Past exam of the gauge field theory course of the University of Cambridge / 2012 / Paper 1 / 3

A bare Dirac mass couples left and right chirality, so the distinct electroweak representations prevent writing it as a gauge-invariant term before symmetry breaking. Instead the Yukawa interactions areWeak indices are contracted to singlets. Substituting the vacuum value gives mass matrices $M_f=Y_{f}v/\sqrt{2}$ and a physical Higgs coupling $-{\bar{f}}_L{M}_f{f}_R\sigma /v$ plus its conjugate. Biunitary diagonalization defines mass eigenstates; a mismatch between the up- and down-quark left rotations appears in the CKM matrix. Gauge-boson masses arise from the Higgs kinetic term, whereas these fermion masses arise from the Yukawa term.

With only the minimal field content there is no right-handed neutrino, so no renormalizable neutrino Yukawa term. Adding a gauge-singlet $N_R$ permits both a Dirac scale $m_D=y_{\nu }v/\sqrt{2}$ and a Majorana scale $M_R$. For one family the symmetric mass matrix in a left-chiral basis isIts signed eigenvalues are $(M_R\pm \sqrt{M_R^2+4m_D^2})/2$. Rephasing gives positive physical masses, and for $|M_R|\gg |m_D|$ the seesaw mechanism yieldsFor $m_D\sim 100\mathrm{GeV}$ and $M_R\sim 10^{14}\mathrm{GeV}$, the light scale is about $0.1\mathrm{eV}$. Integrating out the heavy state generates the dimension-five Weinberg operator, whose Higgs vacuum insertion gives the same suppression. Alternatively a Dirac neutrino can have a very small Yukawa coupling with lepton number conserved. A tiny observed neutrino mass does not by itself distinguish these mechanisms.

In units $\hslash =c=1$, a dimensionless action fixes $[\mathcal{L}]=4$. Canonical kinetic terms then give $[\varphi ]=[A_{\mu }]=1$, $[\psi ]=3/2$ and $[{\partial }_{\mu }]=1$. If an interaction operator has dimension $d_v$, its coefficient has mass dimension $[g_v]=4-d_v$. Thus scalar quartic, Yukawa and gauge couplings are dimensionless; a scalar cubic coupling has dimension one, a four-fermion coupling has dimension minus two, and the Weinberg coefficient has dimension minus one. These are engineering dimensions; anomalous scaling is an additional quantum effect.

For a connected graph with $L$ independent loop momenta, $I_B$ boson lines, $I_F$ fermion lines and $r_v$ derivatives at vertex $v$, ordinary covariant-gauge ultraviolet counting gives the superficial degree of divergenceThe second form follows from $L=I_B+I_F-V+1$ and counting field ends at vertices. It presumes the canonical large-momentum propagators; unitary-gauge vector propagators can obscure the cancellations of a renormalizable gauge theory. A nonnegative $D$ signals a possible overall ultraviolet divergence, not proof of one. Symmetries can cancel it, while a graph with $D<0$ can still contain divergent subgraphs.

With only operators of dimension at most four, increasing the vertex count does not require counterterms of unbounded dimension. Subject to the necessary symmetry identities and absence of gauge anomalies, scalar quartic theory, Yukawa theories and the Standard Model are perturbatively renormalizable when all required interactions are included. In particular, coupling a scalar to other fields can require a scalar quartic counterterm even if its coefficient was initially set to zero. The Higgs mechanism does not destroy renormalizability of the underlying anomaly-free gauge theory.

By contrast, repeated insertions of the dimension-six Fermi interaction or a dimension-five operator allow increasingly high-dimensional counterterms. Such terms are useful in an effective field theory with a stated expansion in energy over a high scale; nonrenormalizable does not mean unpredictive at a fixed order. A scalar ${\varphi }^6$ interaction and perturbative Einstein gravity in four dimensions provide other examples. Renormalizability also does not assert small coupling at every scale or a complete ultraviolet theory.

The SU(5) grand unified theory embeds the strong and electroweak groups in a single simple gauge group. Its 24 gauge generators contain the eight gluons, three weak generators and one hypercharge generator, plus twelve additional generators connecting color and weak directions. A common gauge coupling applies at the unification scale with the properly normalized hypercharge generator; low-energy couplings differ through their renormalization group evolution.

Using left-chiral fields throughout, one Standard Model generation fits intoThe dimensions are $3+2=5$ and $6+3+1=10$, so no right-handed neutrino is required in these multiplets. A singlet neutrino can be added. A suitable adjoint scalar vacuum breaks SU(5) to the Standard Model subgroup; a fundamental scalar contains an electroweak doublet and a colored triplet. The extra heavy vector bosons can mediate transitions between quarks and leptons, hence proton decay in the minimal model. This is a proposed unification, not an experimentally established identification of new particles.

The hierarchy problem concerns keeping an elementary weak-scale scalar light in the presence of much heavier physical scales. A heavy state coupled to it generically induces a threshold correction of order $\delta m_H^2\sim c{M}^2/(16{\pi }^2)$, with $c$ a product of couplings. Without protection, a weak-scale result then needs cancellation against a much larger parameter. In SU(5), this is sharpened by doublet-triplet splitting: the electroweak doublet must stay light while its colored partner must be sufficiently heavy to avoid rapid proton decay. Choosing a small tree-level doublet mass is not by itself a mechanism protecting it from subsequent heavy-particle corrections. Supersymmetry and other proposals change this protection problem, but the minimal SU(5) embedding alone does not solve it.