CP violation and the CKM matrix

Intuition Beginner

Three operations flip a physical process into a related one. Charge conjugation, written C, swaps every particle for its antiparticle: each positive charge becomes negative, each piece of matter becomes antimatter. Parity, P, reflects the system in a mirror, exchanging left and right. Time reversal, T, plays the film of the process backwards. Each operation alone is almost an exact symmetry of the laws of nature, and the three taken together in the order CPT are exactly conserved in any theory that respects special relativity and locality. The shock is that the combined operation CP, which mirrors the system and turns every particle into its antiparticle at once, is broken, very slightly but unmistakably, in the weak interaction.

This tiny crack matters enormously. A universe that treats matter and antimatter perfectly symmetrically would produce the two in equal amounts in the hot early cosmos, watch them annihilate into light, and leave almost nothing behind. For the matter-dominated universe we actually inhabit to emerge from the Big Bang, the laws of physics had to contain a small bias favouring matter over antimatter. CP violation is that bias. Andrei Sakharov argued in 1967 that such a bias is one of three ingredients any successful account of the origin of matter must contain, and the observed excess of matter over antimorphism demands it.

Where does the bias live? In the Standard Model it hides inside a single three-by-three array of numbers, the Cabibbo-Kobayashi-Maskawa matrix, that records how the six quarks mix when they decay through the weak force. Most of the entries of that matrix can be cleaned up by redefining the phases of the quark fields, but one stubborn complex phase refuses every redefinition. That leftover phase is the only source of CP violation in the quark sector, and the whole of this unit is the story of how to isolate it, measure it, and pin down exactly why it cannot be removed.

Visual Beginner

Picture a triangle drawn in the complex plane. Its three sides are three complex numbers that the rules of quantum mechanics force to close head to tail into a triangle. The interior area of this triangle is the same for every such triangle the matrix produces, and that shared area is a direct, rephasing-proof measure of the leftover CP-violating phase. A flat triangle with zero area would mean CP is a perfect symmetry; a triangle with a real interior means CP is broken.

The figure captures the central quantitative claim of the unit: the size of the triangle's interior is the size of CP violation, and a single number J, twice that area, packages the entire effect into one rephasing-invariant quantity.

Worked example Beginner

Take the decay of a neutral B meson, a bound state of an antibottom quark and a down quark, into a kaon and a pion, and compare it with the antimatter twin of that decay. In one sample a detector records 1000 B mesons decaying into a positive kaon and a pion. In a matched sample it records 850 anti-B mesons decaying into the antimatter copy of those same particles.

Step 1. Compute the asymmetry. The standard measure is the difference of the two counts divided by their combined count: $(1000-850)$ over $(1000+850)$, which is $150$ over $1850$, about $0.081$.

Step 2. Interpret the number. If CP were an exact symmetry, matter and antimatter would decay at identical rates and the asymmetry would read zero. The measured value of roughly $0.08$, close to the real-world figure for this decay channel, is direct evidence that the leftover CKM phase is non-zero.

Step 3. Read off the phase. The asymmetry is proportional to the sine of the complex phase hiding in the mixing matrix. A measured asymmetry near $0.08$ corresponds to a phase whose sine is of that order, consistent with the unitarity-triangle angle measured independently by the B-factory experiments.

What this tells us: a small but definite inequality between a decay and its antimatter twin is the fingerprint of the irreducible CP-violating phase. The size of that inequality tracks the area of the unitarity triangle and the value of the single invariant J that the rest of the unit makes precise.

Check your understanding Beginner

Formal definition Intermediate+

Let $(u,c,t)$ denote the up-type quark mass eigenstates and $(d,s,b)$ the down-type mass eigenstates. The charged weak current couples the left-handed quark doublets through a $3\times 3$ unitary matrix $V$, the Cabibbo-Kobayashi-Maskawa matrix ^{[Kobayashi-Maskawa 1973]}:

$$J_{\mathrm{cc}}^{\mu }=(\bar{u},\bar{c},\bar{t}{)}_L{\gamma }^{\mu }V{\left(\begin{array}{c}d\\ s\\ b\end{array}\right)}_L,V^{†}V=1_3.$$

The neutral current is diagonal in the same basis, so $V$ is the only flavour-mixing structure in the Standard Model quark sector. A unitary $3\times 3$ matrix carries $9$ real parameters, which decompose into $3$ rotation angles and $6$ phases. Each quark field may be freely rephased, $q_i\mapsto e^{i{\alpha }_i}{q}_i$, under which $V_{ij}\mapsto e^{i({\alpha }_i-{\beta }_j)}{V}_{ij}$ (writing ${\alpha }_i$ for the up-type rows and ${\beta }_j$ for the down-type columns). Six such rephasings remove five relative phases — one overall phase cancels against the global $U(1)$ of the Lagrangian — leaving $3$ angles and a single physical complex phase ${\delta }_{\mathrm{CKM}}$. That one phase is the sole source of CP violation in the quark sector.

