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An absorptive phase arises from physical intermediate states or rescattering in a decay amplitude. For a pair of CP symmetry conjugate reactions with otherwise symmetric dynamics, this phase is the same in both amplitudes, whereas a weak phase changes sign. Their differing transformation laws allow interference to produce a direct CP asymmetry.

For ${\lambda }_f=(q/p){\bar{A}}_f/A_f$, exchanging both initial flavour and final-state labels defines ${\bar{\lambda }}_{\bar{f}}=(p/q)A_{\bar{f}}/{\bar{A}}_{\bar{f}}$. It equals ${\lambda }_f^{-1}$ exactly when $A_{\bar{f}}/{\bar{A}}_{\bar{f}}=A_f/{\bar{A}}_f$. This holds for a common CP eigenstate final ray. CP symmetry of paired amplitudes alone does not imply it for distinct final states: $p=q$, $A_f={\bar{A}}_{\bar{f}}=2$, ${\bar{A}}_f=A_{\bar{f}}=1$ gives both ratios $1/2$.

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.

In a symmetric initial population of particles and antiparticles, exact CP symmetry equates conjugate decay rates after summing final momenta and helicities. Since parity does not reverse baryon number, this equality prevents a total baryon asymmetry even if charge conjugation alone is violated. Different same-helicity rates can cancel against the opposite-helicity rates.

Under charge conjugation, $\bar{u}{\gamma }^{\mu }(1-{\gamma }^5)d$ becomes $-\bar{d}{\gamma }^{\mu }(1+{\gamma }^5)u$. Parity reverses the chirality again and applies the Lorentz parity matrix. The minus sign cancels that of charge conjugation on the charged vector field. Thus each charged-current operator is exchanged with its Hermitian conjugate under CP symmetry, giving the displayed transformation for the CKM matrix $V$. For nondegenerate quark masses, physical CP conservation means that allowed quark rephasings can make $V$ real. A nonzero Jarlskog invariant obstructs this. A merely complex phase convention is not proof of physical CP violation, and $V=I$ is a CP-conserving counterexample to unconditional noninvariance.

A direct CP asymmetry compares decay widths for a process and its CP symmetry conjugate. Let two decay amplitudes contribute with real nonnegative magnitudes $a_1,a_2$, absorptive phases ${\delta }_i$ and weak phases ${\varphi }_i$. Then $\mathcal{A}=a_1{e}^{i({\delta }_1+{\varphi }_1)}+a_2{e}^{i({\delta }_2+{\varphi }_2)}$ and $\overline{\mathcal{A}}=a_1{e}^{i({\delta }_1-{\varphi }_1)}+a_2{e}^{i({\delta }_2-{\varphi }_2)}$. Expanding the absolute squares givesThus two nonzero contributions and both kinds of relative phase are required in this example. Equal total decay widths under CPT symmetry constrain the sum over channels and do not forbid individual channel asymmetries.

Use the kaon CP eigenstates convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, matching kaon CP mixing parameter. Then $|K_1⟩=(|K^0⟩-|{\bar{K}}^0⟩)/\sqrt{2}$ is CP even, while $|K_2⟩=(|K^0⟩+|{\bar{K}}^0⟩)/\sqrt{2}$ is CP odd. With CPT symmetry and a common small mixing parameter, the physical states can be writtenHere $S,L$ mean short and long mean lifetimes; the kaon CP mixing parameter $ϵ$ depends on the flavor-state phase convention. A positive flavor-exchange convention instead makes the symmetric flavor combination CP even, as explained in exchange-positive CP convention for neutral kaon mixing. Mixing these conventions would reverse the interpretation of the written sums.

