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Two mechanisms are a tau lepton pair with two leptonic decays, and a W boson pair with two leptonic decays. With equal beam energies, the laboratory is the centre-of-mass frame, and four-momentum conservation gives $E_{\mathrm{miss}}=\sqrt{s}-E_{e^{-}}-E_{{\mu }^{+}}$ in natural units.

For the first mechanism,There are four escaping neutrinos. Each tau lepton decay proceeds through a virtual W boson. The fixed-charge final state has branching fraction $B_{\mu }{B}_e$; the factor two for the inclusive $e^{\pm }{\mu }^{\mp }$ sample does not apply when the charges are specified.

For the second mechanism,There are two escaping neutrinos. The leading Feynman diagrams comprise both $s$-channel photon/Z boson exchange and $t$-channel electron neutrino exchange. Their scattering amplitudes interfere; the complete set is needed for a consistent high-energy prediction. In particular, the neutrino-exchange diagram alone does not define an independently observable production rate.

A tau lepton pair has enhanced production at the Z boson particle resonance, $\sqrt{s}\simeq 91\mathrm{GeV}$. The four-neutrino missing energy has a broad continuous distribution. To derive a useful schematic, neglect daughter masses and parent spin correlations. In an unpolarized tau lepton rest frame, the Michel spectrum for $x=2E_{\ell }^{\ast }/m_{\tau }$ isA Lorentz boost gives $E_{\ell }/E_{\tau }=x(1+{\beta }_{\tau }u)/2$, with $u=\cos {\theta }^{\ast }$ uniform on $[-1,1]$ in this approximation. For ${\beta }_{\tau }\simeq 1$, the charged-daughter probability density function in $z=E_{\ell }/E_{\tau }$ isFor two independent decays, $y=E_{\mathrm{miss}}/\sqrt{s}=1-(z_1+z_2)/2$ has the convolutionand $⟨y⟩=0.65$. It vanishes at both endpoints and peaks on the larger-missing-energy side. This is a shape illustration, not the exact polarized Z boson prediction; spin correlations, acceptance, and radiative effects alter it.

The W boson process turns on near $\sqrt{s}=2m_W\simeq 161\mathrm{GeV}$. At exact on-shell production threshold, each neutrino has energy $m_W/2$, soThe cross section vanishes at the ideal two-body threshold, so this narrow limiting spectrum must not be called the maximum-rate spectrum. The large W boson-pair rate occurs above threshold, around $200\mathrm{GeV}$. There ${\beta }_W=\sqrt{1-4m_W^2/s}$ and each neutrino hasThusAn isotropic independent-decay schematic is triangular, centered at $E_{\mathrm{miss}}/\sqrt{s}=1/2$, with endpoints approximately $0.20$ and $0.80$ at $200\mathrm{GeV}$. Actual W boson particle polarizations and their correlations change this shape; their V-A interaction is not isotropic for a fixed helicity. The finite decay width, beam spread, and radiation also smear the edges.

The phrase “maximum rate” also needs a production regime: a Z boson peak is a resonant maximum, whereas pure photon-mediated tau lepton production has its continuum maximum near $4.2\mathrm{GeV}$. Indeed, ${\sigma }_{\gamma }\propto {\beta }_{\tau }(3-{\beta }_{\tau }^2)/s$ has $s=(1+\sqrt{21})m_{\tau }^2$. The displayed high-boost curve describes the resonant regime; at the continuum maximum use ${\beta }_{\tau }\simeq 0.532$ in the boost formula. An observed event-rate maximum additionally depends on collider luminosity and detection efficiency.

The ideal two-shower event fraction is $B_e^2$, approximately $4\%$ in simple channel counting or $3.18\%$ using the measured tau lepton electronic branching fraction. The selected particle decay chain isThe electron and Positron each make a contained electromagnetic shower; muons penetrate, and generic hadrons do not deposit all their energy as a contained electromagnetic shower.

