Callan–Symanzik Scaling and Emergent Conformal Symmetry

The previous page treated scaling dimensions as the language of critical correlators. This page explains how that language arises dynamically. Near a continuous phase transition, perturbation theory first produces logarithms such as $\log(\Lambda/k)$: the same fluctuation physics is being sampled over many length scales. The renormalization group reorganizes those logarithms into scaling laws.

The central object is the critical two-point function of the order-parameter field. In the Gaussian theory,

$$G_0(k)={1\over k^2}, \qquad G_0(x)\sim {1\over |x|^{d-2}},$$

so the engineering dimension of a scalar field is $(d-2)/2$. At an interacting fixed point this becomes

$$G(k)\sim {1\over k^{2-\eta}}, \qquad G(x)\sim {1\over |x|^{d-2+\eta}}, \qquad \Delta_\phi={d-2+\eta\over2}.$$

The exponent $\eta$ is the anomalous part of the field dimension. The Callan–Symanzik equation is the clean differential statement behind this result.

Required background. Lesson 12 supplies the scaling-field and critical-exponent dictionary used throughout this page.

Helpful background. Lesson 4 introduces the upper critical dimension, while Lesson 5 develops the Wilson–Fisher fixed point and anomalous dimensions perturbatively.

Engineering dimension and the Gaussian propagator

Renormalization convention

We work in Euclidean $d$ dimensions. The scalar field theory used as the standard example is

$$S=\int d^d x\,\left[{1\over2}(\partial\phi)^2+{1\over2}r\phi^2+{\mu^\epsilon g\over4!}\phi^4\right], \qquad \epsilon=4-d.$$

The parameter $r$ is tuned to its critical value. We write

$$G(k)=\langle \phi(k)\phi(-k)\rangle_c, \qquad \Gamma^{(2)}(k)=G(k)^{-1}.$$

For a renormalized field $\phi_R$, we use

$$\phi_0=Z_\phi^{1/2}\phi_R, \qquad \gamma_\phi(g)={1\over2}\mu{\partial\over\partial\mu}\log Z_\phi\bigg|_{\rm bare}, \qquad \beta(g)=\mu{\partial g\over\partial\mu}\bigg|_{\rm bare}.$$

With this convention, the fixed-point dimension is

$$\Delta_\phi={d-2\over2}+\gamma_\phi(g_*), \qquad \eta=2\gamma_\phi(g_*).$$

Some references define $\gamma_\phi$ with the opposite sign. The physical exponents are, of course, convention-independent.

The free massless scalar action is

$$S_0={1\over2}\int d^d x\,(\partial\phi)^2.$$

Under a dilation $x\mapsto \lambda x$, the kinetic term remains invariant if

$$\phi(x)\mapsto \lambda^{\Delta_\phi^{(0)}}\phi(\lambda x), \qquad \Delta_\phi^{(0)}={d-2\over2}.$$

Equivalently, the free propagator satisfies

$$G_0(\lambda x)=\lambda^{-(d-2)}G_0(x).$$

The explicit Fourier transform is

$$G_0(x)=\int {d^d k\over(2\pi)^d}\,{e^{ik\cdot x}\over k^2} ={\Gamma(d/2-1)\over4\pi^{d/2}}{1\over |x|^{d-2}} \qquad (d>2).$$

Thus

$$G_0(x)\sim {1\over |x|^{2\Delta_\phi^{(0)}}}.$$

At this level, scaling is just dimensional analysis. The lesson of critical phenomena is that the true long-distance field can scale with a dimension different from its engineering dimension.

Logarithms before powers

Interactions first appear perturbatively as corrections to the inverse propagator and to the interaction vertex. Write the exact inverse propagator schematically as

$$G^{-1}(k)=k^2+r+\Sigma(k),$$

where $\Sigma(k)$ is the self-energy. At criticality the mass-like term is tuned so that

$$G^{-1}(0)=0.$$

The remaining momentum dependence can contain logarithms. In four dimensions, $g\phi^4$ is classically marginal, and typical loop integrals behave like

$$\int_k^\Lambda {d^4p\over p^4} \sim \log {\Lambda\over k}.$$

Therefore a critical inverse propagator can have the schematic form

$$G^{-1}(k)=k^2\left[1+A g^2\log {\Lambda\over k}+\cdots\right].$$

The exact power of $g$ depends on the quantity being studied. In ordinary $\phi^4$ theory the field anomalous dimension begins at two loops; the energy operator $\phi^2$ already receives an anomalous dimension at one loop. The structural point is the same: when $k\ll\Lambda$, the logarithm becomes large and fixed-order perturbation theory is not the right expansion.

