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A momentum-shell renormalization group step has three parts. First split the Fourier transform of the field into slow modes ${\varphi }_{<}$ with $|k|<\Lambda /\zeta$ and fast modes ${\varphi }_{>}$ with $\Lambda /\zeta <|k|<\Lambda$, then perform the functional integral over ${\varphi }_{>}$. Second rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, to restore the cutoff from $\Lambda /\zeta$ to $\Lambda$. Third rescale the field so that the coefficient of $(\nabla \varphi {)}^2/2$ again has its chosen normalization. The effective free energy contains every operator allowed by the symmetries, with transformed coefficients. Repeating the step composes these coefficient maps and produces a renormalization-group flow.

For $d=4-ϵ$, define the dimensionless couplings $r={\mu }^2/{\Lambda }^2$ and $u=\tilde{g}={\Lambda }^{-ϵ}g$. Since $K_4=1/(8{\pi }^2)$, the leading epsilon expansion of the flow isThere is a Gaussian fixed point $(r_{\ast },u_{\ast })=(0,0)$. Its thermal eigenvalue is $y_t=2$, so its correlation-length critical exponent is $\boxed{{\nu }_{\mathrm{G}}=1/2}$.

The interacting Wilson-Fisher fixed point isLinearizing the renormalization-group flow gives the thermal eigenvalueTaking its reciprocal gives

A gauge anomaly breaks a redundancy required to remove unphysical polarizations. It violates the Ward identities and makes the quantum gauge theory inconsistent, so all gauge anomalies must cancel. A chiral anomaly, or Adler-Bell-Jackiw anomaly, instead breaks a classically conserved global axial symmetry; it is physically allowed and explains effects such as anomalous pseudoscalar decays and instanton-induced charge violation. A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is invariant under renormalization-group flow, so 't Hooft anomaly matching constrains the infrared theory to reproduce it through massless fields, symmetry breaking, or a topological sector.

A Wilsonian renormalization group step first splits the field into slow and fast Fourier modes and integrates out the shell $\Lambda /b<|k|<\Lambda$. It then rescales coordinates, or momenta, to restore the cutoff to $\Lambda$, and finally rescales the field to normalize the kinetic term. The resulting local effective free energy has the same allowed operators with changed coefficients. Repeating the operation therefore composes maps on the set of couplings and defines a renormalization-group flow.

If ${\lambda }_0=0$, the action has the exact discrete symmetry $\varphi \mapsto -\varphi$. Integrating out modes and rescaling preserve this symmetry, so no odd operator such as ${\varphi }^3$ can be generated from the even quartic coupling. Thus $g_0$ corrects $\lambda$ at no order in perturbation theory: the hypersurface $\lambda =0$ is invariant under the renormalization-group flow.

A gauge anomaly makes gauge redundancy inconsistent and must cancel. A chiral anomaly is the quantum nonconservation of a classically conserved axial current and may be a physical effect. A 't Hooft anomaly is an obstruction to gauging a global symmetry; it is preserved by renormalization-group flow and must be reproduced by the infrared theory, through massless degrees of freedom, symmetry breaking, or topological order.

A Wilsonian effective action is obtained by integrating out field modes above a chosen momentum scale while retaining lower-momentum modes as backgrounds. Its local couplings change with the scale according to a renormalization-group flow.