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Bottom-quark tagging identifies jets compatible with a bottom hadron, often through displaced tracks, a secondary decay vertex or a lepton within the jet. The finite hadron lifetime gives the displacement. Efficiencies and mistaken tags from charm or light jets must be calibrated; a top quark itself decays before hadronization.

The Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$ compares the left-handed unitary matrices that diagonalize the up-type and down-type Yukawa matrices. Its rows label up quarks, charm quarks and top quarks; its columns label down quarks, strange quarks and bottom quarks. In the weak charged current, the coefficient of ${\bar{u}}_{Li}{\gamma }^{\mu }{d}_{Lj}{W}_{\mu }^{+}$ is $V_{ij}$. A single CKM matrix therefore connects quark masses, flavour-changing decay amplitudes, and CP violation.

Write $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$. An exactly unitary matrix with three mixing angles and one CP-violating phase isThis convention agrees with Makoto Kobayashi's account of three-generation quark mixing.^[ref]

The Christenson-Cronin-Fitch-Turlay kaon CP violation experiment established the phenomenon that required an explanation. Makoto Kobayashi and Toshihide Maskawa showed that two-generation quark mixing cannot supply it, whereas adding a third quark generation permits an irreducible weak phase. Their 1973 paper also considered additional fields as alternative mechanisms; the three-generation conclusion concerns CP violation from the minimal unitary quark mixing mechanism.^[ref]

A common weak phase makes CP violation across mesons testable through the BABAR time-dependent CP violation experiment. It does not explain the cosmic baryon asymmetry.^[ref]

Neglect final-state masses and QCD corrections. Each of the three leptonic W boson decay channels contributes one common unit of width. Each open quark channel contributes $3|V_{ij}{|}^2$, with the factor three counting final colours. The top quark is kinematically inaccessible in an on-shell W decay because $m_t>m_W$. Thus $B(W\to \ell \nu )=1/D\simeq 1/9$ and $B(W^{-}\to b\bar{u},b\bar{c})=3(|V_{ub}{|}^2+|V_{cb}{|}^2)/D$. The large $V_{tb}$ does not open a forbidden top-quark channel.

In an on-shell convention, the Standard Model relates the W boson mass to measured electroweak inputs throughThe radiative corrections in $\Delta r$ depend on the top quark and Higgs boson masses. The leading heavy-top contribution to the electroweak rho parameter is $\Delta \rho \simeq 3G_F{m}_t^2/(8\sqrt{2}{\pi }^2)$, reflecting the large mass splitting in the top-bottom weak doublet. The Higgs dependence in the precision corrections is mainly logarithmic over the relevant range. Direct measurements of $m_t$ and $m_H$ therefore allow an indirect prediction of $m_W$; comparing it with a direct measurement tests loops as well as tree-level electroweak symmetry breaking. Any tension must include input uncertainties, experimental systematics and higher-order corrections.^[ref]

Production of a Higgs boson from two gluons, mediated at leading order in the Standard Model by a quark loop. The large top quark Yukawa coupling makes the top loop important. There is no direct renormalizable tree-level coupling of the colorless Higgs field to two gluons.

There are two one-loop Feynman diagrams: the two orders of the labelled photons around a top quark fermion loop, with one Yukawa interaction insertion for the outgoing Higgs boson. They are added, rather than identified by a Feynman-diagram symmetry factor. Each has the same color charge trace $N_c=3$.

In the first routing, traverse the fermion loop from the $p_2$ insertion to the $p_1$ insertion with momentum $l$, then to the Higgs boson insertion with momentum $l+p_1$, then back to the $p_2$ insertion with momentum $l-p_2$. Conservation at the Higgs boson insertion gives $p_h=p_1+p_2$. The second routing exchanges $(p_1,\mu )$ and $(p_2,\nu )$: its three internal momenta are $l$, $l+p_2$ and $l-p_1$. Reversing the orientation is another way to represent this second ordering. Two vector insertions give no Furry theorem cancellation.