Why does the two-pion final state test CP symmetry? A spin-zero kaon decaying into two spin-zero pions has relative orbital angular momentum $\ell =0$ by angular momentum conservation. Two negative intrinsic parities multiply to plus, and $(-1{)}^{\ell }=1$. Charge conjugation exchanges ${\pi }^{+}$ and ${\pi }^{-}$; the $\ell =0$ state is even under this exchange. Hence this two-pion state has CP eigenvalue $+1$. If a decay transition operator $T$ preserves CP symmetry, inserting CP on both sides of $⟨\pi \pi |T|K_2⟩$ changes its sign without changing its value: that decay amplitude must vanish.

Assume for illustration that CP violation occurs only through the state admixture and the decay transition preserves CP symmetry. Put $A_1=⟨\pi \pi |T|K_1⟩$. ThenNeglecting the tiny difference in two-body Lorentz-invariant phase space between the physical masses gives ${\Gamma }_L^{\pi \pi }/{\Gamma }_S^{\pi \pi }\simeq |ϵ{|}^2$. A branching fraction, however, divides a partial decay width by the total decay width. With $\hslash =1$, ${\tau }_i=1/{\Gamma }_i$, soFor chosen values $|ϵ|=0.01$ and ${\tau }_L/{\tau }_S=100$, the branching fraction ratio is $0.01$, whereas the partial decay width ratio is $0.0001$. This is original arithmetic, not a historical fit. The longer observation time amplifies the integrated fraction relative to the instantaneous partial decay rate.

Neutral kaon regeneration is the production of a short-lived component when a neutral kaon beam passes through matter. In the phase convention of kaon CP eigenstates, let the forward transmission amplitudes of $K^0$ and ${\bar{K}}^0$ be $t$ and $\bar{t}$. Expanding $t|K^0⟩+\bar{t}|{\bar{K}}^0⟩$ in the kaon CP eigenstates gives the displayed equation. Different forward scattering amplitudes, including absorption, can therefore generate a two-pion signal even with CP symmetry. Ordinary matter is not a CP-symmetric preparation.

Tungsten regeneration calibrated the signal; scanning its position sampled acceptance. Helium regeneration was insufficient, bag walls lay outside acceptance, vertex distributions excluded collimator regeneration, and hydrogen data cross-checked the effect.^[ref] These controls are central: the question is whether the long-lived component itself reaches two pions, rather than whether material has prepared a fresh short-lived component.

Let $\epsilon =i{\sigma }_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, so ${\epsilon }^{-1}=-\epsilon$. Direct multiplication of the Pauli matrices gives $\epsilon {\sigma }_i^{\ast }\epsilon ={\sigma }_i$, equivalently $\epsilon {\sigma }_i^{\ast }{\epsilon }^{-1}=-{\sigma }_i$. For $U=\exp (i{\theta }_i{\sigma }_i/2)\in SU(2)$ this impliesTherefore ${\varphi }^c=\epsilon {\varphi }^{\ast }$ transforms as the same weak fundamental representation doublet as $\varphi$, while complex conjugation reverses its hypercharge to $-1/2$. This is the Higgs conjugate doublet. The Yukawa coupling product $\bar{L}{\varphi }^c{u}_R$ is an $SU(2)$ singlet, and its hypercharges add toIt is consequently gauge-invariant and can supply an up-quark mass.

In unitary gauge, ${\varphi }^c=((v+H)/\sqrt{2},0{)}^T$. Writing $y_u$ for the full arbitrary coefficient of the Yukawa coupling term, its physical form isA one-generation field rephasing makes the mass positive, and its Higgs boson coupling is $m_u/v$. The printed Yukawa expression has an additional trailing $m$ after $R^{+}$ which is never defined. If it denotes a constant, absorb it into $y_u=f^{+}m$; in the conventional dimensionless Yukawa notation without that factor, $y_u=f^{+}$. It does not affect the gauge-transformation argument.

