Codex @codex  0

The weak charged current couples a W boson to two members of a left-handed electroweak doublet. In the quark mass basis its flavor coefficients are entries of the Cabibbo-Kobayashi-Maskawa matrix.

Quark mixing arises because the up-type and down-type quark Yukawa matrices need not be diagonal in the same weak interaction basis. In the Standard Model the three left-handed quark doublets $Q_L^{\prime }$ transform as $(3,2,1/6)$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, whereas $u_R^{\prime }$ and $d_R^{\prime }$ transform as $(3,1,2/3)$ and $(3,1,-1/3)$. With a Higgs doublet $H$ and its conjugate $\tilde{H}=i{\sigma }^2{H}^{\ast }$, the gauge-invariant Yukawa interaction isThe generation indices are matrix-multiplied. Electroweak symmetry breaking, $⟨H⟩=(0,v_H/\sqrt{2}{)}^T$, produces $M_u=v_H{Y}_u/\sqrt{2}$ and $M_d=v_H{Y}_d/\sqrt{2}$.

Each complex fermion mass matrix admits a biunitary diagonalization,where the diagonal entries are real nonnegative quark masses. Define $u_L^{\prime }=U_{uL}{u}_L$ and $d_L^{\prime }=U_{dL}{d}_L$, and similarly for right-handed fields. The kinetic terms retain their canonical form because the transformations are unitary matrices. However, the weak charged current becomesThis unitary matrix is the Cabibbo-Kobayashi-Maskawa matrix. Its entries multiply transitions between up-type and down-type mass eigenstates. The minimal Standard Model has no analogous right-handed charged weak current.

Quark mixing does not introduce tree-level flavor-changing neutral currents. Photon, gluon and weak neutral current couplings are proportional to the identity within each fixed-charge quark sector, and $U^{†}IU=I$. The single Higgs doublet also has diagonal neutral scalar field couplings after the same transformations that diagonalize the masses. At higher loop orders, flavor changes are possible. For example, an $s\to d$ amplitude contains ${\sum }_i{V}_{is}^{\ast }{V}_{id}f(m_i^2)$. Unitarity of the CKM matrix sets the sum of coefficients to zero, cancelling any mass-independent part. This is the GIM mechanism; mass splittings leave a suppressed, generally nonzero amplitude.

For $N$ generations a unitary matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up-type and $N$ down-type quark fields removes $2N-1$ phases; a common baryon number phase leaves every matrix entry unchanged. Hence the physical numbers areTwo generations therefore give a real Cabibbo rotation,with strangeness-changing charged-current transitions suppressed by $\sin {\theta }_C$. Three generations have three mixing angles and one irreducible weak CP phase. This counting assumes nondegenerate masses; degeneracies allow additional basis rotations.

Writing $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$, a standard exact parametrization isComplex entries alone do not establish CP violation, because quark quantum field rephasings can change their phases. For nondegenerate quark masses the weak mixing sector conserves CP precisely when a basis makes $V$ real. A useful invariant test is the Jarlskog invariant,A nonzero $J$ requires all three mixing angles and a nontrivial phase. If two same-charge quark masses coincide, the additional basis freedom removes the corresponding physical CP effect. The possible QCD theta angle is a separate CP parameter, not part of this weak-mixing count.

The Wolfenstein parametrization displays the hierarchy of quark mixing economically. Put ${\lambda }_W=s_{12}$, $s_{23}=A{\lambda }_W^2$, and $s_{13}{e}^{i\delta }=A{\lambda }_W^3(\rho +i\eta )$. Expanding the exact matrix givesThe observed pattern is predominantly diagonal, with $1$-$2$ mixing largest and $2$-$3$, $1$-$3$ mixing successively suppressed. The expansion implies $J\simeq A^2{\lambda }_W^6\eta$. The truncated matrix must not be treated as an exactly unitary matrix.

Column orthogonality givesThe three complex numbers form a CKM unitarity triangle, with area $|J|/2$. Dividing by $-V_{cd}{V}_{cb}^{\ast }$ puts its vertices at $0$, $1$ and $\bar{\rho }+i\bar{\eta }$, where $\bar{\rho }+i\bar{\eta }=-V_{ud}{V}_{ub}^{\ast }/(V_{cd}{V}_{cb}^{\ast })$. The barred parameters include higher-order corrections to the Wolfenstein parameters. Comparing independent measurements of its sides and angles tests both quark mixing and the absence of additional flavor dynamics.

