Past exam of the quantum field theory course of the University of Oxford / 2021 / 3 / c

For the two-point function, the original Feynman diagrams below show both one-vertex graphs and every two-vertex topology with external momentum flowing between distinct interaction vertices. All lines represent the same real scalar field. The lower row labels the number of parallel connecting lines; a small closed line at a vertex is a tadpole diagram.

With a single quartic interaction vertex, the two unused line ends close into one tadpole diagram. With a single sextic interaction vertex, the four unused ends pair into two tadpole diagrams. These are the only one-vertex one-particle irreducible Feynman diagrams for the two-point function.

Define the Minkowski Feynman integralsThe amputated two-point function contributions arewhere $C_4,C_6$ are the reciprocals of the corresponding Feynman-diagram symmetry factors. Both are independent of external momentum, and hence affect the mass counterterm rather than wave-function renormalization. No numerical Feynman-diagram symmetry factors are needed here.

To include all momentum-dependent contributions to the two-point function, put one external line at each interaction vertex. Let $r$ be the number of lines connecting the vertices, and $t_A,t_B$ their numbers of tadpole diagrams. For vertex valences $v_A,v_B\in \{4,6\}$,The integer $r$ is odd, and $r=1$ gives a one-particle-reducible Feynman diagram and is excluded. Thus the complete list is

The coefficient of the mixed graph includes the reflected external attachment as well as its reciprocal Feynman-diagram symmetry factor. If both external lines attach to the same interaction vertex, no external momentum enters the connecting internal lines, and that Feynman diagram cannot contribute to wave-function renormalization.

For incoming external momentum $p$, assign the connecting lines momenta $k_1,\dots ,k_{r-1},p-{\sum }_j{k}_j$. DefineThe momentum space Feynman rules give, in the order listed,Each $C$ is a reciprocal Feynman-diagram symmetry factor, including any equivalent external attachments. There are respectively $2,3,4,4$ independent loop momenta.

If “contribute to wavefunction renormalization” means requiring a divergent kinetic counterterm after subtraction of all subdivergences, only the last topology qualifies. The first three still have nontrivial external momentum dependence and give finite contributions to a normalization condition; their raw divergent factors arise from mass counterterms and interaction vertex counterterms. Keeping all four here makes this distinction explicit. The ultraviolet divergence analysis in d establishes it directly.