Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 44 / 2

In the Wilsonian effective action picture, integrate out field modes in a high-momentum shell and encode their effects in the action for the retained modes. Rescale lengths and fields to compare the resulting theory at the original cutoff. The couplings then follow a renormalization-group flow, while low-energy predictions are preserved. In general all local interactions allowed by the symmetries are generated, even if only a few appear in the initial action.

Near a renormalization-group fixed point, a perturbation $u\int d^{d}x\mathcal{O}(x)$ with scaling dimension $\Delta$ has linearized eigenvalue $y=d-\Delta$. Under coarse graining by $b>1$, its dimensionless coupling scales as $u^{\prime }\simeq b^{y}u$. A relevant operator has $y>0$, so its perturbation grows toward long distances; an irrelevant operator has $y<0$ and decreases; a marginal operator has $y=0$ and needs nonlinear flow to determine its behavior. Marginality at linear order need not mean exact scale independence. Interactions can be marginally relevant or marginally irrelevant operators.

Choose $\varphi (x)=\int d^{4}p(2\pi {)}^{-4}{e}^{ip\cdot x}\tilde{\varphi }(p)$ in Euclidean signature. The inverse quadratic kernel is the smooth-cutoff scalar propagatorThe subscript records the explicit cutoff dependence implied by the condition $f_{\Lambda }(z)={\Lambda }^{2}z$ at small $z$. For $p^2+m^2\le {\Lambda }^2$, this reduces to $1/(p^2+m^2)$, the usual Euclidean scalar propagator. For $p^2+m^2\gg {\Lambda }^2$, the kernel $f_{\Lambda }$ grows rapidly and its inverse tends to zero.

High-momentum modes are strongly suppressed. For a finite smooth regulator they are not literally identically zero; that statement would require a sharp cutoff. The field variance carried by those Fourier transform modes is correspondingly negligible.

It is useful to regulate the number of modes first, so the functional integral identities reduce to ordinary integration by parts. Let $C$ be the Gaussian covariance, $F=C^{-1}$, $W=e^{-S_1}$ and put a dot for $\Lambda {\partial }_{\Lambda }$. The matrix identity $\dot{F}=-F\dot{C}F$ givesThe second field derivative of the Gaussian isHence, after two integrations by parts,The imposed flow makes the bracket vanish. The remaining trace is independent of the fields and only changes the Gaussian normalization. Since the free Gaussian normalization is $\mathcal{N}=(\det (2\pi C){)}^{1/2}$, $\dot{\log \mathcal{N}}=\mathrm{Tr}(F\dot{C})/2$. ThereforeEquivalently, $Z$ is cutoff independent after discarding the stated overall rescaling. This is the Gaussian covariance differentiation identity behind the Polchinski equation.

With the Fourier convention above and functional derivatives satisfying $\delta \tilde{\varphi }(p)/\delta \tilde{\varphi }(q)={\delta }^{(4)}(p-q)$, contraction with $\dot{C}$ becomes $\int d^{4}p(2\pi {)}^4{\dot{C}}_{\Lambda }(p){\delta }^2/[\delta \tilde{\varphi }(p)\delta \tilde{\varphi }(-p)]$. Thus the numerator $(2\pi {)}^4$ in the printed flow is consistent with this derivative convention; it must not be changed independently of the convention.

Using functional derivatives of the interaction functional,Dividing the flow by $e^{-S_1}$ therefore givesIn the first term, remove one leg from each of two interaction vertices and join them with a line weighted by ${\dot{C}}_{\Lambda }(p)$. This produces the tree joining of two vertices. In the second, remove two legs from one vertex and contract them with that line, raising the loop order by one, with tadpole diagrams as the simplest example. The factor one half accounts for interchanging the contracted ends; the displayed signs are the signs in the interaction-action flow.

The varying cutoff replaces an internal propagator by its cutoff derivative. Repeated tree joins and loop closures express how eliminated high-momentum fluctuations generate the vertices of the Wilsonian effective action. The action is not restricted to one-particle-irreducible Feynman diagrams: connected tree joins also occur. Field-independent vacuum contributions can again be absorbed into normalization.