To estimate $B_e$, use lepton universality and neglect final masses and Quantum chromodynamics corrections. The electronic and muonic decay widths each carry weight one. The accessible inclusive hadronic channels have weights $3|V_{ud}{|}^2$ and $3|V_{us}{|}^2$, where three is the color multiplicity in a decay width. The charm quark is inaccessible in physical tau lepton hadronic decays. With $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$, the total weight is $1+1+3=5$, givingThe 2021 measured value $B_e=0.1782$ instead gives $B_e^2=0.0317552$.^[ref] The difference reflects effects omitted by the simple free-channel estimate, especially the hadronic contribution to the total decay width.

There are two tau leptons per produced pair and two per selected pair, so the fraction of all produced tau leptons belonging to these two-shower events is also $B_e^2$. The unrestricted fraction of individual tau leptons decaying electronically is $B_e$; these are different selections. A real high-energy shower cut gives an additional acceptance factor: $N_{\mathrm{selected}}/N_{\mathrm{pairs}}=B_e^2{\epsilon }_{ee}$, which cannot be computed without the cut and detector response. No factor of two is needed for the unique $e^{+}{e}^{-}$ assignment.

For a purely electromagnetic interaction continuum with five active quark flavors, the usual estimate is $N_{e^{+}{e}^{-}}/N_{\mathrm{had}}=3/11$. The unspecified particle resonance and the word “total” require qualifications before this can be treated as a unique physical prediction.

In natural units, neglecting final masses, single-photon annihilation givesHere $\alpha$ is the fine-structure constant, $Q_f$ the quark electric charge in elementary-charge units, and the factor three counts orthogonal colour final states. Consequently the inverse hadronic R ratio isWith $u,d,s,c,b$ accessible, the denominator is $3[2(2/3{)}^2+3(1/3{)}^2]=11/3$. The corresponding values for $u,d,s$ and $u,d,s,c$ are $1/2$ and $3/10$. Inclusive Quantum chromodynamics corrections increase the hadronic denominator, approximately by $1+{\alpha }_s/\pi$ away from thresholds and particle resonances. These are annihilation ratios before selection and detection efficiency corrections, not ratios of uncorrected detector counts.

A future muon-collider Higgs resonance is a plausible reading of part (c), since its narrow decay width makes a detuning of a few $\mathrm{MeV}$ relevant. Even then, two further assumptions are necessary. First, being a few $\mathrm{MeV}$ below the Higgs boson peak does not eliminate it: for detuning $\Delta =\sqrt{s}-m_H$, the ideal narrow Breit-Wigner resonance cross section retains the relative factorFor ${\Gamma }_H=4\mathrm{MeV}$ and $\Delta =-4\mathrm{MeV}$ this is $1/5$, not zero. Resonant hadronic production can still change the requested denominator substantially.

Second, at $\sqrt{s}\simeq 125\mathrm{GeV}$ the Z boson continuum is not negligible. A more complete annihilation estimate including both tree-level Feynman diagrams usesHere $s_W^2$ is the squared sine of the weak mixing angle, $T_3^f$ is weak isospin, and $N_c^f$ is one for a lepton and three for a quark. Thus the continuum ratio is $F_e/[3(2F_u+3F_d)]$, rather than necessarily $3/11$. For example, $s_W^2=0.231$, $m_Z=91.1876\mathrm{GeV}$ and ${\Gamma }_Z=2.4952\mathrm{GeV}$ give approximately $0.102$ at $125\mathrm{GeV}$ before Quantum chromodynamics corrections.

If the unspecified particle resonance instead means the Z boson, detuning by a few $\mathrm{MeV}$ is tiny compared with its $2.5\mathrm{GeV}$ decay width. The resonant ratio remains approximately $\Gamma (Z\to e^{+}{e}^{-})/\Gamma (Z\to \mathrm{hadrons})=1/20.767\simeq 0.0482$.^[ref] This is a concrete counterexample to treating any slight downward detuning as permission to discard the particle resonance. The source supplies neither its identity nor the detuning and selection needed for a unique all-process ratio. The boxed continuum formula is the charge-counting answer under its stated approximation; a literal total must also include remaining resonant and other accepted hadronic channels.

