Past exam of the particle and nuclear physics course of the University of Cambridge / 2022 / Section B / 6

The charged pion has spin zero. In its rest frame, the outgoing muon or Positron and the neutrino travel in opposite directions. With a nearly massless left-helicity neutrino, total angular momentum zero requires the positive charged antilepton to have negative helicity. The weak charged current instead creates a massless positive antilepton with positive helicity. A nonzero charged-lepton mass supplies the required opposite-helicity component.

Thus the decay amplitude is proportional to $m_{\ell }$: this is helicity suppression. Explicitly, contracting the pion's axial-current matrix element, proportional to its four-momentum $p_{\pi }^{\mu }$, with the leptonic current and applying the Dirac equation leaves a factor $m_{\ell }$. The leptonic pseudoscalar decay width is, in the $f_{\pi }$ convention,The phase space advantage of the lighter Positron cannot offset its much stronger helicity suppression. The common factors cancel in the pion electron-to-muon decay ratio:This tree-level estimate explains why the muonic branch dominates by roughly four orders of magnitude. The paper's table gives an electronic branching fraction $1.230\times 10^{-4}$; electromagnetic corrections change the simple ratio by a few percent without changing the mechanism.

The original table gives the dimensionless branching fractionsThus the four-lepton channel is about $517$ times as frequent, despite its smaller available phase space.

The dominant neutral-pion interaction is with two photons. In double Dalitz decay, both are virtual and convert to separate Electron-Positron pairs:This requires two electromagnetic conversions but does not force the whole final state into a helicity-suppressed two-fermion configuration. The two virtual photons and four final leptons allow the necessary angular momentum correlations.

For the two-lepton mode, a single virtual photon cannot mediate a $0^{-+}$ pion into $e^{+}{e}^{-}$: the electromagnetic vector current has the wrong quantum numbers. The leading electromagnetic mechanism involves two virtual photons attached to the same lepton line, with a loop integration. The resulting pseudoscalar two-lepton amplitude vanishes as $m_e\to 0$, so its decay width has an additional helicity suppression factor proportional to $m_e^2/m_{\pi }^2$.

Both channels involve extra electromagnetic couplings relative to ${\pi }^0\to \gamma \gamma$. The crucial distinction is the extra leptonic loop and helicity suppression of the two-lepton channel; counting final particles or couplings alone misses it. The pion-two-photon interaction itself already contains strong-interaction structure, so “extra loop” here compares the diagrams after representing that common interaction by an effective ${\pi }^0\gamma \gamma$ vertex.

Both formal processes use one weak charged current transition, with a virtual W boson producing the lepton pair. Two representative lowest-order quark-line diagrams are shown below. They use the $d\bar{d}$ component of ${\pi }^0$; attachment to the antiquark line in the $u\bar{u}$ component gives the other contribution to the same hadronic current.

In the first diagram, $d\to u+W^{-}$ and $W^{-}\to e^{-}{\bar{\nu }}_e$, with $\bar{d}$ a spectator quark. In the second, $u\to d+W^{+}$ and $W^{+}\to e^{+}{\nu }_e$, again with a spectator $\bar{d}$. The diagram for the first process describes the coupling but cannot make an energetically forbidden on-shell decay occur.

From the original tables,Therefore the first process has Q valueand is forbidden for an isolated pion. The second hasThe table therefore lists only the second, pion beta decay, withIts tiny branching fraction combines a weak interaction with an exceptionally small available energy. An allowed semileptonic transition has a decay width scaling approximately as $G_F^2|V_{ud}{|}^2{\Delta }^5$, where $\Delta =m_{{\pi }^{+}}-m_{{\pi }^0}$, with electron-threshold and recoil corrections. The fifth power makes the few-MeV gap especially restrictive. Unlike direct ${\pi }^{+}\to e^{+}{\nu }_e$, this semileptonic channel is not forced to have an amplitude proportional to $m_e$; its primary extra suppression is the small phase space. Its approximately Cabibbo-favoured $u\leftrightarrow d$ current does not supply a further large flavour suppression. The observed branching fraction corresponds to a partial decay rate of about $(1.036\times 10^{-8})/(2.6033\times 10^{-8}\mathrm{s})=0.398{\mathrm{s}}^{-1}$, competing with the much faster dominant muonic decay.