Continue the loop momentum to $D=4-\epsilon$ dimensions, keeping $\mathrm{tr}1=4$ and the external physical polarization vectors four dimensional. Introduce Feynman parameters $x,y\ge 0$, $x+y\le 1$, denoting this integration triangle by $\mathcal{T}$. Completing the square givesOdd terms in $k$ integrate to zero. Lorentz invariance of dimensional regularization gives $k_{\mu }{k}_{\nu }\mapsto g_{\mu \nu }{k}^2/D$. The general projected numerator therefore becomes, under the Feynman integral,Only the $k^2$ term can give an ultraviolet divergence. Define, with the conventional scale factor implicit,The gamma function formula givesThus the pole of the numerator integral, including $2{\int }_{\mathcal{T}}dxdy=1$, isThe coefficients $(4,1,1)$ satisfy this condition. A nonzero local divergence would have to be removed by a counterterm. The Ward identity forbids a local $h{A}_{\mu }{A}^{\mu }$ term, and the first electromagnetic gauge-invariant coupling with one Higgs boson and two photons has the form $h{F}_{\mu \nu }{F}^{\mu \nu }$, of engineering dimension five. It is not an independent counterterm of the Standard Model, a renormalizable quantum field theory. The top quark contribution is separately electromagnetic gauge-invariant; no additional W boson loop Feynman diagram is needed to cancel this pole.

The finite remainder must still be retained. For $A=4C$,Putting $D=4$ before evaluating the divergent Feynman integral would incorrectly set this finite contribution to zero.

Use the Feynman parameters and $J_3$ defined in Solution. With $A=4C$, the finite term just found is $-C\Delta J_3$. Combining it with the remaining $g_{\mu \nu }$ coefficient givesThe other coefficient is $(B-4Cxy)J_3$. Thus, writing ${\mathcal{I}}_{\mu \nu }$ for the Feynman integral without the overall prefactor,For $T_{\mu \nu }=\frac{s}{2}{g}_{\mu \nu }-p_{2,\mu }{p}_{1,\nu }$, the two coefficients must have the corresponding ratio. The mismatch is simply $(B-C)J_3$, independent of $xy$. Therefore the required relation iswhich is satisfied. In the primitive normalization the result isand the full top quark scattering amplitude, in our stated $S$ convention, isIndeed $p_1^{\mu }{T}_{\mu \nu }=p_2^{\nu }{T}_{\mu \nu }=0$: replacing either polarization vector by its photon momentum makes the scattering amplitude vanish. This is the Ward identity, expressing electromagnetic gauge invariance. The result also is symmetric under exchange of the two photons. The finite term in Solution is essential for this Ward identity.

As an independent low-energy check, for $s/m^2\to 0$ the Feynman parameter integral isThe coefficient is consequently $-N_{c}y{e}^2{Q}^2/(6{\pi }^{2}m)$ at leading order. It has the inverse-mass dimension required for an effective $h{F}_{\mu \nu }{F}^{\mu \nu }$ coupling. If $y=m/v$, this coefficient stays finite in the large-$m$ limit at fixed $v$; that limit increases the Yukawa coupling, so it is not a fixed-coupling heavy-field decoupling limit.

The Standard Model uses the local gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its Higgs field is a complex Higgs doublet of hypercharge $Y=1/2$. A bare mass term for a weak gauge boson, or for a charged fermion joining inequivalent left- and right-handed gauge representations, would not respect the unbroken electroweak construction. Instead introduce a gauge-invariant Higgs potentialIts minima satisfy ${\Phi }^{†}\Phi =v^2/2$, $v^2={\mu }^2/\lambda$. Choosing a representative $⟨\Phi ⟩=(0,v{)}^T/\sqrt{2}$ leaves $Q=T_3+Y$ unbroken. In this description electroweak symmetry breaking is $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{\mathrm{em}}$; the underlying local gauge symmetry remains a redundancy of the theory. A vacuum representative is not a physical explicit violation of gauge invariance.

