; the first nontrivial low-energy single-trace operator is the conserved stress-energy tensor, with $\Delta =3$ in three dimensions; and one translation raises the dimension by one. Taking the scheme-dependent vacuum energy to vanish,All other single-trace primaries are heavy by assumption, while the first two-graviton state begins at $6/R$.

The engineering value follows from the Gaussian kinetic term. At an interacting renormalization-group fixed point, momentum-dependent self-energy diagrams change the kinetic coefficient, and restoring its normalization requires wave-function renormalization. The resulting anomalous dimension $\eta$ changes the full scaling dimension to

An operator $\mathcal{O}(w,\bar{w})$ is a primary operator of conformal weights $(h,\tilde{h})$ when its OPEs with the stress tensors arewith no more singular terms. Its scaling dimension and two-dimensional spin are

Let $|\mathcal{O}⟩$ be the state associated with a scalar conformal primary operator of scaling dimension $\Delta$. In radial quantization,and $P_a^{†}=-K_a$. The norm of a level-one conformal descendant isUnitarity first gives $\Delta \ge 0$. If $\Delta =0$, every $P_a|\mathcal{O}⟩$ is null, so the local operator is translation invariant and belongs to the identity conformal family. Excluding the identity therefore gives $\Delta >0$.

Now consider the scalar level-two descendant $P^2|\mathcal{O}⟩$. The conformal algebra and the scalar-primary conditions giveApplying the second $K_a$ and summing over $a$ yieldsPositivity of this norm, together with $\Delta >0$, proves the scalar conformal unitarity boundAt equality the level-two descendant is null; in position space this is the free scalar equation of motion.

A scalar primary operator transforms without a spin rotation and is characterized by its scaling dimension. Conformal symmetry fixes its two- and three-point functions up to constants, while a four-point function can depend on conformal cross-ratios.

A discrete renormalization group with length factor $b>1$ and positive relevant thermal multiplier ${\lambda }_t>1$ sends $t$ to ${\lambda }_{t}t$ and the correlation length to $\xi /b$. Combining this with $\xi \sim |t{|}^{-\nu }$ gives the displayed correlation-length critical exponent. The corresponding scaling dimension is $y_t=\log {\lambda }_t/\log b=1/\nu$. Relevance of discrete multipliers is determined by magnitude relative to one, unlike the sign criterion for continuous-flow eigenvalues.