The same issue appears in the four-point vertex. In $d=4$ one finds, schematically,

$$\Gamma_4(k)=g-Bg^2\log {\Lambda\over k}+\cdots.$$

The logarithm says that the coupling measured at scale $k$ is not the same as the coupling defined at the cutoff. Physics at different scales is being compared.

Perturbation theory near a critical point produces logarithms. RG improvement sums the leading logarithms. At an interacting fixed point the result is a power law $G(k)\sim k^{-2+\eta}$; at the upper critical dimension it is a mean-field power multiplied by logarithms.

A useful cartoon is

$$1+A g\log {\Lambda\over k}+{1\over2}A^2g^2\log^2 {\Lambda\over k}+\cdots \approx \exp\left(A g\log {\Lambda\over k}\right) =\left({\Lambda\over k}\right)^{Ag}.$$

This cartoon is not a substitute for RG, because the coupling itself runs. But it captures the key transition: a tower of logarithms can reorganize into a noninteger power.

The upper critical dimension and logarithmic scaling

The engineering dimension of the quartic coupling is

$$[g]=4-d.$$

Thus $d=4$ is the upper critical dimension of the Ising universality class. Above four dimensions the quartic interaction is irrelevant and mean-field exponents are correct. Below four dimensions the interaction flows to the Wilson–Fisher fixed point. Exactly at four dimensions it is marginally irrelevant in the infrared, so powers acquire logarithmic corrections.

Near $d=4$, the beta function has the form

$$\beta(g)=\mu {dg\over d\mu}=-\epsilon g+b g^2+O(g^3), \qquad b>0.$$

For $d=4-\epsilon$, there is an interacting fixed point

$$g_*={\epsilon\over b}+O(\epsilon^2).$$

At this fixed point the critical correlators are pure powers. At exactly $d=4$, however, $\epsilon=0$, and the infrared running is logarithmic:

$$g(k)\simeq {1\over b\log(\Lambda/k)}.$$

Because the coupling dies only as $1/\log$, thermodynamic quantities inherit logarithmic violations of mean-field scaling. For the four-dimensional Ising universality class, the singular specific heat has the characteristic form

$$C_{\rm sing}\sim \left(\log {1\over |\tau|}\right)^{1/3},$$

up to nonuniversal normalizations and subleading logarithms. Here $\tau$ is the reduced temperature. This is a useful warning: writing the specific-heat exponent as $\alpha=0$ is incomplete at a marginal dimension. The difference between a finite jump, a logarithm, and a fractional power of a logarithm is real physics.

At $d=4$, the quartic coupling is marginally irrelevant and flows as $g(L)\sim1/L$ in infrared time $L=\log(\Lambda/k)$. The slow flow leaves multiplicative logarithms in thermodynamic quantities, including $C_{\rm sing}\sim[\log(1/|\tau|)]^{1/3}$ for the Ising scalar.

The Callan–Symanzik equation

Let

$$G_R^{(n)}(x_1,\ldots,x_n;g,\mu) =\langle \phi_R(x_1)\cdots\phi_R(x_n)\rangle_c$$

be a renormalized critical correlator. The auxiliary scale $\mu$ is introduced by renormalization. Bare quantities do not depend on $\mu$, so changing $\mu$ must be compensated by changing the renormalized coupling and the normalization of the fields. This gives the Callan–Symanzik equation

$$\boxed{ \left[\mu {\partial\over\partial\mu} +\beta(g){\partial\over\partial g} +n\gamma_\phi(g)\right]G_R^{(n)}=0. }$$

For a product of multiplicatively renormalized operators $O_i$, the last term becomes $\sum_i\gamma_{O_i}(g)$. If operators with the same quantum numbers mix, the anomalous dimensions form a matrix and the equation acts on the corresponding vector of correlators.

