Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 47 / 4

Use the weak hypercharge normalization $Q=T_3+Y$. Write $U(x)\in SU(2{)}_L$ and the hypercharge transformation as $e^{i\beta (x)Y}$. A left-handed doublet transforms as ${\psi }_L\mapsto e^{i\beta Y_L}U{\psi }_L$, while a right-handed singlet transforms as ${\psi }_R\mapsto e^{i\beta Y_R}{\psi }_R$. The Higgs doublet transforms as $\varphi \mapsto e^{i\beta /2}U\varphi$.

The representations per fermion generation are:

There is no right-handed neutrino in the minimal Standard Model. If such a sterile singlet were added, its hypercharge would be zero. The two hypercharge columns give equivalent normalizations, not different physical assignments. This is the electroweak representation and hypercharge table.

Yukawa interactions use $\varphi$ for down-type quarks and charged leptons, but $\tilde{\varphi }=i{\sigma }^2{\varphi }^{\ast }$ for up-type quarks. The conjugate doublet transforms as $e^{-i\beta /2}U\tilde{\varphi }$, using the pseudoreality of the $SU(2)$ fundamental representation. For example, the hypercharge sums in the down, charged-lepton, and up terms areThe doublet indices contract between ${\bar{\psi }}_L$ and the scalar, and colour indices contract in the quark terms. This verifies gauge invariance.

The indices $i,j=1,2,3$ label fermion generations, specifying which left-handed family and right-handed family are coupled. They are not weak-isospin indices. Each fermion type has its own complex Yukawa matrix. After symmetry breaking these matrices determine masses; the mismatch of the two quark left-handed mass rotations produces the CKM matrix. Left fields are weak doublets, right fields are weak singlets, and the scalar is a hypercharge-$1/2$ doublet in the stated convention.

Allowed via a charged-current weak decay. The quark emits a virtual $W^{-}$ in $b\to u{W}^{-\ast }$, followed by $W^{-\ast }\to e^{-}{\bar{\nu }}_e$. The relevant terms are the quark and lepton weak charged currents, with amplitude proportional to the CKM matrix element $V_{ub}$. At low energy their product gives a four-fermion interaction of the form $({\bar{u}}_L{\gamma }^{\mu }{b}_L)({\bar{e}}_L{\gamma }_{\mu }{\nu }_{eL})$ plus its conjugate.

Allowed through second-order weak interactions, not a tree-level neutral-current vertex. This is the flavour-changing transition responsible for neutral kaon mixing, with change of strangeness by two. Box Feynman diagrams contain two W bosons and internal up-type quarks. Their vertices come from the weak charged current and contain CKM matrix factors. At low energy they generate an operator $({\bar{s}}_L{\gamma }_{\mu }{d}_L)({\bar{s}}_L{\gamma }^{\mu }{d}_L)$ plus its conjugate. The GIM mechanism cancels the flavour-independent part of the internal-quark sum; unequal quark masses leave a nonzero loop amplitude.

The coupling is allowed, with an on-shell threshold condition. The Higgs field kinetic term producesand hence a tree-level Feynman vertex. A decay into two real W bosons requires $m_h>2m_W$ for nonzero phase space. Below that threshold the vertex still mediates $h\to W{W}^{\ast }$ or decays through two virtual bosons into fermions. Thus an allowed Lagrangian coupling does not by itself guarantee an on-shell two-body decay.

Absent in the minimal Standard Model. The single charged-lepton Yukawa matrix supplies both the fermion mass matrix and the Higgs coupling, so the same left/right rotations diagonalize both. There is no off-diagonal $h\tau \mu$ vertex. With massless neutrinos, the separate charged-lepton family numbers prevent this transition perturbatively. If neutrino masses and mixing are added, loop-induced charged-lepton flavour violation can occur but is extremely suppressed; a sizeable rate would require an additional source of flavour violation. Part (c) provides one such source.

A larger-than-predicted rate can be described by adding new fields and symmetry-respecting renormalizable interactions, or, when those fields are much heavier than the energy being probed, by an effective field theory. In the latter description, add gauge-invariant operators with Wilson coefficients divided by appropriate powers of a heavy scale. Their interference with existing amplitudes, or their leading contribution to an otherwise forbidden amplitude, can enhance a decay. The operators should respect the Standard Model gauge group and be consistent with other observables; writing an arbitrary flavour-changing term without its electroweak completion is insufficient.

For charged leptons, the proposed operator has mass dimension six: the fermion bilinear contributes three, the scalar doublet one, and ${\varphi }^{†}\varphi$ two. Its hypercharge vanishes by the same calculation as the renormalizable lepton Yukawa interaction, and the extra factor ${\varphi }^{†}\varphi$ is a gauge singlet. Thus its coefficient scales as ${\Lambda }^{-2}$.

In unitary gauge, $\varphi =(0,(v+h)/\sqrt{2}{)}^T$, so the renormalizable and dimension-six terms combine intoThe charged-lepton fermion mass matrix and the single-Higgs Yukawa matrix are thereforeThis is the dimension-six Higgs Yukawa misalignment. Let $U_L^{†}M{U}_R=\mathrm{diag}(m_e,m_{\mu },m_{\tau })$ and ${\hat{\lambda }}^{\prime }=U_L^{†}{\lambda }^{\prime }{U}_R$. In the mass basis,The two matrices need not be aligned. Choosing a nonzero off-diagonal ${\hat{\lambda }}_{\mu \tau }^{\prime }$ or ${\hat{\lambda }}_{\tau \mu }^{\prime }$ produces the required charged-lepton flavour violation while keeping the fermion mass matrix diagonal.

For example, the amplitude for $h\to {\mu }^{-}{\tau }^{+}$ is, up to an overall phase,Neglecting the final lepton masses, summing spins gives $m_h^2(|({\hat{Y}}_h{)}_{\mu \tau }{|}^2+|({\hat{Y}}_h{)}_{\tau \mu }{|}^2)$. The conjugate charge channel has the same rate. The summed width isThis demonstrates an enhancement that is absent in the minimal renormalizable theory. It requires flavour misalignment: an added matrix aligned with the original Yukawa matrix would not generate these decays. The same operator also predicts double-Higgs and triple-Higgs lepton interactions through the remaining powers of $(v+h{)}^3$.