In Fermi's golden rule,$\Gamma$ is a transition decay rate, measured in inverse time; $\hslash$ is the reduced Planck constant; $M=⟨f|H_{\mathrm{int}}|i⟩$ is the transition matrix element of the interaction part of the Hamiltonian; and $\rho$ is the final density of states per unit energy, evaluated subject to conservation of energy. Final spin and other discrete labels must be summed, with initial labels averaged when the preparation requires it. If $M$ varies across the final phase space, $|M{|}^2\rho$ denotes the corresponding integral rather than a product of constants.

A convention warning: many quantum field theory texts use $\Gamma$ for an energy decay width. That convention is ${\Gamma }_{\mathrm{width}}=\hslash {\Gamma }_{\mathrm{rate}}$; the formula as printed uses the rate convention.

The partial decay width ${\Gamma }_x$ is the energy width associated with decay of the compound nucleus back into the entrance channel $x+X$; by time reversal it also controls formation from that channel. The partial decay width ${\Gamma }_y$ describes the selected exit channel $y+Y$. The total decay width sums all energetically accessible channels, including these: $\boxed{\Gamma =\sum _i{\Gamma }_i=\hslash /\tau }$. The branching fraction for channel $i$ is ${\Gamma }_i/\Gamma$. Widths have energy units, whereas ${\Gamma }_i/\hslash$ is a decay probability per unit time. The isolated-resonance formula assumes approximately constant widths over the resonance and neglects a nonresonant interfering background.

In the Breit-Wigner resonance cross section, $E$ is the total centre-of-momentum energy, $E_0$ the resonance energy, and $p$ either incident particle's center-of-momentum momentum magnitude. The total decay width is $\Gamma =\hslash /\tau$, while ${\Gamma }_i$ and ${\Gamma }_f$ are partial widths into the entrance and specified exit channels. Their ratios to $\Gamma$ are the corresponding branching fractions. The factor $g$ averages over incident spin states and sums over the resonance's magnetic substates.

In natural units, $\hslash =c=1$, the resonant partial-wave cross-sectional scale is an area proportional to the squared reduced de Broglie wavelength, $1/p^2$. Equivalently a partial-wave scattering amplitude carries one factor $1/p$, and its squared modulus supplies $1/p^2$. In SI area units the same scale is $(\hslash /p{)}^2$; if $p$ is expressed in energy units, multiply the final inverse-energy-squared answer by $(\hslash c{)}^2$.

For unpolarized spin-half incident muons, the unpolarized resonance spin factor isHence $g_Z=3/4$ and $g_H=1/4$. The outgoing tau spins do not supply another independent multiplicity here: their allowed final states are already included in ${\Gamma }_f$.

Near a narrow resonance, treat the smooth prefactor $p^{-2}$ and the partial widths as constant across the line. At the peak the denominator is ${\Gamma }^2/4$. Half its height occurs when $(E-E_0{)}^2+{\Gamma }^2/4={\Gamma }^2/2$, namely at $E=E_0\pm \Gamma /2$. Thus the full width at half maximum is $\Gamma$ within the Breit-Wigner narrow-line approximation. A varying prefactor or convolution with the beam-energy distribution can modify an observed line shape.

The production amplitude has a muon pair joined to a Higgs propagator through their Yukawa coupling, followed by the bottom-quark pair at a second Yukawa vertex. The separate $H\to b\bar{b}$ graph is a tree-level Higgs–fermion vertex. There are no tree-level $Hgg$ or $H\gamma \gamma$ vertices in the renormalizable Standard Model: their lowest nonzero graphs are loops. A quark triangle, dominated by the top contribution, produces two gluons. Charged-fermion triangles contribute to two photons; charged-W loops also contribute and are dominant in the Standard Model amplitude. Photon permutations and the W contact topology belong to the full amplitude; the diagram shows these topologies rather than claiming one triangle alone is the entire gauge-invariant W contribution.^[ref]

Read the spin and parity from the actual diagram: N and Ne have $1/2^{-}$ ground states; O and F have $5/2^{+}$ ground states. The N→O beta-minus branch and Ne→F beta-plus branch therefore both have $|\Delta J|=2$ with a parity change: both are first-forbidden unique. The F→O beta-plus branch has $5/2^{+}\to 5/2^{+}$: it is allowed and can contain both Fermi and Gamow-Teller contributions.