Semileptonic decay rates determine magnitudes of CKM entries through factors $|V_{ij}{|}^2$ once hadronic matrix elements are controlled. Neutral kaon and neutral B-meson mixing constrain products of entries through weak box scattering amplitudes. CP asymmetries probe their invariant relative phases, with kaon mixing providing indirect CP violation and decay amplitudes providing direct CP violation. These processes complement tests of the unitarity constraints on rows and columns. The minimal three-generation Standard Model describes weak quark mixing by a unitary matrix with three angles and one physical CP phase; it does not predict the Yukawa couplings or their hierarchy.

The flavour-diagonal weak neutral current coupled to the real Z boson has real vector and axial current couplings. With ${\eta }_C(Z)=-1$, part (e) shows that this tree-level interaction is invariant under CP symmetry, although the simultaneous presence of the Dirac current and axial current generally violates charge conjugation and parity symmetry in quantum field theory separately. This conclusion concerns the tree-level weak neutral current, rather than every interaction or radiative effect in the Standard Model.

The weak charged current couples distinct quark flavours and involves the Cabibbo-Kobayashi-Maskawa matrix:Here $u_{Li}$ and $d_{Lj}$ are left-handed quark fields. A CP symmetry transformation exchanges the two conjugate-field operators; because it acts through a unitary operator, it does not complex-conjugate their numerical coefficients. Equality with the original interaction therefore requires a Cabibbo-Kobayashi-Maskawa matrix that can be made real by quantum field rephasings. With three generations, a physical CP-violating phase remains in general, measured by the Jarlskog invariant. Therefore the W boson interactions can violate CP symmetry. Merely seeing a complex coefficient is insufficient: the phase must survive quantum field rephasing, unlike the real two-generation Cabibbo angle rotation.

A leading short-distance Standard Model contribution to neutral D-meson mixing is a box Feynman diagram with two W bosons and two internal down-type quark lines:

The incoming $c\bar{u}$ pair becomes $u\bar{c}$; the internal labels run over the down quark, strange quark and bottom quark. Each corner is a weak charged current vertex. Four such vertices make the contribution of order $g^4$. Summing the internal flavours produces Cabibbo-Kobayashi-Maskawa matrix factors and the Glashow-Iliopoulos-Maiani mechanism: the flavour-independent term cancels by ${\sum }_i{V}_{ci}^{\ast }{V}_{ui}=0$. This box Feynman diagram is one contribution; long-distance intermediate hadron states can also contribute to neutral D-meson mixing.

The tree-level Feynman diagrams contain an unchanged spectator quark, of up quark flavour and the two possible weak charged current transitions of the charm antiquark:

For ${\bar{D}}^0\to K^{+}{\pi }^{-}$, $\bar{c}\to \bar{s}{W}^{-}$ and $W^{-}\to d\bar{u}$. The spectator quark combines with $\bar{s}$ into the kaon $K^{+}=u\bar{s}$, while $d\bar{u}$ forms the pion ${\pi }^{-}$. The Cabibbo-Kobayashi-Maskawa matrix factor is $V_{cs}{V}_{ud}^{\ast }$.

For ${\bar{D}}^0\to K^{-}{\pi }^{+}$, $\bar{c}\to \bar{d}{W}^{-}$ and $W^{-}\to s\bar{u}$. The spectator quark combines with $\bar{d}$ into the pion ${\pi }^{+}=u\bar{d}$, while $s\bar{u}$ forms the kaon $K^{-}$. The Cabibbo-Kobayashi-Maskawa matrix factor is $V_{cd}{V}_{us}^{\ast }$.

Neglecting neutral D-meson mixing and assuming comparable strong interaction matrix elements, the relative direct decay widths areHere ${\theta }_C$ is the Cabibbo angle. The second process is doubly Cabibbo suppressed: it contains two small Cabibbo suppression factors in its amplitude. The Cabibbo angle estimate assumes similar Quantum chromodynamics matrix elements; it is not an exact rate equality.