The standard parametrisation writes $V$ as a product of three rotations, the third of which carries the phase:

$$V=R_{12}({\theta }_{12})R_{23}({\theta }_{23})U_{13}({\theta }_{13},{\delta }_{\mathrm{CKM}}),$$

with $U_{13}$ a complex rotation. The Wolfenstein parametrisation expands $V$ in the small parameter $\lambda =\sin {\theta }_{12}\approx 0.2253$ ^{[PDG 2022]}:

$$V=\left(\begin{array}{ccc}1-\frac{{\lambda }^2}{2}& \lambda & A{\lambda }^3(\rho -i\eta )\\ -\lambda & 1-\frac{{\lambda }^2}{2}& A{\lambda }^2\\ A{\lambda }^3(1-\rho -i\eta )& -A{\lambda }^2& 1\end{array}\right)+O({\lambda }^4),$$

with $A\approx 0.83$, $\rho \approx 0.14$, $\eta \approx 0.35$. Every entry in which $\eta$ appears is complex; $\eta =0$ would make $V$ realisable over $\mathbb{R}$ and CP would be a symmetry.

The orthogonality relations $V{V}^{†}=1$ give, for each distinct pair of columns $j\ne k$, the closure relation ${\sum }_i{V}_{ij}{V}_{ik}^{\ast }=0$. This is a complex equation whose three terms are three complex numbers closing head to tail into a triangle in the complex plane: a unitarity triangle. Six such triangles arise (one for each ordered pair of rows or columns), and unitarity forces every one of them to carry the same signed interior area, equal to half the Jarlskog invariant defined below. The most studied triangle, built from $V_{ud}{V}_{ub}^{\ast }+V_{cd}{V}_{cb}^{\ast }+V_{td}{V}_{tb}^{\ast }=0$, has its three interior angles denoted $\alpha ,\beta ,\gamma$ with $\alpha +\beta +\gamma =\pi$.

The Jarlskog invariant is the rephasing-invariant quartic ^{[Jarlskog 1985]}

$$J=\mathrm{Im}\left(V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast }\right),i\ne k,j\ne l,$$

which takes the same absolute value $|J|\approx 3.0\times 10^{-5}$ for every admissible index choice. The signed area of each unitarity triangle equals $\frac{1}{2}J$, so the triangle is degenerate, with zero area, precisely when $J=0$; in that case $V$ can be made entirely real and CP is conserved.

The discrete operations act on a Dirac field $\psi$ as

$$\hat{C}\psi {\hat{C}}^{-1}={\eta }_{C}C{\bar{\psi }}^T,\hat{P}\psi (t,\mathbf{x}){\hat{P}}^{-1}={\eta }_P{\gamma }^0\psi (t,-\mathbf{x}),\hat{T}\psi (t,\mathbf{x}){\hat{T}}^{-1}={\eta }_T\mathcal{T}\psi (-t,-\mathbf{x}),$$

with $C=i{\gamma }^2{\gamma }^0$, $\mathcal{T}=i{\gamma }^3{\gamma }^1\hat{K}$ (up to representation) and intrinsic phases ${\eta }_C,{\eta }_P,{\eta }_T\in \{\pm 1,\pm i\}$. The operator $\hat{T}$ is antiunitary, treated in 12.15.01; $\hat{C}$ and $\hat{P}$ are unitary. The CPT theorem states that the combined antiunitary $\Theta =\hat{C}\hat{P}\hat{T}$ is a symmetry of every Lorentz-invariant local quantum field theory; because CP is violated in the weak interaction, $\hat{T}$ must be violated to the same degree. This is the bridge from the CKM phase to the time-reversal framework of 12.15.01.

Sakharov's three conditions for a matter-antimatter asymmetry generated dynamically in the early universe ^{[Sakharov 1967]}: (i) baryon number non-conservation, (ii) C and CP violation, and (iii) departure from thermal equilibrium. CP violation is necessary because without it the rate of every baryon-producing process equals the rate of its CP-conjugate and the net baryon number remains zero.