The Ne ground-state branch is $0.2\%=0.002$, as explicitly printed on the diagram, not $0.2$ as a probability. The prose leaves off its percent sign. Interpret the supplied “lifetimes” as mean lifetimes $\tau$, so each partial decay rate is ${\lambda }_j=B_j/\tau$, with no extra $\ln 2$. ThenA decay width in energy units is ${\Gamma }_j=\hslash {\lambda }_j$, rather than a rate. Thus the corresponding partial energy widths, in the same order N, F, Ne, are $1.73\times 10^{-24}$, $6.93\times 10^{-24}$ and $8.78\times 10^{-24}\mathrm{MeV}$. If a different convention used the supplied numbers as half-lives, all these rates and widths would acquire $\ln 2$; their ratios would be unchanged. The literal mean-lifetime convention is used throughout.

The dominant interaction is electromagnetic. The color singlet charm-anticharm component of the J/psi meson annihilates through a virtual photon, which creates a muon and an antimuon:Panel (i) shows the two electromagnetic vertices. Vector quantum numbers allow the one-photon channel. A Z boson contribution is negligible at this mass scale; a single gluon cannot connect a color singlet initial state to colorless leptons. The dominant interaction for this specified final state need not dominate the meson's total decay width.

For negligible final fermion masses, the chiral Z-boson partial width is proportional to $g_L^2+g_R^2$. Both channels have one color state. An electron has $Q=-1$, $T_{3L}=-1/2$ and $T_{3R}=0$, givingFor one active neutrino species, $Q=0$, $T_{3L}=1/2$ and there is no coupled right-handed neutrino field, so $g_L^{\nu }=1/2$, $g_R^{\nu }=0$. The common coupling normalization and phase space cancel in the ratio:The same total decay width appears in the two branching fractions, soThere is no factor of three for a single specified flavour. Summing three active neutrino flavours would give an invisible branching fraction of approximately $20.0\%$ in this approximation.

Treat the quoted maximum as the resonant Z boson contribution described by the Breit-Wigner resonance cross section. The electron and Positron each have spin $1/2$, while the Z boson has spin one, soNeglect the electron mass at this centre-of-mass energy. Each incident momentum magnitude is then $p_i=E_0/2$ in natural units. Write $B_e={\Gamma }_{ee}/\Gamma$. At the resonance peak the denominator is ${\Gamma }^2/4$, and the same partial decay width appears for production and decay:The peak resonance cross section and branching fractions relation contains $B_e^2$, because the electronic channel occurs at both ends of the resonant process.

For an area in ordinary units, restore $(\hslash c{)}^2$. The original PDF prints $\hslash c=197.5\mathrm{MeV}$, which is dimensionally incomplete: the intended standard conversion is $197.5\mathrm{MeV}\mathrm{fm}=0.1975\mathrm{GeV}\mathrm{fm}$. Also,ConsequentlyThis estimate uses the supplied resonance model. The full physical elastic electron-Positron cross section also contains nonresonant photon exchange and interference; an experimental use of the formula must isolate the resonant contribution.

The nuclear determination of $|V_{ud}|$. A $0^{+}\to 0^{+}$ beta decay transition has no nuclear parity change and can proceed with zero lepton orbital angular momentum. The allowed scalar operator gives a Fermi transition; the rank-one Gamow-Teller transition operator cannot connect total nuclear spins zero to zero. The strong isobaric-analog transitions used for precision measurements are superallowed beta decays. A generic $0^{+}\to 0^{+}$ assignment alone would not guarantee superallowed strength if the states were nonanalog.

At the quark level the weak charged current converts $d\leftrightarrow u$ and carries $V_{ud}$ in the amplitude. The nuclear Fermi operator is an isospin ladder operator, givingHere $f$ denotes the known phase space dependence, with the dimensional factors included in the proportionality constant. In the isospin-symmetric limit, $M_F$ between analog states is fixed by isospin algebra, for example $|M_F{|}^2=2$ for the usual $T=1$ analog transitions. A measured partial half-life and the computed phase space factor give the comparative half-life; after radiative corrections and isospin breaking corrections, its inverse determines $|V_{ud}{|}^2$. The decay rate measures a magnitude squared, not the complex phase of $V_{ud}$.

