Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 1

The superficial degree of divergence of this scalar electrodynamics three-point vertex is one. Its ultraviolet pole is consequently a local expression at most linear in the external four-momenta. The allowed counterterm Lagrangian, with the charge-conjugation and Hermiticity properties of scalar electrodynamics, contains a single one-derivative scalar electrodynamics three-point vertex:Thus the coefficient ${\delta }_1$ is independent of the chosen external four-momenta. Setting $k^{\prime }=0$ leaves $-ie{\delta }_1{k}^{\mu }$, so a nonzero generic $k$ still determines that coefficient. It does not determine the full finite scattering amplitude at other four-momenta.

Keep $m\ne 0$ and generic off shell $k$ while extracting the ultraviolet pole. These choices avoid spurious infrared divergences; setting all scales to zero would instead produce scaleless integrals that can conceal the ultraviolet pole.

TODO (source gap): Figure 1(a), including its internal connections and four-momentum labels, is absent from the recoverable source. The local PDF renderer reports missing image objects, and the matching official PDF's figure could not be retrieved for visual inspection. Therefore the individual Feynman diagram called (a) cannot be assigned an integrand and its separate independence of $\xi$ cannot be verified from the available diagram. A guessed replacement drawing would not resolve this gap.

An independent determination of the complete ultraviolet pole, given in part (iv), remains possible from the scalar QED Ward identity. Together with the textually supplied integral in part (iv), it fixes the sum of the contributions outside that integral to $3i{e}^3{C}_{ϵ}{k}^{\mu }$ in the stated scalar electrodynamics three-point vertex convention. That sum alone does not identify the contents of panel (a).

TODO (source gap): Figure 1(c) and its four-momentum labels are unreadable. The assertion about this individual Feynman diagram therefore cannot be checked against its actual integrand. In particular, one must not identify the missing panel with a convenient vanishing seagull vertex graph by inference.

A useful criterion, once the drawing is recovered, is that dimensional regularization annihilates a vector integral of the formby the change of variables $l\mapsto -l$. This would prove vanishing for every $\xi$ if all remaining factors were functions of $l^2$; a denominator involving $l+k$ must first be checked rather than discarded. Whether this criterion applies to panel (c) is precisely the unavailable information.

The integral printed in the text can be evaluated without reconstructing Figure 1. Put $A=l^2-m^2+i0$ and $B=(l+k{)}^2-m^2+i0$, and let $I^{\mu }$ denote that integral without its factor $\xi$. The numerator contraction responsible for its covariant gauge dependence isWith the scalar electrodynamics three-point vertex $-ie(p+p^{\prime }{)}^{\mu }$, three such vertices, two scalar propagators and one photon propagator have overall factor $(-ie{)}^3{i}^2(-i)=-e^3$. Thus the supplied integrand gives $V_{\mathrm{supplied}}^{\mu }=-e^3\xi I^{\mu }$.

First isolate the numerator term $l^2$. Its contribution isUnder $l\mapsto -l-k$ the two denominators exchange and the numerator changes sign. A shift-invariant dimensional regularization therefore gives $\boxed{I_0^{\mu }=0}$, including the finite part.

For the remaining numerator $2(l\cdot k)(2l+k{)}^{\mu }$, only its term $4l^{\mu }(l\cdot k)$ contributes to the ultraviolet pole. Corrections from expanding the denominators, and the numerator $2k^{\mu }(l\cdot k)$, are ultraviolet finite by power counting. Use auxiliary-mass extraction of ultraviolet poles with any $M>0$ rather than literally integrating a massless scaleless integral. Lorentz covariance then givesTODO (source gap): The algebra of the supplied expression is complete, but its graphical identification with Figure 1(b), including the routing at each internal vertex, cannot be visually verified from the unreadable panel.

The complete renormalization constant can nevertheless be obtained independently. Here is a short scalar QED wave-function pole calculation, to avoid assuming either missing panel's value. For an external four-momentum $p$, the exchange contribution to the scalar self-energy has numeratorIts denominator is $l^2[(p-l{)}^2-m^2+i0]$. The necessary ultraviolet poles, with the measure $d^{d}l/(2\pi {)}^d$, are respectivelyThe first two follow by a Feynman parameter and a shift of $l$; the third is a tadpole integral; the last follows by the same large-$l$ extraction used above. The seagull vertex contributes only a photon tadpole integral, which is a scaleless integral, and in any case cannot affect the coefficient of $p^2$. Multiplication by the exchange graph's factor $-e^2$ gives the scalar self-energy insertionThe kinetic counterterm $i{\delta }_2{p}^2$ therefore requires ${\delta }_2=e^2(3-\xi )C_{ϵ}$. The scalar QED Ward identity sets ${\delta }_1={\delta }_2$, soIndeed the full unrenormalized scalar electrodynamics three-point vertex has ultraviolet pole $i{e}^3(3-\xi )C_{ϵ}{k}^{\mu }$, which is cancelled by $-ie{\delta }_1{k}^{\mu }$. Subtracting the supplied contribution gives the gauge-independent sum stated in part (ii), without assigning either unreadable panel an integrand.

The scalar QED Ward identity equates the vertex renormalization and wave-function renormalization constants. Consequently, using part (iv),The renormalization constants describe gauge-dependent fields and off shell vertices, so their dependence on $\xi$ is permitted. A physical scattering amplitude includes all Feynman diagrams, the counterterms, and the external-state factors from LSZ reduction. The Ward identities ensure that the dependence on $\xi$ cancels for physical external states. Infrared-sensitive probabilities also require the usual inclusive treatment of unobserved soft photons. Thus no dependence on $\xi$ is implied for consistently calculated physical observables.