The equation is best read by the method of characteristics. Define the running coupling by

$${d\bar g(\ell)\over d\ell}=\beta(\bar g(\ell)), \qquad \bar g(0)=g,$$

where increasing $\ell$ corresponds to changing the reference scale $\mu\mapsto e^\ell\mu$. Then

$$G_R^{(n)}(x_i;g,\mu) = \exp\left[+n\int_0^\ell d\ell'\,\gamma_\phi(\bar g(\ell'))\right] G_R^{(n)}(x_i;\bar g(\ell),e^\ell\mu).$$

The plus sign follows directly from transporting the correlator from the endpoint of the characteristic back to its starting point. Equivalently,

$$G_R^{(n)}(x_i;\bar g(\ell),e^\ell\mu) = \exp\left[-n\int_0^\ell d\ell'\,\gamma_\phi(\bar g(\ell'))\right] G_R^{(n)}(x_i;g,\mu).$$

These expressions are exact but are not yet scaling laws. The scaling law appears when the running coupling approaches a fixed point. It is often more intuitive to use infrared RG time $L=-\ell=\log(\mu/\mu_{\rm IR})$; then $d\bar g/dL=-\beta(\bar g)$. The figure uses this infrared convention.

The Callan–Symanzik equation transports correlators along RG characteristics. The diagram uses infrared time $L=\log(\mu_0/\mu)$, so $dg/dL=-\beta(g)$. Near a fixed point, the running coupling stops changing and the remaining effect is a definite scaling dimension.

At a fixed point,

$$\beta(g_*)=0, \qquad \gamma_\phi(g_*)=\gamma_\phi^*.$$

Combining the Callan–Symanzik equation with ordinary dimensional analysis gives

$$G_R^{(n)}(\lambda x_1,\ldots,\lambda x_n) =\lambda^{-n\Delta_\phi}G_R^{(n)}(x_1,\ldots,x_n),$$

where

$$\boxed{ \Delta_\phi={d-2\over2}+\gamma_\phi^*. }$$

For the two-point function,

$$G(x)=\langle\phi(x)\phi(0)\rangle_c \sim {1\over |x|^{2\Delta_\phi}} ={1\over |x|^{d-2+\eta}},$$

with

$$\boxed{\eta=2\gamma_\phi^*.}$$

Momentum space gives the equivalent form

$$\boxed{ G(k)\sim {1\over k^{2-\eta}}, \qquad G^{-1}(k)\sim k^{2-\eta}. }$$

This is the cleanest interpretation of the anomalous exponent: it is the fixed-point value of the field-renormalization part of the RG flow.

How logarithms become powers

The quickest way to see the logarithm-to-power mechanism is to pretend that the coupling has already reached a fixed point and that the leading logarithms exponentiate. Suppose

$$G^{-1}(k)=k^2\left[1+a\log {\Lambda\over k}+{a^2\over2!}\log^2 {\Lambda\over k}+\cdots\right].$$

Then

$$G^{-1}(k)=k^2\exp\left(a\log {\Lambda\over k}\right) =k^2\left({\Lambda\over k}\right)^a.$$

The cutoff factor $\Lambda^a$ is not universal; it can be absorbed into the normalization of the field. The $k$ dependence is the important long-distance information:

$$G^{-1}(k)\propto k^{2-a}.$$

In a real calculation, $a$ is replaced by the fixed-point anomalous exponent, with signs depending on whether one discusses $G$ or $G^{-1}$. The invariant statement is

$$\text{large logarithms} \quad\longrightarrow\quad \text{RG flow} \quad\longrightarrow\quad \text{anomalous power law}.$$

Skeleton equations and self-consistent scaling

There is another way to think about anomalous exponents. Instead of computing a few diagrams with bare propagators, write self-consistent equations for dressed quantities. The exact two-point function satisfies a Dyson equation

$$G^{-1}(k)=G_0^{-1}(k)+\Sigma(k),$$

where $\Sigma$ is the self-energy. The exact four-point vertex satisfies a schematic skeleton equation

$$\Gamma_4=g+\Phi[\Gamma_4,G],$$

where $\Phi$ denotes loop functionals built from dressed propagators and dressed vertices.

At criticality, dressed propagators and vertices can be inserted into skeleton equations. A power-law ansatz turns scale-invariant integral equations into consistency conditions for anomalous exponents.