For three families, use a singular value decomposition of the two mass matrices: $U_{uL}^{†}{M}_u{U}_{uR}=D_u$ and $U_{dL}^{†}{M}_d{U}_{dR}=D_d$, with positive diagonal entries. The original left-handed weak current then becomes a current between mass eigenstates withA complex $3\times 3$ matrix has eighteen real parameters, and $U^{†}U=I$ imposes nine independent real conditions, leaving nine. A generic unitary matrix can be built from three complex plane rotations and three diagonal phases: each plane rotation supplies one angle and one phase. Thus these are three rotation angles and six phases. Rephase each of the six Dirac quark mass eigenfields, applying the same phase to its two chiralities so the diagonal masses stay real. A common phase changes no entry of $V$, leaving five effective rephasings. Removing five of the six phases leaves three angles and one physical phase. This CKM parameter counting assumes generic nondegenerate masses; degeneracies give additional basis freedom.

Exactly massless neutrinos have an arbitrary unitary flavor rotation that changes no mass term. After diagonalizing the charged-lepton mass matrix, choose the neutrino basis with the same left-handed rotation. The apparent lepton charged-current mixing matrix becomes the identity. Thus mixing of exactly massless neutrinos is removable; it is not an analogue of physical CKM mixing.

For the requested CP symmetry calculation, write $j_{ij}^{\mu }={\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)d_j$. The supplied charge conjugation transformations, with the fermion-reordering sign, giveFor $\Gamma ={\gamma }^{\mu }(1-{\gamma }^5)$, use $C({\gamma }^5{)}^T{C}^{-1}={\gamma }^5$ and the given gamma matrix relation to obtainUnder parity, the current acquires the Lorentz parity matrix and $1+{\gamma }^5$ becomes $1-{\gamma }^5$. Under charge conjugation the charged vector also changes sign and becomes its conjugate. The two minus signs cancel, and the parity matrices cancel in the contracted operator. ConsequentlyWith $k=g/(2\sqrt{2})$, the interaction is $-k\sum (V_{ij}{O}_{ij}+V_{ij}^{\ast }{O}_{ij}^{†})$. Thereforewith the transformed density evaluated at $x_P$. This is the CP transformation of a charged quark current.

Physical noninvariance requires a phase that cannot be removed by quark rephasings. The invariant quartetis the Jarlskog invariant, where $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$ in the standard three-angle parametrization. If $J\ne 0$, no allowed row/column phases can make the matrix real, and the charged-current interaction violates CP symmetry. For example all three angles $\pi /4$ with $\delta =\pi /2$ give $J=1/(8\sqrt{2})\ne 0$.

The printed noninvariance claim is conditional on a nonzero physical phase. Arbitrary Yukawa matrices do not force it: $V=I$, or any real CKM matrix, gives a CP-invariant charged-current interaction. The correct statement is generic CP violation when the irreducible phase is nontrivial, as explained by quark rephasing and CP conservation.

Finally a right-handed neutrino has gauge representation $(1,1{)}_0$. Its Lorentz-invariant same-chirality bilinear with the field obtained by charge conjugation is therefore a gauge singlet and permits a Majorana mass term, conventionallyThe factor $1/2$ is the usual convention for a field paired with itself; it may be absorbed into the printed coefficient. A Higgs Yukawa interaction can independently give a Dirac mass term. The Majorana term violates lepton number by two units but preserves the gauge symmetries. Corresponding terms for the other charged Standard Model fermions carry nonzero electric charge, and quarks also carry nontrivial color charge. Thus this gauge-invariant right-handed neutrino Majorana mass is allowed while those charged-fermion analogues are forbidden. Merely introducing a right-handed neutrino permits mass; vanishing mass coefficients would still leave it massless.

The baryon asymmetry is an excess of net baryon number over zero, conveniently measured by ${\eta }_B=(n_B-n_{\bar{B}})/n_{\gamma }$. Ordinary matter contains abundant baryons, whereas no comparable primordial antibaryon population is observed. Large nearby domains of matter and antimatter would produce radiation through particle-antiparticle annihilation at their boundaries. Antiparticles produced in energetic collisions are not evidence for a cosmological antimatter domain. The small nonzero baryon-to-photon ratio inferred from Big Bang nucleosynthesis is independently tested by baryon-density consistency between nucleosynthesis and the microwave background.