Counterexamples to common slips

• A single complex entry of $V$ does not signal CP violation. The phase of any individual entry is a rephasing convention: rephasing any row or column redistributes the phase among the entries. Only the rephasing-invariant quartic $J$ is physical. Inferring CP violation from "$\mathrm{Im}{V}_{ij}\ne 0$" for one entry is meaningless.
• A $2\times 2$ mixing matrix cannot violate CP. With two generations the single Cabibbo angle ^{[Cabibbo 1963]} exhausts the mixing and every phase rephases away; the 1964 observation of CP violation was therefore evidence that a third generation existed, an inference Kobayashi and Maskawa turned into a prediction of the charm and top quarks.
• The QCD $\theta$-term is a separate CP-violating parameter, unrelated to the CKM phase. The observed smallness of $\theta$ is the strong-CP problem, addressed in the Master tier; conflating it with the CKM phase double-counts the sources of CP violation.

Key theorem with proof Intermediate+

Theorem (Jarlskog's CP-violation criterion, 1985). Let $V\in U(3)$ be the CKM matrix. The following are equivalent:

1. $V$ can be made real by rephasing the quark fields, that is, CP is conserved in the quark sector;
2. the Jarlskog invariant $J=\mathrm{Im}(V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast })$ vanishes.

Moreover, $|J|$ is independent of the index choice $i\ne k$, $j\ne l$, so $J$ is the unique rephasing-invariant measure of the CKM phase, and a single phase ${\delta }_{\mathrm{CKM}}$ satisfies $J=c_{12}{c}_{23}{c}_{13}{s}_{12}{s}_{23}{s}_{13}\sin {\delta }_{\mathrm{CKM}}$ with $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. Here ^{[Jarlskog 1985]}, ^{[PDG 2022]}.

Proof. Rephasing invariance. Under $V_{ij}\mapsto e^{i({\alpha }_i-{\beta }_j)}{V}_{ij}$ the quartic transforms as

$$V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast }\mapsto e^{i({\alpha }_i-{\beta }_j+{\alpha }_k-{\beta }_l-{\alpha }_i+{\beta }_l-{\alpha }_k+{\beta }_j)}{V}_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast }=V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast },$$

the inserted phase being identically zero. Hence $J$ depends on $V$ alone and not on the arbitrary field phases, so it is a genuine physical quantity.

Independence of index choice. Rephase the first row and first column real, which is always possible: $V_{11},V_{12},V_{13},V_{21},V_{31}\in {\mathbb{R}}_{\ge 0}$. The remaining $2\times 2$ submatrix is then fixed by unitarity up to a single complex phase in its lower-right block, and the imaginary part of every $2\times 2$ minor of $V$ reduces to the imaginary part of one and the same entry. Concretely, orthogonality of the first and third columns gives $V_{11}{V}_{13}^{\ast }+V_{21}{V}_{23}^{\ast }+V_{31}{V}_{33}^{\ast }=0$, and since $V_{11}$ and $V_{13}$ are real the imaginary part of the first term vanishes, forcing $\mathrm{Im}(V_{21}{V}_{23}^{\ast })=-\mathrm{Im}(V_{31}{V}_{33}^{\ast })$. Chasing this identity around the matrix shows every quartic $\mathrm{Im}(V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast })$ equals $\pm J$.

Equivalence of (1) and (2). If $V$ is realisable over $\mathbb{R}$, every quartic is real and $J=0$. Conversely, suppose $J=0$. Rephase the first row and column real as above; the four real entries $V_{11},V_{12},V_{21},V_{31}$ together with the unitarity constraints $V{V}^{†}=1$ determine $V_{22},V_{23},V_{32},V_{33}$ up to a common phase in the lower-right $2\times 2$ block. The single imaginary invariant of that block is $J$; with $J=0$ the block is real, so every entry of $V$ is real and $V\in O(3)\subset U(3)$. Thus CP is conserved.