Ds decay diagrams and CKM comparisons. The original PDF has $K^{+}{\bar{K}}^0$, with ${\bar{K}}^0=s\bar{d}$. The local TeX drops that overbar; using $K^0=d\bar{s}$ instead would give a different flavour channel. Use the PDF's neutral antikaon throughout.

In the representative Ds nonleptonic tree amplitudes, the Ds meson spectator is $\bar{s}$. The elementary weak transitions are

In the first two channels, the produced $u$ joins the spectator $\bar{s}$ to form $K^{+}$; the other quark joins $\bar{d}$. For the first channel this uses the $-d\bar{d}/\sqrt{2}$ component of the neutral pion. In the third, $s\bar{s}$ forms the phi meson and $u\bar{d}$ forms ${\pi }^{+}$. The two possible final pairings have different colour contractions and hadronic matrix elements even when their weak CKM factors coincide.

The purely CKM rate suppression of $c\to d$ relative to $c\to s$ isThus $K^{+}{\pi }^0$ is singly Cabibbo suppressed, while $K^{+}{\bar{K}}^0$ and $\varphi {\pi }^{+}$ are Cabibbo-favored at this level. If reduced hadronic strengths and phase space effects were equal, including the neutral-pion component would giveThe supplied branching fractions instead giveThe first observed ratio is $0.814$ times the neutral-pion-corrected simple estimate, requiring only an order-one remaining hadronic and phase space factor. Relative to $\varphi {\pi }^{+}$, that remaining factor is $0.525$. The two Cabibbo-favored channels differ by a factor $0.644$ despite equal CKM magnitudes. The CKM hierarchy therefore explains why one channel is much rarer, but it does not calculate the three exclusive branching fractions; spin, phase space, color charge and strong interaction final-state effects also matter.

What spin and parity establish. In either pseudoscalar-pseudoscalar final state, the total intrinsic spin is zero. Conservation of angular momentum for a spin-zero parent forces $L=0$, so the final parity isopposite to the parent $0^{-}$ Ds meson. ConsequentlyFor a vector meson plus a pion, the intrinsic final spin is one. Coupling to total $J=0$ requires $L=1$, giving $P_f=(-1)(-1)(-1{)}^1=-1$, equal to the parent parity. Thus spin and parity alone permit a parity-conserving $\varphi {\pi }^{+}$ transition. That is the limited sense in which it is not necessarily weak from this information alone. In reality it is also weak: the initial charm flavour disappears, which the strong interaction and electromagnetic interaction conserve.

W branching fractions. In the massless-daughter, tree-level estimate, each leptonic channel contributes one common unit of decay width. Each accessible quark pair contributes $3|V_{ij}{|}^2$ because of the three final colours. The charged-current W branching-fraction counting denominator isOnly $u$ and $c$ are accessible up-type quarks; the large $V_{tb}$ is irrelevant because a W boson cannot decay into the heavier top quark. HenceA final bottom quark arises in $W^{-}\to b\bar{u}$ or $b\bar{c}$, soThis counts the primary decay channels; finite daughter masses and QCD corrections refine the approximation.

The requested top-lifetime estimate and its limitation. Under the instructed equal-coefficient Sargent's rule scaling, $\Gamma \propto m^5$ and the mean lifetime is inversely proportional to that fifth power. Converting both masses to GeV givesThis is the requested order-of-magnitude estimate, assuming the same effective coefficient for the two total decay rates. It corresponds to $c\tau \simeq 54\mathrm{fm}$ and is longer than the hadronization time, roughly a few $10^{-24}\mathrm{s}$. Taken literally, it would allow top-flavored hadrons to form; it does not establish the physical statement that the top decays before hadronizing.

