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Past exam of the gauge field theory course of the University of Cambridge / 2013 / Paper 1 / 3 / Solution by

Codex  0 2026-10-20

Interactions produce Feynman diagrams with internal lines whose independent loop momenta are integrated without an upper bound. Their large-momentum contributions can have an ultraviolet divergence. Two explicit examples in a scalar quartic theory are a one-loop tadpole diagram contributing to the two-point function and a one-loop four-point bubble diagram. After a Wick rotation, a spherical cutoff

Λ

gives the representative integrals

I_{1} = \int_{∣ k_{E} ∣ < Λ} \frac{d ^{4} k _{E}}{( 2 π ) ^{4}} \frac{1}{k _{E}^{2} + m ^{2}} = \frac{1}{16 π ^{2}} [Λ^{2} - m^{2} ln (1 + \frac{Λ ^{2}}{m ^{2}})],

(1)

I_{2} (0) = \int_{∣ k_{E} ∣ < Λ} \frac{d ^{4} k _{E}}{( 2 π ) ^{4}} \frac{1}{( k _{E}^{2} + m ^{2} ) ^{2}} = \frac{1}{16 π ^{2}} [ln (1 + \frac{Λ ^{2}}{m ^{2}}) + \frac{m ^{2}}{Λ ^{2} + m ^{2}} - 1] .

(2)

Combinatorial factors and couplings multiply these integrals in a particular diagram. The first is quadratically divergent by cutoff power counting; the second is logarithmically divergent. Other schemes package the divergent pieces differently.

In renormalization, introduce a regularization, separate bare parameters into renormalized parameters and counterterms, and fix the latter by stated normalization conditions. Local divergent pieces of the scalar two- and four-point functions are absorbed into mass, field normalization and quartic-coupling counterterms, with all other operators allowed by the chosen symmetries included as needed. A perturbatively renormalizable quantum field theory requires this finite counterterm family to remain closed at all orders. It does not require every individual bare diagram to be finite. For gauge theories, identities expressing gauge symmetry and absence of a gauge anomaly are essential in addition to superficial counting.

The action is dimensionless in natural units, so in four spacetime dimensions

[L] = 4

and

[\partial_{μ}] = 1

. The engineering dimensions of relativistic fields follow from matching the free kinetic terms:

2 [ϕ] + 2 = 4, 2 [ψ] + 1 = 4, 2 [A_{μ}] + 2 = 4,

(3)

and hence

[ϕ] = 1, [ψ] = \frac{3}{2}, [A_{μ}] = 1.

(4)

For a local operator

O

of mass dimension

d_{O}

, its coupling has dimension

4 - d_{O}

. Operators with dimension at most four are candidates for ordinary perturbative power-counting renormalizability. This test presupposes consistent kinetic terms, symmetries and the full set of induced counterterms; it is not a proof of physical consistency for an arbitrary vector interaction.

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