Past exam of the particle physics course of the University of Oxford / 2017 / 6

The dominant semileptonic decay is $t\to b{W}^{+}\to b{\ell }^{+}{\nu }_{\ell }$. Replacing $b$ by $s$ or $d$ weights the amplitude by $V_{ts}$ or $V_{td}$ rather than $V_{tb}$. Neglecting the down-type masses,with the source's rounded matrix entries. For the $W$, three leptonic widths and two open up-type quark rows with three colours give nine common width units. Thus, ignoring QCD and final-mass corrections,Summing all three flavours gives approximately one third; summing only electron and muon modes gives approximately $22.2\%$. These are per-top branching ratios, not the separate branching ratio of a specified two-top event topology.

An isolated high-transverse-momentum charged lepton, missing transverse momentum from the neutrino, and a tagged bottom jet form a much cleaner signature than an entirely hadronic final state amid QCD jets. Electrons and muons are the easiest charged-lepton flavours to identify cleanly. Tau decays add neutrinos and often hadrons, making identification and reconstruction more difficult. The figure contains the quark/lepton-level charged-current diagram as well as the rare-loop examples used below.

Top decay occurs before a colour-bound hadron can form. Using representative scales ${\Gamma }_t\sim 1$–$1.5\mathrm{GeV}$ and ${\Lambda }_{\mathrm{QCD}}\sim 0.2\mathrm{GeV}$,Thus top quarks do not live long enough to form ordinary top-containing mesons. The decay products still participate in QCD radiation and hadronization, so “free decay” here means decay before binding, not a stable observable free coloured asymptotic particle.

The weak charged current couples an up-type quark to the rotated down field $d_{iL}^{\prime }={\sum }_j{V}_{ij}{d}_{jL}$. A neutral current is flavour universal within equal-charge quarks. Rotating it into the mass basis therefore givesThe off-diagonal terms vanish by CKM unitarity, so no tree-level flavour-changing neutral current is generated. In the two-generation description, the orthogonal strange-charm partner completes the rotation whose single down-strange combination alone would have produced an off-diagonal neutral coupling. The complete multiplet supplies the cancellation.

At loop level, a flavour-changing amplitude has the structureIf all internal masses were equal, $F$ would be common and ${\sum }_i{V}_{ai}{V}_{bi}^{\ast }=0$. Subtracting an arbitrary common value makes the suppression explicit:This is the Glashow-Iliopoulos-Maiani mechanism. Unequal masses leave small loop-level flavour-changing neutral currents, so the statement that they are never observed would be too strong: the mechanism suppresses them and forbids them at tree level. The cancellation requires complex CKM products; the rounded table of absolute values alone must not be summed as if every phase were positive.

The literal $t\to qZ\gamma$ final state has a three-vertex loop using the Standard Model quartic vertex $W^{+}{W}^{-}Z\gamma$. The two other vertices are $t\to d_i{W}^{+}$ and $d_i{W}^{+}\to q$, with $q=u,c$. The internal down-type quark joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the photon. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.

The companion single-$Z$ penguin diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal quark line and has three cubic vertices. These are different final states. A complete gauge-invariant amplitude requires all companion attachments; for $qZ\gamma$ these include photon radiation from charged lines as well as the quartic graph.

Both flavour sums have the formwhere the loop function also depends on the final-state kinematics. After cancellation of a mass-independent part by GIM mechanism, the bottom-quark mass gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}{V}_{cb}|\simeq 0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final charm. Schematically,The same flavour preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative flavour preference, not an absolute branching ratio. Loop factors and mass-difference suppression remain, so these channels are fully compatible with GIM. A resolved extra photon also introduces another electromagnetic coupling and three-body phase space; an experimental $Z\gamma$ definition needs photon-energy and isolation cuts to distinguish hard radiation from the soft limit.

For a $t\to cZ$ search, the cleanest neutral-boson reconstruction is $Z\to e^{+}{e}^{-}$ or ${\mu }^{+}{\mu }^{-}$: require an isolated opposite-sign same-flavour pair near the $Z$ mass and a jet forming the top mass with that pair. In top-pair events, tagging the other top through its ordinary $bW$ decay provides a bottom jet and, if its $W$ is leptonic, another isolated lepton and missing transverse momentum. Kinematic reconstruction and heavy-flavour information help reject vector-boson-plus-jets and other backgrounds. Hadronic or invisible $Z$ decays have higher rate but less distinctive backgrounds or incomplete mass reconstruction.

For the literal $t\to cZ\gamma$ channel, additionally require a prompt isolated photon and reconstruct the top with $Z+\gamma +$jet. For the alternative $t\to c\gamma$, use an isolated energetic photon and jet near the top mass, again exploiting the other top as a tag. These are search signatures; no present-day observation claim is needed to interpret the historical question.