Starting with zero net baryon number, the Sakharov conditions require baryon-number violation, C and CP violation, and departure from thermal equilibrium. Conservation of baryon number would keep the initial value zero. C symmetry or CP symmetry would pair baryon production with equal antibaryon production; CP rate cancellation in baryogenesis explains why C violation alone does not suffice. In thermal equilibrium with zero imposed charge chemical potentials, CPT symmetry forbids the required net baryon number. An asymmetric rate in one reaction must not be mistaken for net production after all reverse reactions are included.

A simple example is heavy-boson decay baryogenesis. Take equally many heavy $X$ bosons and their antiparticles, with decay channelsThe symbols in parentheses are branching fractions. The two $X$ channels carry baryon numbers $2/3$ and $-1/3$, so both cannot conserve one assigned baryon number for $X$. Their mean baryon number is $r-1/3$, while the mean for $\bar{X}$ is $-\bar{r}+1/3$. Consequently,where $ϵ$ is the asymmetry per decaying particle. CP violation permits $r\ne \bar{r}$ without violating the equality of total decay widths required by CPT symmetry. Microscopically a direct CP asymmetry requires interference between decay amplitudes with both relative weak phases and absorptive phases; specifying complex couplings alone would not establish unequal partial decay widths.

For radiation domination, a sufficient decay condition is ${\Gamma }_X\lesssim H(T=M_X)=1.66\sqrt{g_{\ast }}{M}_X^2/m_P$. The heavy population can then survive as the equilibrium abundance drops, before decaying while inverse production is suppressed. Its actual abundance differs from its thermal equilibrium value. If subsequent baryon-number washout is negligible and cosmological entropy conservation holds, an initial total abundance $Y_X=(n_X+n_{\bar{X}})/s$ producesHere $s$ is the entropy density. Residual baryon-number washout multiplies this expression by an efficiency factor below one. The example realizes all three Sakharov conditions; it illustrates a mechanism, rather than identifying the actual cosmological source of the baryon asymmetry.

Start with the Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$. The equation for the Dirac adjoint is $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }=0$. Transpose it and multiply by the charge-conjugation matrix C to obtainFor this to be the original Dirac equation multiplied by minus one, the necessary matrix identity isUsing $C^{†}=C^{-1}$, the charge conjugation of the Dirac adjoint follows directly:The final step uses $({\gamma }^0{)}^T{C}^{-1}=-C^{-1}{\gamma }^0$ and $({\gamma }^0{)}^2=1$.

Let $j_V^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$, $j_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$, and $j_L=j_V-j_A$. For anticommuting fields, or normal-ordered operator bilinears, the minus sign in the transformed Dirac adjoint cancels the minus sign from interchanging the two fields. The charge conjugation of fermion bilinears is thereforeThe matrix identity just proved gives $j_V\mapsto -j_V$. Also $C({\gamma }^5{)}^T{C}^{-1}={\gamma }^5$, obtained by transposing its four gamma matrices and reversing their order. Henceso $j_A\mapsto j_A$ and $j_L\mapsto -j_R$, where $j_R=j_V+j_A$. Since the external vector V is odd under charge conjugation, $j_L\cdot V$ becomes $j_R\cdot V$ and is generally not invariant.

For parity symmetry in quantum field theory, write $P^{\mu }{}_{\nu }=\mathrm{diag}(1,-1,-1,-1)$. The given spinor transformation gives $j_V\mapsto P{j}_V$, $j_A\mapsto -P{j}_A$, and $V\mapsto PV$, with arguments evaluated at the reflected point. The contracted interaction again becomes $j_R\cdot V$. Each of parity and charge conjugation interchanges the two chiral contractions; applying both restores $j_L\cdot V$. This particular neutral current interaction is CP invariant, although neither P nor C is separately a symmetry. This conclusion uses the stipulated C-odd vector transformation; it is not a claim that all interactions of the Standard Model preserve CP symmetry.