In unitary gauge, write $\Phi =(0,v+H{)}^T/\sqrt{2}$. The gauge-covariant kinetic term $|D_{\mu }\Phi {|}^2$ produceswith $\tan {\theta }_W=g^{\prime }/g$ and $e=g\sin {\theta }_W$. Three angular scalar directions supply the longitudinal gauge-boson polarizations of $W^{+},W^{-},Z$, while one real radial Higgs boson survives with $m_H^2=2\lambda v^2$. Counting physical degrees of freedom checks the Higgs mechanism: the electroweak gauge sector and scalar doublet have $4\times 2+4=12$ before the rearrangement and $3\times 3+2+1=12$ afterwards.

The Higgs potential also fixes the scalar self-interactions:The color gauge group is unbroken, so gluons remain massless. The same electroweak kinetic term fixesGauge-invariant Yukawa couplings such as $-y_d{\bar{Q}}_L\Phi d_R-y_u{\bar{Q}}_L\tilde{\Phi }{u}_R+h.c.$ give $m_f=y_{f}v/\sqrt{2}$ and the physical coupling $-m_{f}H\bar{f}f/v$. These relations predict stronger Higgs couplings to heavier massive particles; they do not determine the Yukawa parameters themselves. The minimal Standard Model does not supply observed neutrino masses. The surviving Higgs boson is neutral and has spin zero; its mass and self-coupling are tied to the curvature of the Higgs potential, not fixed by the gauge couplings alone. Higgs exchange also provides the Higgs cancellation in longitudinal vector scattering, maintaining the required high-energy behavior of weak amplitudes.

At a proton collider, important production mechanisms are gluon-fusion Higgs production through a heavy-quark loop, vector-boson fusion with two outgoing quark jets, associated $WH$ or $ZH$ production, and production with a top quark pair. At an electron–positron collider, Higgsstrahlung $e^{+}{e}^{-}\to ZH$ is a clean example. Tree-level decays include Higgs decay to a fermion pair and, when allowed, Higgs decay to two W bosons or Higgs decay to two Z bosons. Below an on-shell vector-pair threshold, one or both vectors can be virtual. The $gg$ and $\gamma \gamma$ channels arise through loops; neutral colorless $H$ has no corresponding direct tree-level gauge coupling. Rates and branching fractions depend on both couplings and kinematics, so “heaviest available particle” is a guide rather than a complete rule.

The Higgs boson discovery in 2012 supplied an experimental test. The ATLAS experiment and CMS experiment reconstructed two-photon and four-lepton invariant masses in Large Hadron Collider proton collisions. Electromagnetic calorimeters measure photon energies, while tracking and muon systems identify charged leptons; narrow reconstructed peaks can be distinguished from smooth backgrounds with control samples and a likelihood analysis. The ATLAS discovery paper reported a new resonance near $126.5\mathrm{GeV}$. An observed mass peak supports the existence of a new state; production channels, decay rates and angular distributions test whether it has the particular Higgs couplings and quantum numbers.^[ref]

Finally, the hierarchy problem asks why the electroweak scalar mass scale remains much smaller than a putative heavy physical scale without delicate cancellations. A cutoff estimate suggests corrections $\delta m_H^2\sim {\Lambda }^2/(16{\pi }^2)$ times couplings, but the quadratic cutoff term itself depends on the regularization convention. The physical concern persists when a heavy state of mass $M$ couples to the Higgs field and produces threshold corrections of order its coupling times $M^2/(16{\pi }^2)$. Matching a much smaller Higgs mass may then require cancellation between independent parameters. Supersymmetry, compositeness and other protective structures propose different resolutions; renormalizability of the Standard Model does not by itself resolve this sensitivity or establish that new particles must occur at one particular mass.