At a fixed point, the only scale is momentum itself. It is therefore natural to try

$$G(k)\sim {1\over |k|^{2-\eta}}.$$

The four-point vertex has its own scaling. With the overall momentum-conserving delta function stripped off, the momentum-space 1PI vertex has dimension

$$[\Gamma_4]=d-4\Delta_\phi =d-2(d-2+\eta) =4-d-2\eta.$$

Thus a single-scale four-point vertex behaves schematically as

$$\Gamma_4(k)\sim k^{4-d-2\eta}.$$

Different normalizations may move powers between external legs and the vertex, but the invariant statement is that once every object is assigned a scaling dimension, each skeleton equation must be homogeneous under $k\mapsto \lambda k$.

Relevant perturbations and scaling forms

A critical point is a fixed point only after relevant perturbations have been tuned. For the Ising class the two most important perturbations are the temperature-like coupling and the magnetic field:

$$S=S_*+\tau\int d^d x\,\epsilon(x)-h\int d^d x\,\sigma(x)+\cdots.$$

Near the fixed point their RG equations are

$${d\tau\over d\ell}=y_t\tau+\cdots, \qquad {dh\over d\ell}=y_h h+\cdots,$$

where

$$y_t=d-\Delta_\epsilon, \qquad y_h=d-\Delta_\sigma.$$

Solving the corresponding Callan–Symanzik equation gives the scaling form

$$\langle O(x)O(0)\rangle_\tau ={1\over |x|^{2\Delta_O}} \Phi_O\left(\tau |x|^{y_t}\right).$$

The correlation length is the value of $|x|$ at which the scaling variable becomes order one:

$$\tau \xi^{y_t}\sim1, \qquad \boxed{\xi\sim |\tau|^{-1/y_t}.}$$

Therefore

$$\nu={1\over y_t}={1\over d-\Delta_\epsilon},$$

which matches the scaling-dimension dictionary from the previous page. The Callan–Symanzik equation supplies its RG derivation.

Emergent scale invariance

A continuum critical point is obtained by taking a limit in which the microscopic lattice spacing $a$ becomes invisible compared with the physical correlation length:

$$a\ll |x|\ll \xi.$$

Exactly at criticality, $\xi=\infty$, and the long-distance theory has no intrinsic scale. This is emergent scale invariance. It does not mean the microscopic lattice was scale invariant. It means the RG flow has forgotten most microscopic details and landed on a fixed point.

The phrase “fixed point” should be taken literally. Under coarse graining and rescaling, the effective action returns to itself, up to field normalization and irrelevant corrections:

$$S_*[\phi]\mapsto S_*[\phi]+\text{redundant changes}.$$

Correlation functions then obey homogeneous scaling laws. For a scalar scaling operator,

$$\langle O(\lambda x_1)\cdots O(\lambda x_n)\rangle = \lambda^{-n\Delta_O} \langle O(x_1)\cdots O(x_n)\rangle.$$

Scale invariance alone does not yet give the full conformal transformation law. The next step is to understand when a scale-invariant local theory also has invariance under transformations that rescale distances by a position-dependent factor.

First hint of conformal symmetry

Consider an infinitesimal coordinate transformation in flat Euclidean space,

$$x^\mu\mapsto x'^\mu=x^\mu+\xi^\mu(x).$$

The flat metric changes according to

$$ds'^2 =\delta_{\mu\nu}\,dx'^\mu dx'^\nu =\left(\delta_{\rho\sigma} +\partial_\rho\xi_\sigma +\partial_\sigma\xi_\rho\right)dx^\rho dx^\sigma +O(\xi^2).$$

The transformation is conformal if it preserves angles, equivalently if the metric changes only by a local scale factor:

$$ds'^2=\rho(x)\,ds^2.$$

To first order this means

$$\partial_\rho\xi_\sigma+ \partial_\sigma\xi_\rho=2\lambda(x)\delta_{\rho\sigma}.$$

Taking the trace gives

$$2\partial\cdot\xi=2d\lambda(x),$$

so

$$\boxed{ \partial_\mu\xi_\nu+ \partial_\nu\xi_\mu ={2\over d}(\partial\cdot\xi)\delta_{\mu\nu}. }$$

This is the conformal Killing equation. Translations and rotations solve it with $\partial\cdot\xi=0$; dilations solve it with $\xi^\mu=cx^\mu$; special conformal transformations give the remaining solutions in $d>2$.