The phase form. Substituting the standard parametrisation into the quartic and using $c_{ij}\cos \delta +is\sin \delta$ expansions of the complex rotation $U_{13}$ yields $J=c_{12}{c}_{23}{c}_{13}{s}_{12}{s}_{23}{s}_{13}\sin {\delta }_{\mathrm{CKM}}$. CP violation vanishes exactly when ${\delta }_{\mathrm{CKM}}\in \{0,\pi \}$, and its magnitude is bounded by the product of sines of the three mixing angles — the geometric origin of the smallness of $|J|\approx 3\times 10^{-5}$. $\square$

Bridge. This criterion builds toward the experimental programme of the Master tier, where the three independent measurements of $J$ and the unitarity-triangle angles come from the neutral-kaon, B-factory, and D-meson systems, and appears again in the CPT theorem of 12.06.04, where the violation of CP forces a compensating violation of $\hat{T}$. The foundational reason a single number packages all CP violation is that the rephasing quotient of a $3\times 3$ unitary matrix by its row-and-column phases is a space of three angles plus one phase, and this is exactly the quotient that $J$ measures. The central insight is that CP violation is a property not of any one decay amplitude but of the closed consistency relations among all of them. Putting these together, the bridge is the recognition that the antiunitary $\hat{T}$ of 12.15.01 and the CKM phase are two faces of one broken discrete symmetry, mediated by CPT, and the whole construction generalises to $N$ generations through the $(N-1)(N-2)/2$ phase count developed in the exercises.

Exercises Intermediate+

Advanced results Master

Theorem (Jarlskog commutator formula, 1985). Let $H_u=M_u{M}_u^{†}$ and $H_d=M_d{M}_d^{†}$ be the Hermitian mass products in a generic weak basis. Then

$$\det [H_u,H_d]=2iJ\prod _{1\le i<j\le 3}(m_{u_i}^2-m_{u_j}^2)\prod _{1\le i<j\le 3}(m_{d_i}^2-m_{d_j}^2),$$

so CP is violated if and only if $J\ne 0$ and no two quarks of the same charge are mass-degenerate ^{[Jarlskog 1985]}. The formula relocates $J$ from the mixing matrix to an invariant of the mass matrices themselves, exposing that CP violation is a joint property of flavour mixing and the quark-mass spectrum. Because the real quark masses are widely separated, the degeneracy factors never vanish, and the entire question of CKM CP violation reduces to whether $J$ is non-zero.

Theorem (unitarity-triangle area). The signed interior area of every non-degenerate unitarity triangle equals $\frac{1}{2}J$, independent of which pair of rows or columns defines the triangle. The six triangles therefore share a common area, and the over-determination (six triangles, one number) is the consistency test the global CKM fit exploits ^[Buras]. The area can be read off directly from any side and opposite angle: $\frac{1}{2}|V_{ij}{V}_{ik}^{\ast }||V_{lj}{V}_{lk}^{\ast }|\sin \varphi$ for the appropriate vertex angle $\varphi$. Because every triangle has the same area, measuring the three angles $\alpha ,\beta ,\gamma$ of the "bd" triangle and checking $\alpha +\beta +\gamma =\pi$ is a direct experimental test of the unitarity of $V$.

Theorem (the three experimental sources). CP violation has been established in three distinct quark systems, each probing a different combination of the CKM phase.

• Neutral kaons. Christenson, Cronin, Fitch, and Turlay (1964) observed the decay $K_L^0\to {\pi }^{+}{\pi }^{-}$, forbidden if CP were conserved ^{[Cronin-Fitch 1964]}. The mixing-decay parameter $|ϵ|\approx 2.23\times 10^{-3}$ measures indirect CP violation in $K^0$-${\bar{K}}^0$ mixing.
• B mesons. The B-factory experiments BaBar and Belle (2001) measured $\sin 2\beta \approx 0.699$ in the decay $B^0\to J/\psi K_S^0$, determining the unitarity-triangle angle $\beta$; later measurements of $B^0\to K^{+}{\pi }^{-}$ established direct CP violation with asymmetry $A_{CP}\approx -0.08$.
• D mesons. The LHCb collaboration (2019) established CP violation in the charm system, measuring $\Delta A_{CP}\approx -1.5\times 10^{-3}$ in $D^0\to {\pi }^{+}{\pi }^{-},K^{+}{K}^{-}$, the first observation of CP violation in up-type quarks.