For (i), CP symmetry relates the negative- and positive-pion radiative channels to excellent accuracy. The original positive-pion table therefore predictsThe comparison requires the same photon-energy and event-selection definition. The table attaches an unexplained footnote to this radiative channel; it does not specify those cuts in the supplied excerpt. An arbitrarily soft photon is not a separately infrared-finite exclusive observable, so the numerical value must retain the table's definition.

Part (ii) contains an apparent notation error in the original PDF as well as the TeX: it prints $e^{-}{\nu }_e{e}^{-}{e}^{+}$, rather than $e^{-}{\bar{\nu }}_e{e}^{-}{e}^{+}$. Taken literally, the charged leptons carry net electron-family lepton number $2-1=+1$, and the printed ${\nu }_e$ adds another $+1$. The initial pion has zero lepton number. This is a $\Delta L_e=2$ channel, not the charge conjugate of the listed positive-pion decay. In the ordinary lepton-number-conserving Standard Model approximation,Allowing lepton-number-violating masses for Majorana neutrinos would require additional physical assumptions and does not justify assigning the tabulated number to that channel.

For the likely intended charge-conjugate process, the prediction is insteadThis distinction follows from a lepton-number test of a decay channel; replacing the printed neutrino by an antineutrino must be stated explicitly.

The table gives the Xi baryon mean lifetime ${\tau }_{{\Xi }^0}=2.90\times 10^{-10}\mathrm{s}$. At $t_1=10^{-12}\mathrm{s}$, an initially resting particle has survival probabilityThus the most probable contents are one undecayed ${\Xi }^0$. The initial momentum is unspecified. For a moving parent, time dilation replaces the denominator by ${\gamma }_{\Xi }{\tau }_{{\Xi }^0}$, making the survival probability at least as large. The times below are characteristic proper mean lifetimes; laboratory times depend on the particle velocities.

The principal first step isIt typically occurs on the $10^{-10}\mathrm{s}$ scale. The neutral pion almost immediately converts to two photons, with ${\tau }_{{\pi }^0}=8.43\times 10^{-17}\mathrm{s}$ and $B_{\gamma \gamma }=0.98823$. The Lambda baryon has ${\tau }_{\Lambda }=2.632\times 10^{-10}\mathrm{s}$ and two dominant routes.

In the charged-pion route,The muon lifetime is a standard additional input, not present in the supplied five table pages.^[ref] The successive contents are approximately $\Lambda +2\gamma$, then $p+{\pi }^{-}+2\gamma$, then $p+{\mu }^{-}+{\bar{\nu }}_{\mu }+2\gamma$. After a few muon lifetimes the dominant stable endpoint isThe leading-path probability including the tabulated two-photon pion branch isHere the last factor is the tabulated dominant muonic charged-pion fraction; rare radiative corrections are being neglected in describing a fixed photon count. Before selecting pion decay channels, the probability of reaching the $p{\pi }^{-}{\pi }^0$ history is $0.99524\times 0.639=0.63596$.

In the neutral-pion route,This produces $n+4\gamma$ by roughly the nanosecond scale. It is a long-lived intermediate state, not the final endpoint: the free neutron has ${\tau }_n=879.4\mathrm{s}$ in the original table and eventually undergoes beta-minus decay,Thus after several thousand seconds the dominant stable endpoint isBefore selecting the two ${\pi }^0\to 2\gamma$ decays, the $n{\pi }^0{\pi }^0$ history has probability $0.99524\times 0.358=0.35630$. The familiar rough description is therefore a $64\%$ charged-pion route and a $36\%$ neutron route; including the specified leading pion branches gives approximately $62.8\%$ and $34.8\%$ for the two displayed endpoints. The remainder includes Dalitz decays and other rare channels, not a failure of probability conservation.

These are particle decay cascades, treating the separated final proton, Electron, photons and neutrinos as stable asymptotic particles and neglecting subsequent collisions and rare extra radiation. An infinitely large box prevents escaping particles from being removed from the inventory. Neutrino labels specify their production flavours. Each lifetime describes a stochastic waiting time, so there is no finite sharp instant at which every unstable particle has certainly decayed.