The holomorphic map $w=z+0.18z^2$ gives a genuine two-dimensional example: its image grid is curved, yet the tangent directions remain orthogonal wherever $f'(z)\neq0$. In any dimension, the infinitesimal condition is the conformal Killing equation $\partial_\mu\xi_\nu+\partial_\nu\xi_\mu=(2/d)(\partial\cdot\xi)\delta_{\mu\nu}$.

The connection to RG is through the stress tensor. A scale-invariant theory has a conserved dilation current of the form

$$D_\mu=x^\nu T_{\mu\nu}-V_\mu,$$

where $V_\mu$ is a possible virial current. Conservation gives

$$T^\mu_{\ \mu}=\partial^\mu V_\mu.$$

If the virial current is absent, or if it can be removed by improving the stress tensor, then

$$T^\mu_{\ \mu}=0.$$

A conserved traceless stress tensor supplies the conformal currents. Thus a removable virial current is the precise bridge from scale to conformal invariance. This bridge is established under broad hypotheses in two dimensions and in many higher-dimensional settings, but it is not a consequence of scale invariance alone without assumptions on locality, unitarity, the spectrum, and the stress tensor.

In two dimensions, the conformal Killing equation becomes especially strong. Writing

$$z=x+iy, \qquad \xi(z)=\xi^1(x,y)+i\xi^2(x,y),$$

one obtains

$$\partial_x\xi^1=\partial_y\xi^2, \qquad \partial_x\xi^2=-\partial_y\xi^1.$$

These are the Cauchy–Riemann equations,

$$\partial_{\bar z}\xi(z)=0.$$

Thus two-dimensional infinitesimal conformal transformations are locally holomorphic functions. This is the first glimpse of the infinite-dimensional symmetry that will dominate the CFT part of the course.

Example: fixed-point solution of the two-point equation

Take the critical two-point function in momentum space. Dimensional analysis says that before anomalous scaling is included, the propagator has momentum dimension $-2$:

$$G_R(k;g_*,\mu)=\mu^{-2}F(k/\mu).$$

At the fixed point, the Callan–Symanzik equation is

$$\left[\mu {\partial\over\partial\mu}+2\gamma_\phi^*\right]G_R(k;g_*,\mu)=0.$$

Since $G_R$ depends on $\mu$ through $k/\mu$ and through the engineering prefactor,

$$\mu {\partial G_R\over\partial\mu} =-2G_R-k{\partial G_R\over\partial k}.$$

The fixed-point equation becomes

$$-k{\partial G_R\over\partial k}-2G_R+2\gamma_\phi^*G_R=0.$$

Hence

$$k{\partial G_R\over\partial k}=(-2+2\gamma_\phi^*)G_R.$$

Solving gives

$$G_R(k)\propto k^{-2+2\gamma_\phi^*}.$$

Using $\eta=2\gamma_\phi^*$,

$$\boxed{G_R(k)\propto {1\over k^{2-\eta}}.}$$

Fourier transformation gives

$$G_R(x)\propto {1\over |x|^{d-2+\eta}}.$$

The anomalous exponent is therefore not an independent assumption; it is the fixed-point value of the RG field-renormalization function.

Summary

Perturbation theory near a critical point produces logarithms such as $\log(\Lambda/k)$. These logarithms are warnings that the same physics is being compared at several scales. The Callan–Symanzik equation turns that warning into a tool: it tracks how couplings and operator normalizations change with the renormalization scale.

At an RG fixed point,

$$\beta(g_*)=0,$$

and the Callan–Symanzik equation becomes a scaling equation. The field dimension becomes

$$\Delta_\phi={d-2\over2}+\gamma_\phi^*,$$

and the critical propagator behaves as

$$G(k)\sim {1\over k^{2-\eta}}, \qquad G(x)\sim {1\over |x|^{d-2+\eta}}.$$

Relevant perturbations such as the reduced temperature produce scaling variables like $\tau |x|^{y_t}$ and correlation lengths $\xi\sim|\tau|^{-1/y_t}$. At the upper critical dimension, where a coupling is marginally irrelevant, pure powers are modified by logarithms.

Finally, a local fixed point often does more than scale. If the stress tensor can be improved to be traceless, scale symmetry extends to conformal symmetry. Infinitesimally, conformal transformations are the solutions of

$$\partial_\mu\xi_\nu+ \partial_\nu\xi_\mu={2\over d}(\partial\cdot\xi)\delta_{\mu\nu}.$$

The next page studies these transformations directly.