Theorem (Sakharov conditions and the baryogenesis shortfall). The CP violation supplied by the CKM phase, $|J|\approx 3\times 10^{-5}$, is too small by roughly ten orders of magnitude to generate the observed baryon asymmetry $n_B/s\approx 8.6\times 10^{-11}$ of the universe through Standard Model electroweak baryogenesis ^{[Sakharov 1967]}. The Standard Model satisfies all three Sakharov conditions in principle — baryon number is anomalous, CP is violated, and the electroweak phase transition was a departure from equilibrium — but the single CKM phase combined with the small quark-mass splittings gives a baryon asymmetry of order $10^{-20}$. This shortfall is among the strongest quantitative hints of CP violation beyond the Standard Model, motivating sources in the lepton sector (leptogenesis), extended Higgs sectors, or supersymmetry.

Theorem (the strong-CP problem). QCD admits a CP-violating term ${\mathcal{L}}_{\theta }=\theta \frac{g_s^2}{32{\pi }^2}{G}_{\mu \nu }^a{\tilde{G}}^{a\mu \nu }$ in its Lagrangian, yet the experimental bound on the neutron electric dipole moment, $|d_n|<1.8\times 10^{-26}e\cdot \mathrm{cm}$, constrains the effective angle $\bar{\theta }$ to $|\bar{\theta }|<10^{-10}$ ^{[Peccei-Quinn 1977]}. The parameter $\bar{\theta }=\theta +\arg \det M_q$ is a second, independent source of CP violation in the strong interaction, unrelated to the CKM phase. The absence of any symmetry requiring $\bar{\theta }\approx 0$ is the strong-CP problem; the leading resolution is the Peccei-Quinn mechanism, which promotes $\bar{\theta }$ to a dynamical field relaxed to zero by a spontaneously broken $U(1{)}_{\mathrm{PQ}}$ symmetry, whose pseudo-Nambu-Goldstone boson is the axion. A detection of a non-zero neutron electric dipole moment at forthcoming experiments would signal CP violation from $\bar{\theta }$ or beyond-the-Standard-Model sources rather than from the CKM phase.

Theorem (global CKM fit). The four Wolfenstein parameters $(\lambda ,A,\rho ,\eta )$ are over-determined by more than a dozen independent measurements — charm and top decays, ${ϵ}_K$, $\sin 2\beta$, $|V_{ub}|/|V_{cb}|$, $B_s$ mixing, and more — and the global fits of the CKMfitter and UTfit collaborations converge on a self-consistent unitarity triangle, with no significant tension at present precision ^{[PDG 2022]}. The agreement is a substantive consistency check of the Standard Model: had any of the six unitarity triangles failed to share the common area $J/2$, the CKM picture of CP violation would be incomplete and new-physics contributions to mixing or rare decays would be indicated.

Synthesis. The single rephasing-invariant $J$ is the foundational reason every result in this unit coheres, and putting these together they form one structure rather than a list. The central insight is that CP violation is a property of the closed orthogonality relations of $V$, not of any single decay, and this is exactly what the unitarity triangles visualise as a shared interior area $J/2$. The Jarlskog commutator formula identifies the analyst's $J$ with the algebraist's invariant $\det [H_u,H_d]$, tying CP violation to the non-degeneracy of the quark-mass spectrum, so the same number that governs $K^0$-${\bar{K}}^0$ mixing also controls baryogenesis. The bridge is that the CKM phase, the antiunitary $\hat{T}$ of 12.15.01, and the parity framework of 12.15.02 are three faces of one broken discrete symmetry mediated by CPT, and this pattern generalises to $N$ generations through the $(N-1)(N-2)/2$ count that forces CP violation to require at least three quark doublets. The shortfall between the CKM-driven baryon asymmetry and the observed one, together with the unrelated smallness of $\bar{\theta }$, points to additional CP-violating phases beyond the Standard Model.

Full proof set Master

Proposition 1 (rephasing invariance of $J$). The quartic $V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast }$ is invariant under $V_{ij}\mapsto e^{i({\alpha }_i-{\beta }_j)}{V}_{ij}$.

Proof. The four factors acquire phases $e^{i({\alpha }_i-{\beta }_j)}$, $e^{i({\alpha }_k-{\beta }_l)}$, $e^{-i({\alpha }_i-{\beta }_l)}$, $e^{-i({\alpha }_k-{\beta }_j)}$ respectively, whose product is $\exp i[({\alpha }_i-{\beta }_j)+({\alpha }_k-{\beta }_l)-({\alpha }_i-{\beta }_l)-({\alpha }_k-{\beta }_j)]=\exp (0)=1$. The quartic is unchanged and so is its imaginary part $J$. $\square$

Proposition 2 (signed area of a unitarity triangle). The triangle formed by the three complex numbers $V_{i_{0}j}{V}_{i_{0}k}^{\ast }$, $i_0=1,2,3$, for fixed distinct columns $j,k$, has signed area $\frac{1}{2}J$.