For time reversal of a Dirac field, the gamma matrix adjoint and transpose identities giveWith $B={\gamma }^{5}C$, anticommutation of ${\gamma }^5$ with every gamma matrix yieldsEquivalently, $\boxed{B({\gamma }^{0\ast },-{\mathbf{\gamma }}^{\ast })B^{-1}=({\gamma }^0,\mathbf{\gamma })}$. Since time reversal is an antiunitary operator, the factor i in ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ must also be conjugated. ThusUsing the stated transformations of the field and its Dirac adjoint, the transformed bilinear consequently isThe external vector has the same component signs. Their contraction is unchanged, proving time reversal of a real neutral chiral vector interaction. A real coupling is understood; an antiunitary operator also conjugates coefficients, and relative complex weak phases need not satisfy this invariance.

A charged weak box contribution to kaon mixing is shown below. The incoming $d\bar{s}$ and outgoing $s\bar{d}$ connect through two charged W bosons and two internal up-type quarks. The black arrows are fermion flow, so the lower line's arrows oppose the momentum of the corresponding antiquark. The blue arrows specify the direction of positive charge carried by the charged propagators. Each vertex is a charged weak current vertex, with the appropriate CKM matrix element; the internal flavors are summed over $u,c,t$.

Use precisely the printed exchange-positive CP convention for neutral kaon mixing. In the flavor basis, CP symmetry acts by $S=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. Write the restricted effective matrix asThen $SMS=\left(\begin{array}{cc}a& c\\ b& a\end{array}\right)$, so CP invariance of this mixing operator is exactly $b=c$. Therefore CP noninvariance in this neutral-kaon mixing block means $M_{12}\ne M_{21}$. Violation elsewhere in a larger Hamiltonian would not, by itself, prove an inequality for this particular block.

For neutral-kaon mass matrix diagonalization, choose correlated roots $r^2=b$, $s^2=c$, and eigenvalues ${\lambda }_{\pm }=a\pm rs$. The vectors $(r,s{)}^T$ and $(r,-s{)}^T$ are the corresponding eigenvectors, as direct multiplication verifies. Taking orthonormal flavor states and writing the printed CP eigenstates as $K_1=(K^0+{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0-{\bar{K}}^0)/\sqrt{2}$, one findswhere the kaon CP mixing parameter isOverall phases have been absorbed. The kaon mixing square-root branch convention must be chosen continuously so that $r=s$ in the CP-conserving limit for the eigenstate labeled plus. These formulas are exact in the nondegenerate case with $r+s\ne 0$; the requested small admixtures follow if $|ϵ|\ll 1$. Smallness is an additional weak-CP-violation assumption, not a consequence of $b\ne c$ alone. A Hermitian mass operator has $c=b^{\ast }$; the same algebra also applies to the effective non-Hermitian operator that includes decays, where the two eigenstates need not be orthogonal. At a fully degenerate $b=c=0$, a CP eigenbasis can simply be chosen.

Define ${\Pi }^{\mu }{}_{\nu }=\mathrm{diag}(1,-1,-1,-1)$ and $x_P=(x^0,-\mathbf{x})$. In a conventional intrinsic-phase choice, the charged W boson transforms under CP symmetry asThus its time component maps to minus the conjugate time component, and its spatial components map to plus the conjugate spatial components. Charge is exchanged, the Lorentz index undergoes parity, and the overall minus sign is a consistent phase convention shared with the charged-current transformation. The analogous formula exchanges $W^{-}$ with $W^{+}$.

An experimentally observed example of CP violation is the long-lived neutral kaon decay $K_L\to {\pi }^{+}{\pi }^{-}$. The two-pion spin-zero final state is CP even, whereas $K_L$ would be the CP-odd neutral-kaon state in the CP-conserving limit. Its two-pion decay reveals CP violation. The historical experimental account is cern-courier.web.cern.ch/a/cp-violations-early-days/ .