Common pitfalls

A logarithm in perturbation theory is not itself an anomalous dimension. The anomalous dimension is obtained after RG improvement, and at a true fixed point it becomes a number.

The cutoff factor in expressions such as $G^{-1}(k)\sim \Lambda^\eta k^{2-\eta}$ is not universal. It reflects the normalization of the microscopic field. The exponent of $k$ is the universal part.

The upper critical dimension is special. In four-dimensional Ising/$\phi^4$ theory, the interaction is marginally irrelevant, so mean-field powers are dressed by logarithms rather than replaced by generic Wilson–Fisher powers.

The self-energy $\Sigma(k)$ must be interpreted after mass tuning. A constant term in $\Sigma(0)$ shifts the critical temperature; the anomalous critical propagator comes from the nonanalytic momentum dependence at the tuned point.

Scale invariance and conformal invariance are not identical statements. The bridge between them involves locality, the stress tensor, and the possibility of improving the trace.

Exercises

Exercise 1: Gaussian field dimension

Derive the engineering dimension of a free scalar field in $d$ Euclidean dimensions from

$$S_0={1\over2}\int d^d x\,(\partial\phi)^2.$$

Then show that the Gaussian two-point function behaves as $G_0(x)\sim |x|^{-(d-2)}$.

Solution

Under $x\mapsto \lambda x$, the measure scales as

$$d^d x\mapsto \lambda^{-d}d^d x$$

if we rewrite the transformed action in terms of the original coordinate. The derivative scales as

$$\partial\mapsto \lambda\partial.$$

Let

$$\phi(x)\mapsto \lambda^\Delta\phi(\lambda x).$$

Then the kinetic term scales as

$$\int d^d x\,(\partial\phi)^2 \mapsto \lambda^{-d}\lambda^{2+2\Delta} \int d^d x\,(\partial\phi)^2.$$

Scale invariance requires

$$-d+2+2\Delta=0,$$

so

$$\Delta={d-2\over2}.$$

The two-point function of a scalar scaling field has exponent twice the scaling dimension:

$$G_0(x)=\langle\phi(x)\phi(0)\rangle\sim {1\over |x|^{2\Delta}} ={1\over |x|^{d-2}}.$$

Exercise 2: Fixed-point propagator from Callan–Symanzik

Assume the fixed-point Callan–Symanzik equation for the critical two-point function is

$$\left[\mu {\partial\over\partial\mu}+2\gamma_\phi^*\right]G_R(k;\mu)=0,$$

and that dimensional analysis gives $G_R(k;\mu)=\mu^{-2}F(k/\mu)$. Show that

$$G_R(k)\propto k^{-2+\eta}, \qquad \eta=2\gamma_\phi^*.$$

Solution

Since

$$G_R(k;\mu)=\mu^{-2}F(k/\mu),$$

we have

$$\mu{\partial G_R\over\partial\mu} =-2G_R-k{\partial G_R\over\partial k}.$$

Substitute this into the fixed-point Callan–Symanzik equation:

$$-k{\partial G_R\over\partial k}-2G_R+2\gamma_\phi^*G_R=0.$$

Thus

$$k{\partial G_R\over\partial k}=(-2+2\gamma_\phi^*)G_R.$$

Solving,

$$G_R(k)=Ck^{-2+2\gamma_\phi^*}.$$

With $\eta=2\gamma_\phi^*$,

$$G_R(k)=Ck^{-2+\eta}={C\over k^{2-\eta}}.$$

Exercise 3: Multiplicative logarithms from a marginal coupling

Let $L=\log(|x|/a)$ be infrared RG time. Suppose a marginally irrelevant coupling obeys

$${dg\over dL}=-b g^2, \qquad b>0,$$

and let a composite operator have anomalous dimension $\gamma_O(g)=c g+O(g^2)$. Show that its two-point function receives a multiplicative logarithmic correction of the form

$$\langle O(x)O(0)\rangle \sim {1\over |x|^{2\Delta_O^{(0)}}} \left(\log {|x|\over a}\right)^{-2c/b},$$

up to convention-dependent signs in the definition of $c$.

