Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 (Advanced Quantum Field Theory)

A single quintic vertex has five half-edges, so Wick theorem cannot contract all of them in pairs. The leading connected Vacuum Feynman diagrams therefore have two quintic vertices and are of order $g^2$. If $c$ propagators join the two vertices, each vertex has $5-c$ half-edges left for tadpoles, so $c$ must be odd. This gives exactly three topologies:

The three diagrams are connected; every other pairing is either isomorphic to one of them or disconnected.

For $n=2$, the coupling is a mass squared, $m^2=g$. After integration by parts, the quadratic action has kernel $K=-(\square +g-i0)$. Completing the square in the Gaussian integral and choosing the source-independent normalization so that $Z_2[0]=1$ givesEquivalently, in momentum space,The $i0$ prescription selects the Feynman propagator.

Inside the generating functional, multiplication by a field can be replaced by a functional derivative of the source factor:Expanding the interaction exponential, making this replacement in every term, and resumming givesThis formal identity assumes a common regulator, a source-independent normalization, and permission to interchange the path integral, power series, and functional derivatives. A normalized functional with $Z_n[0]=1$ requires division by the same expression evaluated at $J=0$, which removes connected Vacuum Feynman diagrams.

For $n=2$, the operator formula becomesTwo source derivatives of the free Gaussian produce a $J$-independent coincident propagator and a source-dependent insertion between two free propagators. The former is a vacuum bubble and disappears when $Z_2[0]=1$. ThusExpanding the exact denominator from part (b),gives the same source-dependent correction.

The first three terms are the renormalized kinetic, mass, and quartic interaction terms. The remaining three are counterterms: $\delta Z$ gives wave-function renormalization, $\delta m^2$ renormalizes the mass, and $\delta \lambda$ renormalizes the four-point coupling. With all momenta entering a vertex, the momentum-space Feynman rules aretogether with momentum conservation at every vertex and an integral $\int d^{d}p/(2\pi {)}^d$ for each independent loop momentum.

The amputated connected four-point function through one loop contains the tree quartic vertex, the quartic-counterterm vertex, and three bubble diagrams in the $s$, $t$, and $u$ channels. The unamputated connected function also has one-loop self-energy and two-point-counterterm insertions on each external propagator, which do not determine $\delta g$.

For channel momentum $P$, the bubble has symmetry factor $1/2$ and the Feynman parameter identity gives an integral proportional toIn dimensional regularization, $\Gamma (ϵ/2)=2/ϵ+O(1)$, so the pole from one channel is $g^2/(16{\pi }^2ϵ)$. Summing the three crossing channels gives $3g^2/(16{\pi }^2ϵ)$. The minimal subtraction scheme cancels only this pole, hence

The bare coupling does not depend on the renormalization scale. Expressing it through the renormalized field and coupling gives, to the stated order,At one loop $\delta Z=0$. Put $a=3/(16{\pi }^2)$ and $\delta g=a{g}^2/ϵ$. If ${\beta }_g=\mu dg/d\mu$, differentiation and expansion through $g^2$ yieldSince $\lambda ={\mu }^{ϵ}g$,and in four dimensions

The inverse Fourier transform is linear:Splitting the integration domain into $p^2<b^2{\Lambda }^2$ and $b^2{\Lambda }^2<p^2<{\Lambda }^2$, then inserting the definitions of $\tilde{\varphi }$ and $\tilde{\chi }$, gives

The disjoint Fourier supports make the quadratic cross term $\int \partial \varphi \cdot \partial \chi$ vanish. ThereforewhereDefine the high-mode free generating functional with source conventionThen inserting $\chi =-\delta /\delta J$ reproduces every high-field factor, andExpanding the interaction exponential and applying Wick theorem evaluates the Wilsonian effective action as a sum of connected diagrams whose internal lines are restricted to the high-momentum shell.

The original cubic vertex gives $g$. At order $g^3$, the one-particle-irreducible correction to the cubic interaction is the triangle diagram: one slow external leg leaves each of three cubic vertices, and three high-mode propagators join the vertices cyclically. Diagrams with a high-momentum bridge carrying only a sum of vanishing external momenta do not contribute to the local zero-momentum coupling; external self-energy diagrams belong to the neglected field rescaling.

The relevant interaction at each vertex is $(g/2)\varphi {\chi }^2$. The third connected cumulant contributesWick's theorem gives $8G_{>}(x-y)G_{>}(y-z)G_{>}(z-x)$. Projecting onto a local interaction therefore givesFor $d=6$, the area of the unit five-sphere is ${\Omega }_5={\pi }^3$, and henceThus

DefineThe gauge-field transformation is precisely the one for which the gauge covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ transforms by $D_{\mu }^{\prime }=U{D}_{\mu }{U}^{†}$. Since $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$, it follows immediately thatFor $U=1+i{\alpha }^c{T}_c+O({\alpha }^2)$,Define the structure constant of a Lie algebra by $[T_b,T_c]=i{f}_{bc}{}^a{T}_a$. Antisymmetry then gives

Writing the source interaction as $\mathrm{Tr}(J^{\mu }{A}_{\mu })$, gauge invariance requires the matrix current to transform in the Adjoint representation,The fermion bilinear with this transformation law isup to a convention-dependent overall coupling or generator normalization. It is the Noether current that appears when the free derivative is replaced by $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$.

Write $\Psi ={\Psi }^a{T}_a$. The infinitesimal form of $\Psi \mapsto U\Psi U^{†}$ isThe adjoint gauge covariant derivative iswhich transforms as $D_{\mu }\Psi \mapsto U(D_{\mu }\Psi )U^{†}$. A gauge-invariant Lagrangian is thereforeThe trace and cyclicity make each term invariant under conjugation.

The gluon two-point function through one loop contains the tree propagator and these one-particle-irreducible insertions: a gluon loop with two three-gluon vertices, a gluon tadpole with one four-gluon vertex, a Faddeev-Popov ghost field loop, a fermion loop, and the gluon two-point counterterm. Gauge fixing is required before these Feynman diagrams and the propagator are defined.

Only the fermion loop changes when the fermion representation changes. Its two gauge vertices contain the representation matrices $T_R^a$, and its color factor isWith the conventional normalization, $T(F)=1/2$ for the fundamental representation and $T(\mathrm{adj})=C_A=N$ for the adjoint representation of $SU(N)$. The momentum and spinor integral is otherwise the same, apart from the number and type of fermion species; the pure-gluon and ghost diagrams are unchanged.