Proof. The closure relation ${\sum }_{i_0}{V}_{i_{0}j}{V}_{i_{0}k}^{\ast }=0$ places the three complex numbers $z_{i_0}:=V_{i_{0}j}{V}_{i_{0}k}^{\ast }$ head to tail in the complex plane. The signed area of the triangle with vertices $0$, $z_1$, $z_1+z_2$ is $\frac{1}{2}\mathrm{Im}({\bar{z}}_1{z}_2)=\frac{1}{2}\mathrm{Im}(V_{1j}^{\ast }{V}_{1k}{V}_{2j}{V}_{2k}^{\ast })$. Relabelling rows and using the index-independence established in the Key theorem, this equals $\pm \frac{1}{2}J$; choosing the orientation set by $(i,k)=(1,2)$, $(j,l)=(j,k)$ gives area $\frac{1}{2}J$ for every triangle. $\square$

Proposition 3 (counting: CP violation requires $N\ge 3$ generations). An $N\times N$ unitary quark-mixing matrix carries $(N-1)(N-2)/2$ physical phases, so CP violation is impossible for $N<3$.

Proof. The group $U(N)$ has dimension $N^2$, decomposing into $N(N-1)/2$ rotation angles and $N(N+1)/2$ phases. Rephasing the $2N$ quark fields acts by $V_{ij}\mapsto e^{i({\alpha }_i-{\beta }_j)}{V}_{ij}$; the $2N$ phases ${\alpha }_i,{\beta }_j$ generate a $(2N-1)$-dimensional orbit (one overall phase acts as the identity), removing $2N-1$ phases. The physical phase count is $N(N+1)/2-(2N-1)=(N^2+N-4N+2)/2=(N^2-3N+2)/2=(N-1)(N-2)/2$. For $N=1$ and $N=2$ this is zero; for $N=3$ it is one. Hence a CP-violating phase first appears at three generations, the inference Kobayashi and Maskawa drew from the 1964 kaon result. $\square$

Proposition 4 (Jarlskog commutator formula, sketch). With $H_u=M_u{M}_u^{†}$ and $H_d=M_d{M}_d^{†}$, $\det [H_u,H_d]=2iJ{\prod }_{i<j}(m_{u_i}^2-m_{u_j}^2){\prod }_{i<j}(m_{d_i}^2-m_{d_j}^2)$.

Proof. Diagonalise $H_u=U_u{D}_u{U}_u^{†}$ and $H_d=U_d{D}_d{U}_d^{†}$ with $D_u=\mathrm{diag}(m_{u_i}^2)$, $D_d=\mathrm{diag}(m_{d_i}^2)$. The commutator $[H_u,H_d]=U_u[D_u,U_u^{†}{U}_d{D}_d{U}_d^{†}{U}_u]U_u^{†}$ depends only on the relative unitary $V_{\mathrm{diag}}=U_u^{†}{U}_d$, which is the CKM matrix in the diagonal basis. Expanding $\det [H_u,H_d]$ in the entries of $V_{\mathrm{diag}}$ and using the antisymmetry of the commutator together with $[D_u,D_d{]}_{ij}=(m_{u_i}^2-m_{u_j}^2)(D_d{)}_{ij}$, every term carries one factor $(m_{u_i}^2-m_{u_j}^2)$ and one factor $(m_{d_i}^2-m_{d_j}^2)$; the antisymmetric product over pairs yields the double Vandermonde determinant ${\prod }_{i<j}(m_{u_i}^2-m_{u_j}^2){\prod }_{i<j}(m_{d_i}^2-m_{d_j}^2)$, and the sole rephasing-invariant imaginary part of $V_{\mathrm{diag}}$ is $J$. Collecting the factors of $2$ and $i$ from the commutator expansion gives the stated formula. The full derivation appears in Jarlskog Phys. Rev. Lett. 55 (1985) ^{[Jarlskog 1985]}. $\square$

Connections Master

• Time-reversal symmetry and Kramers' degeneracy 12.15.01. The CPT theorem ties CP violation directly to the antiunitary $\hat{T}$ developed there: because $\Theta =\hat{C}\hat{P}\hat{T}$ is an exact symmetry, the CKM phase forces $\hat{T}$ to be violated to the same degree. The antiunitary framework (Wigner's structure theorem $\hat{T}=\hat{U}\hat{K}$, the square classification ${\hat{T}}^2=(-1{)}^{2j}$) is the prerequisite operator language for the discrete transformations on Dirac fields used here, and the orthogonal-pairing structure of Kramers degeneracy reappears in the orthogonality relations $V{V}^{†}=1$ that build the unitarity triangles.

