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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.

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.