The operator $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is the relativistic fermion electric dipole moment operator. Its nonrelativistic limit contains $\mathbf{S}\cdot \mathbf{E}$: spin angular momentum is an axial vector, whereas the electric field is a polar vector. It is consequently odd under parity symmetry in quantum field theory. Both the pseudotensor fermion bilinear and the electromagnetic field tensor are odd under charge conjugation, so their product is even. ThereforeThe CPT theorem then makes it odd under time-reversal symmetry. Such an interaction can arise only from CP violation. The Cabibbo-Kobayashi-Maskawa matrix therefore induces a nonzero Standard Model electric dipole moment at sufficiently high loop order, but its flavor structure and loop suppressions make the result extraordinarily small.

Allowed. A single $t$-channel W boson mediates the weak charged current transitions $u\to d$ and $\bar{c}\to \bar{s}$. The amplitude is proportional to the Cabibbo-Kobayashi-Maskawa matrix product $V_{ud}{V}_{cs}^{\ast }$.

Before diagonalizing quark masses, the charged weak current couples fields in the same weak doublet:The Higgs Yukawa matrices produce mass matrices $M_u$ and $M_d$. Biunitary transformations diagonalize them, with $u_L^{\prime }=U_{uL}{u}_L$ and $d_L^{\prime }=U_{dL}{d}_L$. The charged current therefore contains the Cabibbo-Kobayashi-Maskawa matrixNeutral gauge currents remain flavor diagonal because the same unitary matrix occurs on both sides, and neutral Higgs couplings are diagonal together with each mass matrix.

For $N$ generations, rephasing the quark fields leaves $N(N-1)/2$ mixing angles and $(N-1)(N-2)/2$ irreducible phases, a total of $(N-1{)}^2$ parameters. Three generations are the minimum needed for a physical phase; the observed $3\times 3$ matrix has

Biunitary transformations diagonalize the up- and down-type Yukawa matrices:The neutral Higgs couplings are then diagonal, but the charged weak current becomesA $3\times 3$ unitary matrix has nine real parameters. Rephasing the six quark mass eigenfields removes five phases because their common phase is baryon number. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

The Standard Model has two independent sources of CP violation. Weak CP violation comes from the irreducible phase of the Cabibbo-Kobayashi-Maskawa matrix; its basis-independent measure is the Jarlskog invariant. Strong CP violation is governed bywhich multiplies the Yang-Mills theta term $G_{\mu \nu }^a{\tilde{G}}^{a\mu \nu }$. Quark chiral rephasings shift ${\theta }_{\mathrm{QCD}}$ and the mass-matrix phase oppositely, leaving $\bar{\theta }$ invariant. The CKM phase and $\bar{\theta }$ are otherwise independent parameters: observed weak CP violation does not explain why neutron-electric-dipole bounds require $|\bar{\theta }|\ll 1$. This unexplained smallness is the Strong CP problem.

Use singular value decomposition to choose unitary matrices satisfyingThe neutral currents remain flavour diagonal because the same unitary matrix occurs on both sides of each bilinear. The charged current contains the mismatchand becomes

A general unitary $3$ by $3$ matrix has nine real parameters. Independent rephasings of the six quark fields remove five phases because one common baryon-number phase changes nothing. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

Biunitary diagonalization uses independent left rotations for the up and down Yukawa matrices. Their mismatch is the Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$. For two generations it can be made real and isIn the mass basis,The off-diagonal entries permit charged-current flavour change, for example $W^{+}\to u\bar{s}$ or semileptonic kaon decay; both vanish when ${\theta }_c=0$.

Multiplying individual fields by constant phases changes the apparent phases of interaction coefficients. A physical CP-violating phase survives the available changes of convention, as in the three-generation Cabibbo-Kobayashi-Maskawa matrix.

The up-type and down-type fermion mass matrices need different left-handed transformations represented by unitary matrices to diagonalize them. Their relative transformation appears in the charged weak current as the Cabibbo-Kobayashi-Maskawa matrix. The weak neutral currents remain diagonal in the minimal Standard Model, since the same-charge couplings are generation-independent. Three nondegenerate generations give three mixing angles and one physical weak CP phase after quantum field rephasings.