• Parity, discrete-symmetry groups, and the Wigner-Eckart theorem 12.15.02. The parity operator $\hat{P}$ and its selection rules are half of the CP composite; the Laporte rule and the $Wu$ et al. ${}^{60}$Co measurement of maximal parity violation frame why each discrete operation is broken individually before their combination carries the residual CKM phase. The Wigner-Eckart factorisation of matrix elements into geometric and reduced parts is the tool that isolates the phase-dependent from the phase-independent contributions in CP-asymmetric decay amplitudes.

• The Standard Model 12.21.01. The CKM matrix is the only flavour-mixing structure of the Standard Model quark sector, and the single phase ${\delta }_{\mathrm{CKM}}$ is its sole source of CP violation. The global CKM fit — six unitarity triangles constrained to a shared area $J/2$ — is a consistency test of the Standard Model at the precision frontier, and the baryogenesis shortfall is a quantitative pointer to physics beyond it.

• Crossing symmetry and the CPT theorem at the $S$-matrix level 12.06.04. The CPT theorem that mediates between CP and $\hat{T}$ is established there at the level of the $S$-matrix and in/out states; the unitarity of $V$ used throughout this unit is the flavour-sector shadow of the optical-theorem unitarity of the full $S$-matrix. The crossing relations underlie the extraction of CKM elements from scattering and decay data.

• Dirac equation and relativistic spin 12.11.01. The action of $\hat{C}$, $\hat{P}$, $\hat{T}$ on the four-component Dirac spinor, with $C=i{\gamma }^2{\gamma }^0$ and $\mathcal{T}=i{\gamma }^3{\gamma }^1\hat{K}$, is defined on the relativistic field introduced there; the left-handed projector $(1-{\gamma }^5)/2$ that selects the doublets entering the charged current $J_{\mathrm{cc}}^{\mu }$ is the chirality structure that makes the mixing matrix $V$ unitary on the left-handed subspace alone.

Historical & philosophical context Master

The discovery of CP violation came before its mechanism. James Cronin and Val Fitch, with James Christenson and René Turlay, observed in 1964 at Brookhaven that the long-lived neutral kaon $K_L^0$, believed to be the CP-odd eigenstate and hence forbidden to decay into the CP-even two-pion final state, in fact decays to ${\pi }^{+}{\pi }^{-}$ at a branching ratio of about $2\times 10^{-3}$ ^{[Cronin-Fitch 1964]}. The result was a surprise: parity had already fallen in 1957 (Wu et al.), and it was assumed that CP would restore the matter-antimatter symmetry of the weak interaction at the composite level. The Cronin-Fitch measurement shattered that assumption and earned the 1980 Nobel Prize.

The theoretical response was the Kobayashi-Maskawa paper of 1973 ^{[Kobayashi-Maskawa 1973]}. Makoto Kobayashi and Toshihide Maskawa observed that CP violation is impossible with fewer than three quark generations — a two-by-two unitary mixing matrix, as in Nicola Cabibbo's 1963 parametrisation ^{[Cabibbo 1963]}, has no physical phase — and proposed the three-by-three mixing matrix carrying a single irreducible phase that bears their initials. At the time only the $u$, $d$, $s$ quarks were known; the charm quark would be discovered in 1974 and the bottom in 1977, and the top quark — required by the six-quark picture — only in 1995. The CKM matrix was therefore a genuine prediction, extending Cabibbo's two-angle scheme to three angles and one phase in anticipation of particles not yet observed.

The invariant $J$ was isolated by Cecilia Jarlskog in 1985, who showed that every rephasing of the quark fields leaves one and the same quantity $\mathrm{Im}(V_{ij}{V}_{kl}{V}_{il}^{\ast }{V}_{kj}^{\ast })$ invariant and that this quantity is the unique necessary-and-sufficient measure of CP violation, equivalent to the commutator $\det [H_u,H_d]$ of the mass matrices ^{[Jarlskog 1985]}. The B-factory era opened in 2001 when the BaBar (SLAC) and Belle (KEK) collaborations independently measured $\sin 2\beta$ and confirmed the CKM picture of CP violation in the $B^0$ system; charm-sector CP violation was established by LHCb in 2019, completing the three-system programme. Andrei Sakharov's 1967 conditions for baryogenesis ^{[Sakharov 1967]} and the Peccei-Quinn proposal of 1977 for the strong-CP problem ^{[Peccei-Quinn 1977]} frame the modern open questions: the CKM phase is too small to account for the cosmic baryon asymmetry, and the unrelated smallness of the QCD $\bar{\theta }$ angle remains unexplained.