The Standard Model is a chiral gauge theory: left-chiral quarks and leptons form SU(2) weak doublets, while the corresponding right-chiral charged fields are weak singlets. The covariant kinetic term contains $g_W{\bar{Q}}_L{\gamma }^{\mu }{t}^a{Q}_L{W}_{\mu }^a$. With $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$, its off-diagonal generators give the weak charged currentAn overall interaction sign depends on the convention for the covariant derivative and has no effect on this decay rate. The chiral projector $P_L=(1-{\gamma }^5)/2$ explains the vector-minus-axial combination. The Cabibbo-Kobayashi-Maskawa matrix comes from the different left-handed rotations that diagonalize up- and down-type fermion mass matrices. Set $V_{tb}=1$ and $m_b=0$ as stipulated.

Let $P_{\mu \nu }=-g_{\mu \nu }+p_{3\mu }{p}_{3\nu }/m_W^2$, the massive vector polarization sum. Sum over final spins and average over the two equally populated initial top quark spins. The fermion spin sum and the Dirac adjoint of the vertex giveThe $m_t$ term vanishes because $(1-{\gamma }^5){\gamma }^{\nu }(1-{\gamma }^5)=0$. Anticommuting ${\gamma }^5$ through the remaining gamma matrices, using $(1-{\gamma }^5{)}^2=2(1-{\gamma }^5)$, then using cyclicity of the matrix trace and the symmetry of $P_{\mu \nu }$ yieldsThe extra factor two from the chiral algebra compensates the initial spin-average factor. The term with ${\gamma }^5$ is antisymmetric in $\mu ,\nu$ and vanishes upon contraction with $P_{\mu \nu }$. The remaining gamma-matrix trace isIts contraction givesHere $p_1=p_2+p_3$, $p_1^2=m_t^2$, $p_2^2=0$ and $p_3^2=m_W^2$, soThereforeThe two-body Lorentz-invariant phase space has rest-frame momentum $p^{\ast }=(m_t^2-m_W^2)/(2m_t)$. Substitution into $\Gamma =p^{\ast }\overline{|\mathcal{M}{|}^2}/(8\pi m_t^2)$ gives the two-body top-quark decay widthThe squared threshold factor combines the vanishing final momentum with the corresponding kinematic factor in the squared scattering amplitude. The longitudinal W boson polarization contributes the high-mass enhancement proportional to $m_t^3/m_W^2$. The result is at tree level, for an on-shell W boson and $m_t>m_W$; finite bottom mass and radiative corrections change it. Restoring quark mixing multiplies the width by $|V_{tb}{|}^2$.

Adopt the phase convention $CP|K^0⟩=-|{\bar{K}}^0⟩$ and $CP|{\bar{K}}^0⟩=-|K^0⟩$. Changing both phases changes the convention for the mixing parameter, not physical predictions. The flavor contents are $K^0=d\bar{s}$ and ${\bar{K}}^0=s\bar{d}$. A pair of charged weak currents changes strangeness by two units through the following box contractions, with $u_i,u_j=u,c,t$ summed over.

The drawing uses unitary gauge: the W boson gauge-boson propagator contains its longitudinal part. In a gauge of a renormalizable quantum field theory, charged Goldstone boson boxes must also be included to obtain the same gauge-independent result. The external lines represent the valence quarks; the meson matrix element also contains strong interaction binding effects.

