Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 3 / iv

There is an interpretation issue in this clause. A proper quartic vertex, defined as the fourth derivative of the quantum effective action, has no additional tree or reducible contribution in the bare cubic theory: the three boxes exhaust its one-loop diagrams. The strict local zero-momentum vertex of a Wilsonian effective action also has no high-mode exchange bridge, since the momentum carried by that bridge would be zero, outside the shell. Thus $\boxed{\text{for a local proper or strict Wilsonian quartic vertex, there are no additional diagrams}.}$

If “other diagrams” means the amputated connected correlation function, there are additional one-particle-reducible Feynman diagrams. Specify full external-propagator amputation and a vacuum with tadpole subtraction. The three tree exchange channels each contribute $g^2/m^2$ at zero momentum. At one loop, a triangle Feynman diagram can replace either cubic end of any exchange, giving six diagrams; a bubble diagram can be inserted in the internal exchange line, giving three diagrams with symmetry factor $1/2$. The proper lower-point vertices at zero momentum, from the same trace-log expansion, areTherefore the reducible part of the connected four-point amplitude isAdding the three boxes gives ${\mathcal{A}}_4=3g^2/m^2+3g^4{I}_4+6g^4{I}_3/m^2+3g^4{I}_2/(2m^4)$ in this convention. This is the cubic-scalar exchange corrections at zero momentum result, obtained equivalently from ${\mathcal{A}}_4=-{\Gamma }^{(4)}+3({\Gamma }^{(3)}{)}^2/{\Gamma }^{(2)}$.

If the tadpole has not been subtracted, it is an additional one-loop subgraph that must not be forgotten. The proper one-point function is $t=g{I}_1/2$. Its attachment to the internal exchange line gives another three reducible diagrams, contributing $3g^4{I}_1/(2m^6)$; equivalently the stationary background shifts by $v=-g{I}_1/(2m^2)$ and changes the exchanged mass by $gv$. A linear counterterm enforcing $v=0$ cancels this contribution. External self-energy or tadpole diagram attachments are removed by the stated full-propagator amputation; using bare external amputation would retain them instead. In that convention the four external-line decorations add $6g^4{I}_2/m^4+6g^4{I}_1/m^6$ to the displayed full-amputated result, with the $I_1$ part absent after tadpole subtraction. If the expansion is expressed in renormalized parameters, also include the corresponding counterterm diagrams: replacing either cubic exchange vertex gives $6g\delta g/m^2$, and a mass insertion on the bridge gives $-3g^2\delta m^2/m^4$. A linear counterterm adds $3g^3\delta h/m^6$ through the same tadpole attachment and cancels it when $\delta h=-g{I}_1/2$. The kinetic counterterm has zero insertion at zero exchanged momentum. At one loop these exhaust the possibilities: the single cycle has length four, three, two or one, corresponding respectively to box, triangle, bubble or tadpole, with tree branches attached.

The mass dimensions are $[g]=0$, $[I_r]=6-2r$, $[m]=1$. Accordingly $g^2/m^2$, $g^4{I}_4$, $g^4{I}_3/m^2$, $g^4{I}_2/m^4$ and $g^4{I}_1/m^6$ all have dimension $-2$. These connected exchanges are generally nonlocal functions of external momentum before taking the zero-momentum value; they should not be silently reclassified as an independent local quartic coupling.