For the CKM matrix, diagonalize the up- and down-quark mass matrices independently by biunitary transformations. Writewith $L_u^{†}{M}_u^{\prime }{R}_u$ and $L_d^{†}{M}_d^{\prime }{R}_d$ positive diagonal matrices. Only the left transformations enter the weak charged current, because its chiral projector removes the right-handed fields. Substitution givesThe matrix is unitary but need not be diagonal: the two left mass diagonalizations generally differ.

Using the field transformations in the subparts and reordering the anticommuting Dirac fields, the charged bilinear transforms aswhere the fields on the right are both evaluated at $x_P$. The W boson contributes its second minus sign and Lorentz parity factor, so the product maps into its Hermitian-conjugate product. Crucially CP symmetry acts unitarily: it does not separately complex-conjugate the numerical coefficient $V_{ij}$. Thus the transformed interaction isComparing with the original interaction requires $V_{ij}=V_{ij}^{\ast }$ in this canonical CP convention. The physical condition is quark rephasing and CP conservation:A merely complex-looking matrix in an arbitrary mass-eigenstate phase convention is not itself proof of CP violation.

For CKM parameter counting, a unitary $N\times N$ matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up and $N$ down mass eigenfields removes $2N-1$ phases, since their common phase cancels out of $V$. For generic nondegenerate masses this leavesThere is one physical phase for three families and none for one or two. Degenerate masses would permit additional transformations, so the generic count need not apply unchanged.

For the QCD part of the gauge covariant derivative, define the Hermitian colour matrix ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$ and chooseThis sign convention agrees with the field-strength definition ${\mathcal{F}}_{\mu \nu }=-i[D_{\mu },D_{\nu }]/g_s$ in the source. The kinetic term contains $-g_s\overline{u}{\gamma }^{\mu }{\mathcal{A}}_{\mu }u$. A vector bilinear under CP changes sign, gains the Lorentz parity matrix, and transposes its colour matrix upon exchanging quark and antiquark. Invariance of this interaction therefore requires the CP transformation of a non-Abelian gauge connection:Both the time/spatial distinction and the colour transpose are essential; individual colour components need not all share one charge-conjugation sign.

The same gauge covariant derivative givesApply the transformation above and the chain rule ${\partial }_{\mu }f(x_P)={\Pi }_{\mu }{}^{\alpha }({\partial }_{\alpha }f)(x_P)$. The transpose reverses commutator order, $[X^T,Y^T]=-[X,Y{]}^T$, so the nonlinear term transforms with the same overall minus sign as the derivatives. This proves the CP transformation of non-Abelian field strength:Finally $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$ rewrites the theta density as $2\theta {ϵ}^{\mu \nu \rho \sigma }\mathrm{tr}({\mathcal{F}}_{\mu \nu }{\mathcal{F}}_{\rho \sigma })$. The two minus signs cancel, and transposition followed by cyclicity leaves the colour trace unchanged. But the four parity matrices acting on the Levi-Civita symbol contribute $\det \Pi =-1$. HenceThe density is C even and P odd, therefore CP odd. A generic fixed nonzero theta coefficient violates CP symmetry. This is the source of the Strong CP problem, separate from the charged-current phase in the CKM matrix.

Use the metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and the usual Dirac adjoint $\bar{\psi }={\psi }^{†}{\gamma }^0$.

Field transformation. The mode expansion of a Dirac field isCharge conjugation exchanges particle and antiparticle operators. Choose their phases so thatThe conjugate relations for creation and annihilation operators preserve their canonical anticommutation relations. Thus the transformed expansion is ${\eta }_C$ times the same integral with $a_s{u}^s$ replaced by $b_s{u}^s$ and $b_s^{†}{v}^s$ replaced by $a_s^{†}{v}^s$. On the other hand, transposing the adjoint expansion gives $b_{s}C{\bar{v}}^{sT}{e}^{-ipx}+a_s^{†}C{\bar{u}}^{sT}{e}^{ipx}$. The given spinor identities identify these expressions, proving