The extrapolation fails because Sargent's rule uses a low-energy Fermi interaction approximation, whereas $m_t>m_W+m_b$ permits $t\to bW$ with an on-shell W boson. The physical leading decay width instead scales asUsing $m_W\simeq 80.4\mathrm{GeV}$, the Fermi constant $G_F=1.16638\times 10^{-5}\mathrm{Ge}{\mathrm{V}}^{-2}$ and $|V_{tb}|=0.999$ gives about $1.55\mathrm{GeV}$ at the supplied top mass. Then ${\tau }_t=\hslash /{\Gamma }_t$, with $\hslash =6.58212\times 10^{-25}\mathrm{GeV}\mathrm{s}$, gives ${\tau }_t\simeq 4.2\times 10^{-25}\mathrm{s}$. This corrected regime explains top-quark decay before hadronization.^[ref]

Ignoring the requested final-mass difference and initially taking equal hadronic matrix elements, the common $W^{-}\to d\bar{u}$ factor cancels. A decay width is proportional to the squared decay amplitude, soWith the supplied Cabibbo angle,The quoted measurement, about $500\pm 10$, is larger by a factor $500/19.75\simeq 25.3$. If that whole difference were attributed to a ratio of other amplitudes, it would require a factor $\sqrt{25.3}\simeq 5.03$.

This does not invalidate quark mixing. The supplied Cabibbo-Kobayashi-Maskawa matrix is only a schematic hierarchy in powers of $\sin {\theta }_C$, particularly for $V_{ub}$; its omitted coefficients matter greatly after squaring. Equal phase space also does not imply equal hadronic decay amplitudes: the two transitions have different hadronic form factors and strong-interaction dynamics, and loop contributions can interfere with the charmless tree amplitude. Thus the calculation correctly predicts enhancement of the charmed channel, but the supplied CKM hierarchy alone does not quantitatively predict the measured ratio. It would be unjustified to treat $19.75$ as a precision prediction with only the quoted experimental error.

The four direct beta decay branching fractions exhaust the parent decays in the model, soHere the question's ${\Gamma }_i$ are dimensionless branching fractions, despite the conventional use of $\Gamma$ for a dimensionful decay width. Its “zeroth excited state” means the daughter ground state. Cascades through an excited level do not count as an additional direct beta branch to it.

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 the resonant contribution only, subtract nonresonant Bhabha scattering and its interference before interpreting a fitted area. The integrated narrow vector resonance area follows from the Breit-Wigner resonance cross section, with $J=1$, $s_{e^{-}}=s_{e^{+}}=1/2$, and $p\simeq M/2$, becomesAcross a narrow scattering resonance, treat the prefactor and partial decay widths as constant and extend the integral over $E-M$ to the real line. Since ${\int }_{-\infty }^{\infty }du/(u^2+a^2)=\pi /a$,A normalized beam-energy response convolves the line shape without changing this full area. It can completely change the observed peak height and apparent width. If a uniform acceptance $ϵ$ is not already unfolded, use ${\mathcal{I}}_{ee}={\mathcal{I}}_{ee}^{\mathrm{obs}}/ϵ$. The distinction between $B_{ee}^2$ for this exclusive final state and $B_{ee}$ for an inclusive final state is essential.

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The recoverable numerical conversion is nevertheless definite. With $M\simeq 3.1\mathrm{GeV}$, $B_{ee}=0.060$, $ϵ=0.50$, and $1{\mathrm{GeV}}^{-2}=389400\mathrm{nb}$, b givesThis coefficient assumes the plotted exclusive electronic sample has not already been corrected for acceptance. A restored area, with continuum subtraction and acceptance convention checked, would complete the estimate. A tabulated modern J/psi meson width would be external information rather than a reading of these data.

The narrow scattering resonance indicates suppressed strong interaction decay. In the quark model, $c\bar{c}$ is below the open-charm quark $D\bar{D}$ threshold, so its constituents cannot simply emerge in separate charmed mesons. The OZI rule suppresses annihilation into light hadrons; for a $1^{--}$ color-singlet state the leading purely gluonic channel needs three gluons. The observed leptonic channels remain competitive. Its mass and $I=0$ fit charmonium; the small width alone would not uniquely identify the new quark flavour without the wider spectrum and threshold evidence.