Solution

The running coupling satisfies

$${dg\over dL}=-b g^2,$$

so at large $L$,

$$g(L)\simeq {1\over bL}.$$

The RG normalization factor for a two-point function of $O$ is

$$\exp\left[-2\int^L dL'\,\gamma_O(g(L'))\right].$$

Using $\gamma_O(g)=cg+O(g^2)$ gives

$$\int^L dL'\,\gamma_O(g(L')) \simeq \int^L dL'\,{c\over bL'} ={c\over b}\log L.$$

Therefore the normalization factor is

$$L^{-2c/b}.$$

Since $L=\log(|x|/a)$ for a separation $|x|$, the correlator becomes

$$\langle O(x)O(0)\rangle \sim {1\over |x|^{2\Delta_O^{(0)}}} \left(\log {|x|\over a}\right)^{-2c/b}.$$

Changing the sign convention for $\gamma_O$ changes the sign of $c$; the RG integration is the invariant content.

Exercise 4: Derive the conformal Killing equation

Starting from

$$x'^\mu=x^\mu+\xi^\mu(x),$$

show that angle preservation to first order requires

$$\partial_\mu\xi_\nu+ \partial_\nu\xi_\mu ={2\over d}(\partial\cdot\xi)\delta_{\mu\nu}.$$

Solution

The transformed differential is

$$dx'^\mu=dx^\mu+\partial_\rho\xi^\mu dx^\rho.$$

Thus

$$ds'^2=\delta_{\mu\nu}dx'^\mu dx'^\nu =\left(\delta_{\rho\sigma} +\partial_\rho\xi_\sigma +\partial_\sigma\xi_\rho\right)dx^\rho dx^\sigma +O(\xi^2).$$

A conformal transformation preserves angles, so it may only multiply the metric by a scalar function:

$$ds'^2=(1+2\lambda(x))\delta_{\rho\sigma}dx^\rho dx^\sigma.$$

Equating coefficients gives

$$\partial_\rho\xi_\sigma+ \partial_\sigma\xi_\rho=2\lambda(x)\delta_{\rho\sigma}.$$

Taking the trace yields

$$2\partial\cdot\xi=2d\lambda,$$

so

$$\lambda={1\over d}\partial\cdot\xi.$$

Substituting back gives

$$\partial_\mu\xi_\nu+ \partial_\nu\xi_\mu ={2\over d}(\partial\cdot\xi)\delta_{\mu\nu}.$$

Exercise 5: Cauchy–Riemann form in two dimensions

In two Euclidean dimensions, set

$$\xi(z)=\xi^1(x,y)+i\xi^2(x,y), \qquad z=x+iy.$$

Show that the conformal Killing equation is equivalent to $\partial_{\bar z}\xi=0$.

Solution

For $d=2$, the conformal Killing equation is

$$\partial_i\xi_j+ \partial_j\xi_i=(\partial_k\xi_k)\delta_{ij}.$$

The $11$ component gives

$$2\partial_x\xi^1=\partial_x\xi^1+\partial_y\xi^2,$$

so

$$\partial_x\xi^1=\partial_y\xi^2.$$

The $12$ component gives

$$\partial_x\xi^2+\partial_y\xi^1=0.$$

These are precisely the Cauchy–Riemann equations for $\xi^1+i\xi^2$. Therefore

$$\partial_{\bar z}\xi=0.$$

References

• C. G. Callan, “Broken Scale Invariance in Scalar Field Theory,” Physical Review D 2 (1970), 1541–1547.
• J. Polchinski, “Scale and Conformal Invariance in Quantum Field Theory,” Nuclear Physics B 303 (1988), 226–236.
• K. Symanzik, “Small Distance Behaviour in Field Theory and Power Counting,” Communications in Mathematical Physics 18 (1970), 227–246.

Further reading

• J. Cardy, Scaling and Renormalization in Statistical Physics, for scaling forms and the RG derivation of critical exponents.
• A. M. Polyakov, Gauge Fields and Strings, especially the discussion of statistical mechanics, renormalization, and conformal field theory.
• K. G. Wilson and J. Kogut, “The Renormalization Group and the $\epsilon$ Expansion,” Physics Reports 12 (1974), 75–199.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, for Callan–Symanzik equations, anomalous dimensions, and logarithmic corrections at upper critical dimensions.