Symmetry of the spin matrix. Transposing $C{\gamma }^{\mu T}=-{\gamma }^{\mu }C$ gives ${\gamma }^{\mu }{C}^T=-C^T{\gamma }^{\mu T}$. Substitute ${\gamma }^{\mu T}=-C^{-1}{\gamma }^{\mu }C$ and multiply on the right by $C^{-1}$:The complex four-dimensional Clifford algebra representation generated by the gamma matrices is irreducible; equivalently its sixteen independent gamma products span the full matrix algebra. The Schur lemma therefore gives $C^T{C}^{-1}=zI$. Hence $C^T=zC$. Transposing once more gives $C=z^{2}C$, so $z^2=1$ andThis is the transpose symmetry of a charge-conjugation matrix. The given antisymmetric choice is consistent with this conclusion.

Equation of motion. Taking the adjoint of the free Dirac equation yields $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }=0$. Its transpose, multiplied by $C$, isThe defining matrix relation converts this intoThus the charge conjugation of a Dirac field maps free solutions to free solutions. With an external electromagnetic field, its charge would also reverse; no such background is present here.

Chirality. Define the chiral projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$. The operator $\hat{C}$ acts on the field operators and commutes with these numerical matrices:The transformed component is therefore left-chiral, as requested. There is a useful distinction: the algebraic charge conjugate of a left-chiral spinor is right-chiral. Indeed $C{\gamma }^{5T}{C}^{-1}={\gamma }^5$ and $\overline{P_L\psi }=\bar{\psi }{P}_R$ giveThis is the operator conjugation of a chiral field component; it avoids confusing projection after operator conjugation with charge conjugation of the already projected spinor.

Baryogenesis. Consider equal initial populations of a heavy particle $X$ and its antiparticle, with two baryon-number-violating decay channels carrying different final baryon numbers $b_1,b_2$. Let $r$ and $\bar{r}$ be the branching fractions into a channel and its antiparticle channel. The net baryon number produced per initial particle-antiparticle pair isExact charge conjugation symmetry equates the conjugate branching fractions and makes this vanish. Exact CP symmetry also equates the fully summed conjugate decay rates, because parity reverses momenta and helicities without changing baryon number. Thus both violation of charge conjugation and CP violation are necessary for generating an asymmetry from a symmetric initial state.

Violation of charge conjugation alone can generate a helicity asymmetry while leaving total baryon number zero. For example, with $L,R$ denoting helicities, CP symmetry may enforce $\Gamma (X\to q_L{q}_L)=\Gamma (\bar{X}\to {\bar{q}}_R{\bar{q}}_R)$ and the analogous relation with $L,R$ exchanged, even when same-helicity charge-conjugate rates differ. The CP rate cancellation in baryogenesis means that summing over helicities cancels the baryon asymmetry. These are the symmetry requirements in the Sakharov conditions; baryon-number violation and departure from thermal equilibrium are also needed in the conventional decay mechanism.

Keep the color matrix intact: ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Under parity, invariance of its contraction with the vector current requiresUnder charge conjugation, the fermionic reordering in part (b) transposes the color matrix as well as the spinor matrix. For a fixed color matrix $M$, $\bar{\psi }M{\gamma }^{\mu }\psi$ becomes $-\bar{\psi }{M}^T{\gamma }^{\mu }\psi$. Hence invariance of the interaction requiresThis is not a rule that changes the sign of every individual color component: transposition also acts on the generators of the fundamental representation.

