Momentum-shell renormalization group

The momentum-shell renormalization group splits a field into slow modes below $\Lambda /\zeta$ and fast modes in $\Lambda /\zeta <|k|<\Lambda$, integrates out the fast modes, then rescales momenta and fields to restore the cutoff and kinetic normalization.

Integrate field modes with momenta between a fixed microscopic ultraviolet cutoff ${\Lambda }_0$ and a lower cutoff $\Lambda$, retaining the modes below $\Lambda$ as backgrounds. The exact statistical Hamiltonian includes all generated interactions and field-independent constants. Integrating its retained modes reproduces the original partition function, so long-distance observables do not depend on the arbitrary intermediate cutoff. With sources retained, the corresponding coarse observables reproduce long-distance response and connected correlation functions. A local gradient expansion or a finite-coupling truncation is an additional approximation, rather than a consequence of cutoff independence. The momentum-shell construction is also explained in the Cambridge statistical field theory lecture notes.

For centered fast Fourier modes and $V=u\int {\varphi }^4/4!$, the two-fast-field vertex is $(u/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$, producing $-u^2\int {\phi }^2(x){\phi }^2(y)G_{>}(x-y{)}^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u^{\prime }=b^{4-D}(u-3u^2{I}_2/2)+\cdots$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}K(p{)}^{-2}>0$. At $D=4$ its sign makes weak positive quartic coupling marginally irrelevant for increasing length scale, after tuning the critical mass. This local truncation omits generated derivative and higher-order operators; it is not the exact complete finite-shell effective action.

For interaction $V=u\int {\varphi }^4/4!$, split $\varphi =\phi +\eta$ into slow and fast Fourier modes, with a centered fast Gaussian measure. The first cumulant expansion term is $⟨V⟩=(u/24)\int [{\phi }^4+6G_{>}(0){\phi }^2+3G_{>}(0{)}^2]$ by Wick theorem. The constant shifts the free energy, the quadratic term gives $\delta r=u{G}_{>}(0)/2$, and there is no gradient correction at this order. Rescaling gives $r^{\prime }=b^2[r+u{G}_{>}(0)/2]+O(u^2)$ and $u^{\prime }=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.

For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

Splitting a scalar field into slow and fast components $\varphi =\phi +\eta$, the Wick contractions in $\lambda ⟨(\phi +\eta {)}^6⟩$ generate $15\lambda ⟨{\eta }^2⟩{\phi }^4$. A vanishing bare quartic coupling is therefore not generally preserved by renormalization-group flow.

Writing the interaction as $V$ and averaging over the fast modes, which have a Gaussian distribution, gives the cumulant expansion $F_{\mathrm{eff}}=F_0+⟨V⟩-\frac{1}{2}(⟨V^2⟩-⟨V{⟩}^2)+\cdots$. The second cumulant selects connected Feynman diagrams.