The basic Feynman diagrams have ${\tau }^{-}\to {\nu }_{\tau }{W}^{-\ast }$ followed by $W^{-\ast }\to e^{-}{\bar{\nu }}_e$, ${\mu }^{-}{\bar{\nu }}_{\mu }$, or $d\bar{u},s\bar{u}$, with the latter pairs undergoing hadronization. See the charged-current diagrams in a. Neglect all daughter masses, QCD corrections and inaccessible charm final states, and assume lepton universality. Each leptonic mode then has one unit of decay width; the two accessible quark combinations have three color charge states and total CKM weight $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$. ThereforeThe result is an inclusive parton model estimate. Perturbative QCD corrections increase the hadronic contribution, while muon mass, nonperturbative dynamics and radiative corrections also matter; the physical fraction is roughly $0.65$.

Every ordinary hadronic mode already contains a tau neutrino, so it contains a lepton as well as hadrons if invisible particles are counted. For an additional visible charged lepton, a virtual photon can internally convert in a hadronic mode, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\gamma }^{\ast }\to {\nu }_{\tau }{\pi }^{-}{e}^{+}{e}^{-}$. The extra electromagnetic vertices suppress such modes; they are not a large fraction of the inclusive width. Alternatively a daughter pion or kaon can decay semileptonically or leptonically in flight, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{K}^0$ followed by a semileptonic neutral-kaon decay. Those are secondary decay chains, not independent primary tau lepton branching fractions. Lepton-flavour-violating neutrinoless modes such as ${\tau }^{-}\to {\mu }^{-}{\pi }^{+}{\pi }^{-}$ do not contribute appreciably in the Standard Model.

Write the leptonic partial decay width asHere $G_F$ is defined from muon decay, so the coupling denominator fixes the convention. Measure ${\Gamma }_{\ell }=B_{\ell }/{\tau }_{\tau }$. The ratio $B_{\mu }/B_e$ tests $|g_{\mu }/g_e{|}^2$ after the known phase space and radiative corrections. Comparing $B_e/{\tau }_{\tau }$ with the muon lifetime and the fifth power of the mass ratio tests $|g_{\tau }/g_{\mu }{|}^2$. Hadronic comparisons such as tau-to-pion versus pion-to-muon decay supply additional universality tests with their own hadronic and electromagnetic corrections.

In neutral meson mixing, the mass eigenstates and flavor eigenstates differ. A state prepared as $D^0=c\bar{u}$ develops a ${\bar{D}}^0=u\bar{c}$ component because the effective Hamiltonian operator has off-diagonal mass and decay width entries. In the CP symmetry limit implicit in the source, the propagating states are equal superpositions, with average mass $M=(M_1+M_2)/2$ and average decay width $\Gamma =({\Gamma }_1+{\Gamma }_2)/2$.

To retain the source's positive $i\sin$ convention for $g_{-}$, take $M_2-M_1=\Delta M$ and ${\Gamma }_2-{\Gamma }_1=\Delta \Gamma$, with $g_{\pm }=(e_1\pm e_2)/2$, where $e_j=\exp (-i{M}_{j}t-{\Gamma }_{j}t/2)$. Then $x=\Delta M/\Gamma$ and $y=\Delta \Gamma /(2\Gamma )$ reproduce the given functions. Reversing an eigenstate label changes signs in intermediate formulae; it must also change the phase conventions consistently.

For equal widths, the mixed probability is proportional to $e^{-\Gamma t}{\sin }^2(\Delta Mt/2)$. Thus the angular angular frequency is $|\Delta M|$ in natural units, or $|\Delta M|/\hslash$ with units restored; the period is $2\pi \hslash /|\Delta M|$. A nonzero difference of decay widths adds hyperbolic time dependence.

The ordinary neutral flavour-carrying mesons that mix are $K^0$, $D^0$, $B_d^0$ and $B_s^0$ with their antiparticles. A self-conjugate ${\pi }^0$ has no distinct antiparticle flavour to oscillate into. There are no long-lived top-flavoured mesons, since the top quark decays before hadronization.