Put ${\lambda }_i=V_{is}^{\ast }{V}_{id}$, $x_i=m_i^2/m_W^2$ and $P_L=(1-{\gamma }^5)/2$. The charged weak box contribution to kaon mixing has the effective four-fermion interaction structureHere $G_F$ is the Fermi constant, $V$ is the CKM matrix, and $S$ is a dimensionless loop function from a Feynman integral; the estimate suppresses convention-dependent numerical factors. Four weak interaction Feynman vertices and a one-loop Feynman integral give the coupling and loop order suppression. Unitarity of the CKM matrix gives ${\sum }_i{\lambda }_i=0$, so the GIM mechanism cancels the mass-independent part. With a negligible up quark mass, the leading charm quark term has $S(x_c,x_c)\sim x_c$. Define $f_K$ by $⟨0|\bar{s}{\gamma }_{\mu }{\gamma }^{5}d|K^0(p)⟩=i{f}_K{p}_{\mu }$. Estimating the hadronic matrix element by $f_K^2{m}_K{B}_K$, with the dimensionless kaon bag parameter $B_K$ of order unity, givesThe hadronic normalization includes division by $2m_K$ when relativistically normalized states are converted to the effective Hamiltonian for neutral-meson mixing. Since $|V_{cs}^{\ast }{V}_{cd}|\simeq \sin {\theta }_C\cos {\theta }_C$, the Cabibbo suppression is explicit. Illustrative scales $G_F\sim 10^{-5}{\mathrm{GeV}}^{-2}$, $m_c\sim 1\mathrm{GeV}$, $f_K\sim 0.1\mathrm{GeV}$ and $m_K\sim 0.5\mathrm{GeV}$ give an order $10^{-16}$ to $10^{-15}\mathrm{GeV}$ estimate. The full sum includes charm quark-top quark and top quark-top quark terms, important for CP violation; long-distance strong interaction contributions prevent this dimensional estimate from being a precision prediction. A meson matrix element necessarily needs hadronic input in addition to electroweak parameters.

For the decaying kaon system write $H^{\prime }=M-i\Gamma /2$, where $M$ and $\Gamma$ are Hermitian matrices. The entries denoted $M_{ij}$ in the question are entries of this effective Hamiltonian for neutral-meson mixing; they need not themselves form a Hermitian matrix. CPT invariance equates the diagonal masses and diagonal decay widths, givingOne can see why there is no complex conjugation on the right as follows. In the flavor subspace let $J=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and represent the antiunitary CPT operation by $J$ followed by complex conjugation. CPT changes a projected forward propagator into its adjoint. Indeed, if $P$ projects onto the flavor subspace and $H_{\mathrm{full}}$ is the Hermitian microscopic generator, its invariance and the antiunitary reversal of $i$ give $\Theta P{e}^{-i{H}_{\mathrm{full}}t}P{\Theta }^{-1}=P{e}^{+i{H}_{\mathrm{full}}t}P=(P{e}^{-i{H}_{\mathrm{full}}t}P{)}^{†}$. In the time-independent two-state approximation, the effective generator this means $J{H^{\prime }}^{\ast }J={H^{\prime }}^{†}$, whose diagonal entries give $(H_{22}^{\prime }{)}^{\ast }=(H_{11}^{\prime }{)}^{\ast }$; the off-diagonal entries impose no further condition. Equivalently, apply CPT separately to the dispersive and absorptive Hermitian matrices. Treating the decaying effective Hamiltonian for neutral-meson mixing as a Hermitian matrix would incorrectly discard the absorptive part.

In the chosen flavor phase convention CP is represented by $-J$. Its invariance requires $J{H}^{\prime }J=H^{\prime }$. Thus it gives diagonal equality and additionallyThe normalized CP eigenstates areApplying CP directly gives eigenvalues $+1$ and $-1$ respectively.

Now use CPT to put $H^{\prime }=\left(\begin{array}{cc}a& b\\ c& a\end{array}\right)$ and take correlated square roots $\alpha =\sqrt{b}$, $\beta =\sqrt{c}$. Its characteristic polynomial equation is $(a-\mu {)}^2-bc=0$, and direct multiplication showsExpressing these two eigenvectors in the CP eigenstate basis gives, respectively, coefficients proportional to $(\alpha +\beta ,\alpha -\beta )$ and $(\alpha -\beta ,\alpha +\beta )$. Therefore the kaon CP mixing parameter isand, because the CP eigenstate basis is orthonormal, the normalized states at production areThe kaon mixing square-root branch convention correlates the roots continuously with the limit of conserved CP $b=c$, where $\alpha =\beta$ and $ϵ=0$. Reversing one root interchanges the two eigenstates. The labels short and long are fixed by their decay widths: their complex eigenvalues are ${\mu }_{S,L}=m_{S,L}-i{\Gamma }_{S,L}/2$, and subsequent evolution supplies $e^{-i{\mu }_{S,L}t}$. The displayed formula assumes nonzero mixing and $\alpha +\beta \ne 0$, as in the physical kaon system; degenerate or defective matrices require a separate limiting treatment.