Bibliography Master

@article{KobayashiMaskawa1973,
  author  = {Kobayashi, Makoto and Maskawa, Toshihide},
  title   = {CP-Violation in the Renormalizable Theory of Weak Interaction},
  journal = {Progress of Theoretical Physics},
  volume  = {49},
  year    = {1973},
  pages   = {652--657}
}

@article{Jarlskog1985,
  author  = {Jarlskog, Cecilia},
  title   = {Commutator of the Quark Mass Matrices and the {K}obayashi-{M}askawa Matrix},
  journal = {Physical Review Letters},
  volume  = {55},
  year    = {1985},
  pages   = {1039--1042}
}

@article{ChristensonCroninFitchTurlay1964,
  author  = {Christenson, James H. and Cronin, James W. and Fitch, Val L. and Turlay, Ren\'e},
  title   = {Evidence for the $2\pi$ Decay of the $K_2^0$ Meson},
  journal = {Physical Review Letters},
  volume  = {13},
  year    = {1964},
  pages   = {138--140}
}

@article{Cabibbo1963,
  author  = {Cabibbo, Nicola},
  title   = {Unitary Symmetry and Leptonic Decays},
  journal = {Physical Review Letters},
  volume  = {10},
  year    = {1963},
  pages   = {531--533}
}

@article{Sakharov1967,
  author  = {Sakharov, Andrei D.},
  title   = {Violation of {CP} Invariance, {C} Asymmetry, and Baryon Asymmetry of the Universe},
  journal = {Journal of Experimental and Theoretical Physics Letters},
  volume  = {5},
  year    = {1967},
  pages   = {24--27}
}

@article{PecceiQuinn1977,
  author  = {Peccei, Roberto D. and Quinn, Helen R.},
  title   = {{CP} Conservation in the Presence of Pseudoparticles},
  journal = {Physical Review Letters},
  volume  = {38},
  year    = {1977},
  pages   = {1440--1443}
}

@article{Wolfenstein1983,
  author  = {Wolfenstein, Lin},
  title   = {Parametrization of the {K}obayashi-{M}askawa Matrix},
  journal = {Physical Review Letters},
  volume  = {51},
  year    = {1983},
  pages   = {1945--1947}
}

@article{PDG2022,
  author  = {{Particle Data Group} and Workman, R. L. and others},
  title   = {Review of Particle Physics},
  journal = {Progress of Theoretical and Experimental Physics},
  volume  = {2022},
  year    = {2022},
  pages   = {083C01},
  note    = {Sections 12 (CKM quark-mixing matrix) and 14 (CP violation in quark and lepton systems)}
}

@article{Buras1998,
  author  = {Buras, Andrzej J.},
  title   = {Weak Hamiltonian, {CP} violation and rare decays},
  journal = {arXiv preprint},
  year    = {1998},
  eprint  = {hep-ph/9806471},
  note    = {Lectures at the Les Houches Summer School; canonical operator-product-expansion treatment of the weak Hamiltonian and the unitarity-triangle fit}
}

@book{BigiSanda2009,
  author    = {Bigi, Ikaros I. Y. and Sanda, Akira I.},
  title     = {CP Violation},
  publisher = {Cambridge University Press},
  series    = {Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology},
  year      = {2009},
  edition   = {2}
}

@article{AbeBelle2001,
  author  = {Abe, K. and others},
  title   = {Observation of Large {CP} Violation in the {B} Meson System},
  journal = {Physical Review Letters},
  volume  = {87},
  year    = {2001},
  pages   = {091802},
  note    = {Belle Collaboration}
}

@article{BABAR2001,
  author  = {Aubert, B. and others},
  title   = {Measurement of {CP}-violating Asymmetries in {$B^0$} Decays to {CP} Eigenstates},
  journal = {Physical Review Letters},
  volume  = {86},
  year    = {2001},
  pages   = {2515--2522},
  note    = {{BaBar} Collaboration}
}