Write the gauge field strength as ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$. Since $[A^T,B^T]=-[A,B{]}^T$, the charge conjugation rule gives ${\mathcal{F}}_{\mu \nu }\mapsto -{\mathcal{F}}_{\mu \nu }^T$. Including the coordinate reflection in CP symmetry,For generators normalized by $\mathrm{Tr}(T^a{T}^b)=\kappa {\delta }^{ab}$, the color contraction equals ${\kappa }^{-1}\mathrm{Tr}({\mathcal{F}}_{\mu \nu }{\mathcal{F}}_{\rho \sigma })$. The two charge-conjugation minus signs cancel, and transposition reverses the order inside the trace without changing it. The Levi-Civita symbol acquires $\det \Lambda =-1$ under the remaining reflection. ThusThe Yang-Mills theta term is CP-odd; a generic fixed nonzero $\theta$ produces CP violation. Setting $\theta =0$ removes this source. Although the density is a total derivative, it can affect the quantum theory through nontrivial topological sectors; the local transformation test is not rendered irrelevant by that fact.

Use $t_{\mu }=(1,-1,-1,-1{)}_{\mu }$ without summing in individual component transformations. The Dirac current transforms asSince the coupling is real, invariance of its contraction fixes the time reversal of an electromagnetic gauge field:These equations use a compatible gauge; an additional pure gauge transformation would not change the gauge field strength.

Coordinate differentiation at $x_T$ introduces $r_{\mu }=-t_{\mu }$, because the time coordinate is reversed and the spatial coordinates are unchanged. HenceThus the electric field is even and the magnetic field is odd. The chirality matrix is even by part (a). The two tensor signs cancel in the dipole contraction, but antiunitarity conjugates its explicit $i$. Therefore the time-reversal parity of a fermion electric dipole operator isIt is a time-reversal symmetry violating interaction, and, under the usual local relativistic quantum field theory hypotheses of the CPT theorem, it violates CP symmetry. A nonzero coefficient is absent from the renormalizable tree-level Standard Model Lagrangian: the broken-phase fermion electric dipole moment operator has mass dimension five, and its electroweak-invariant completion requires a Higgs field and has dimension six. However, it can arise as a radiatively induced effective interaction in the Standard Model, whose CKM matrix contains a CP-violating phase. It is not forbidden to all orders. An explicit primary calculation of quark dipoles is Czarnecki and Krause's Standard Model calculation.

Starting from a matter-antimatter symmetric ensemble, exact charge conjugation would pair every baryon-producing process with an equally probable antibaryon-producing process. Their contributions to the net baryon number cancel. Thus C violation is necessary. Violating charge conjugation alone is insufficient: if CP symmetry is exact, the parity-reflected conjugate channels still have equal probabilities and cancel in a CP symmetry invariant ensemble. One needs CP violation as well.

The remaining Sakharov conditions are baryon number violation and departure from thermal equilibrium. Conserving baryon number cannot turn an initially zero value into a nonzero one. In thermal equilibrium, under the usual CPT theorem hypotheses and with no imposed chemical potentials for charges odd under CPT symmetry, states and their conjugates have equal thermal weights, so the equilibrium expectation of net baryon number vanishes and thermal stationarity prevents its sustained production. A nonequilibrium history allows different forward and reverse populations to generate an asymmetry. These are the standard necessary conditions for dynamical baryogenesis from symmetric initial conditions, not a guarantee of a sufficiently large asymmetry.

Write $J_V^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ for the Dirac current and $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ for the axial current. Under parity symmetry in quantum field theory, the Dirac spinor phases cancel andThe extra sign for the axial current follows from ${\gamma }^0{\gamma }^5{\gamma }^0=-{\gamma }^5$. Combining these transformations with part (c), both currents transform under CP symmetry as $J_{V,A}^{\mu }(x)\mapsto -{\mathbb{P}}^{\mu }{}_{\nu }{J}_{V,A}^{\nu }(x_P)$. The vector relativistic quantum field transforms as $V^{\mu }(x)\mapsto {\eta }_C{\mathbb{P}}^{\mu }{}_{\nu }{V}^{\nu †}(x_P)$. The two parity matrices cancel in the tensor contraction, givingFor a real vector relativistic quantum field with ${\eta }_C=-1$, the interaction is invariant under CP symmetry. A complex vector relativistic quantum field instead maps to the conjugate-field interaction, which must be compared with its Hermitian conjugation partner.