Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2

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.

The free energy is dimensionless in units with $k_{B}T=1$, so the integrand has momentum dimension $d$. Since a derivative has dimension one, the kinetic term givesThe mass term then givesThese are engineering dimensions.

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

The operator ${\nabla }^{2m}{\varphi }^{2n}$ has engineering dimensionThe action integral is dimensionless, soThus $\alpha$ is a relevant coupling, marginal coupling, or irrelevant coupling according asrespectively.

At order $g_0^2$, use the two-point sunset diagram: two quartic vertices are joined by three internal propagators, with one external line attached to each vertex. Unlike the one-vertex tadpole diagram, its self-energy $\Sigma (p)$ depends nontrivially on the external momentum $p$. The coefficient of $p^2$ in the expansion of $\Sigma (p)$ changes the kinetic term, so normalizing that term requires wave-function renormalization and gives a nonzero field anomalous dimension.

No. With only ${\lambda }_0{\varphi }^6$, the free energy has an exact Z2 symmetry $\varphi \mapsto -\varphi$. Integrating out fast modes and rescaling preserve that symmetry, whereas ${\varphi }^5$ is odd. Therefore no five-point vertex and no correction to $\gamma (\zeta )$ can be generated at any order in ${\lambda }_0$.

Choose four of the six fields at one sextic vertex to be slow and contract the remaining two fast fields into a tadpole diagram. There are $\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15$ choices. Ifthen the first term of the cumulant expansion contributesAfter the canonical coordinate and field rescaling, its contribution to the quartic coupling is

In the displayed connected Feynman diagram, each sextic vertex carries two slow external legs and the four remaining legs at each vertex are paired across the vertices. The second cumulant expansion has a factor $-1/2$, the slow legs can be selected in ${\left(\genfrac{}{}{0ex}{}{6}{2}\right)}^2$ ways, and the four cross-contractions can be paired in $4!$ ways. The coefficient is thereforeWriting ${\mathcal{S}}_{\zeta }=\{q:\Lambda /\zeta <|q|<\Lambda \}$, the local zero-external-momentum contribution iswith the integral restricted further so that $q_1+q_2+q_3\in {\mathcal{S}}_{\zeta }$. The three independent internal momenta agree with the diagram's loop order $L=4-2+1=3$.

Classify the connected contractions by the numbers $(r,4-r)$ of external slow legs on the two sextic vertices and by the number $c$ of fast propagators joining them. Besides the displayed $(2,2;c=4)$ graph, the distinct topologies are:

Exchanging the two vertices gives no new topology. All four are connected Feynman diagrams selected by the logarithm in the cumulant expansion. With an ideal sharp momentum shell and a projection at exactly zero external momentum, the single joining line in the last topology cannot carry shell momentum, so that topology gives zero to the local quartic coupling; it is still the remaining formal connected contraction.