The tree-level Feynman diagram contains one weak charged current vertex. Fermion arrows point along fermion-number flow, so the outgoing antiquark arrow points toward the vertex.

For one fixed matching color, the scattering amplitude isAn overall phase from the vertex has no effect on the width. The fermion spin sums give $\sum u\bar{u}=/k$, $\sum v\bar{v}=/k^{\prime }$ in the massless approximation. For the three initial polarizations, use a spin average of $1/3$. The gamma matrix trace identities giveThe sign of the last term follows the PDF's ${\gamma }^5$ convention and does not affect this decay: the polarization sum is symmetric. Since $p=k+k^{\prime }$ and the daughter masses vanish, $p_{\mu }{T}^{\mu \nu }=0$ and $k\cdot k^{\prime }=M_W^2/2$. ThereforeIn the rest frame, the two-body Lorentz-invariant phase space integrates to $1/(8\pi )$: after the spatial delta function sets ${\mathbf{k}}^{\prime }=-\mathbf{k}$, its radial delta function fixes $|\mathbf{k}|=M_W/2$. Explicitly,The decay formula consequently givesThis is the printed expression, interpreted for one color. For a physical quark-flavor channel there are three orthogonal final color states. Their probabilities add; no initial color average is present for a colorless W boson. ThusThe PDF's formula omits this color multiplicity if read as the ordinary inclusive flavor width. It is also the familiar normalization for a colorless lepton channel.

The six physically accessible quark combinations are $u\bar{d},u\bar{s},u\bar{b},c\bar{d},c\bar{s},c\bar{b}$. A real on-shell W boson cannot produce a top quark; the massless approximation is applied to the accessible daughters and does not open a physically forbidden top channel. CKM matrix unitarity gives ${\sum }_{q^{\prime }=d,s,b}|V_{u{q}^{\prime }}{|}^2={\sum }_{q^{\prime }=d,s,b}|V_{c{q}^{\prime }}{|}^2=1$. Therefore the physical color-summed hadronic width at this order isIf the printed one-color convention is retained for every channel, its sum is instead $G_F{M}_W^3/(3\pi \sqrt{2})$. In the artificial theory where all three up-type flavors, including the top, are kinematically massless, there would be nine channels and the physical sum would be $9G_F{M}_W^3/(6\pi \sqrt{2})$; that is not the on-shell Standard Model channel list.

Production and identification. At a hadron collider, initial partons carry fractions of the proton momenta; observable production rates combine parton-level amplitudes with parton distribution functions.

Why the three masses form a test. At tree level, electroweak symmetry breaking gives $m_W=gv/2$ and $m_Z=\sqrt{g^2+g^{\prime 2}}v/2$. The Higgs mechanism relates masses to the vacuum field, but it does not by itself predict the top Yukawa coupling or the Higgs self-coupling, so the three masses are not fixed independently of measured parameters.

The stronger test comes from radiative corrections. With accurately measured $\alpha$, $G_F$ and $m_Z$, the electroweak precision test with the W, top and Higgs masses usesThe top-bottom loop contributes approximately $\Delta \rho =3G_F{m}_t^2/(8\sqrt{2}{\pi }^2)$, while the Higgs contribution has a characteristic logarithmic mass dependence. Hence direct measurements of $m_t$ and $m_H$ turn the precision calculation into a prediction for $m_W$. Comparing that prediction with the direct W-mass measurement overconstrains the Standard Model; disagreement beyond experimental and theoretical uncertainties could indicate additional particles or interactions in the loops. The test concerns the relation among independently measured quantities, rather than merely the existence of the three particles.

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.