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

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 $\Lambda$ gives the representative integralsCombinatorial 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 $[\mathcal{L}]=4$ and $[{\partial }_{\mu }]=1$. The engineering dimensions of relativistic fields follow from matching the free kinetic terms:and henceFor a local operator $\mathcal{O}$ of mass dimension $d_{\mathcal{O}}$, its coupling has dimension $4-d_{\mathcal{O}}$. 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.

The operator $({\varphi }^{†}\varphi {)}^2$ has mass dimension four, so $[\lambda ]=0$. It is a power-counting renormalizable scalar quartic interaction. It describes scalar self-interactions, including the quartic term in the Higgs potential for the appropriate scalar representation. Its sign in the potential must be chosen for a stable large-field theory; dimension counting alone does not ensure stability.

The operator $({\varphi }^{†}+\varphi )\bar{\psi }\psi$ has mass dimension $1+3=4$, so $[g]=0$. It is a power-counting renormalizable Yukawa interaction between a real scalar combination and a fermion bilinear. A scalar vacuum expectation value contributes to a Dirac mass term; its fluctuations couple to the fermion pair. This is a schematic scalar–fermion theory. In a chiral gauge theory, representations must be combined gauge-invariantly, as in the Standard Model Higgs–fermion interactions. Without additional protecting symmetries, all allowed scalar counterterms must be included as well.

The operator ${\varphi }^{†}\varphi \bar{\psi }\psi$ has mass dimension $2+3=5$, giving $[g]=-1$. It is not power-counting renormalizable as a fundamental interaction treated to arbitrary energy. It is a perfectly usable dimension-five effective operator below its suppression scale. Successive orders in that effective field theory require a derivative and operator expansion rather than one finite set of parameters valid at every order.

The operator $A^{\mu }\bar{\psi }{\gamma }_{\mu }\psi$ has mass dimension $1+3=4$, so $[e]=0$. It is the renormalizable electromagnetic current coupling when completed as quantum electrodynamics: a massless Abelian gauge field couples minimally to the conserved Dirac current, with consistent gauge fixing and Ward identities. Indeed, replacing ${\partial }_{\mu }$ by a gauge covariant derivative in the Dirac equation produces this interaction. The gauge structure, rather than dimension counting alone, ensures the necessary cancellations.^[ref]

The operator $A^{\mu }\bar{\psi }{\partial }_{\mu }\psi$ has mass dimension $1+3+1=5$, so $[e]=-1$. It is not power-counting renormalizable. Hermiticity may require a suitable conjugate or a symmetrized derivative, and electromagnetic gauge invariance would require a further consistent completion; neither repair changes its engineering dimension. Such derivative interactions can occur in an effective field theory, subject to its symmetries and operator redundancies.

Each Dirac current $\bar{\psi }{\gamma }^{\mu }\psi$ has mass dimension three, so their product has dimension six. Its coefficient therefore has dimension $-2$:The Fermi interaction is not power-counting renormalizable, but its success at low energies is explained by effective field theory. Exchange of a heavy W boson between the appropriate chiral weak currents has a propagator that expands as $1/(m_W^2-q^2)=m_W^{-2}[1+q^2/m_W^2+\cdots ]$. The leading term is a local four-fermion interaction and further terms are suppressed by powers of $q^2/m_W^2$. The complete spontaneously broken electroweak gauge theory resolves the pointlike approximation at higher energies. The source's vector-current product illustrates the operator dimension; the actual weak interaction also has its specific flavors and chiral structure.