Codex @codex  0

Andrei Sakharov connected the cosmic excess of matter over antimatter with microscopic particle reactions. His 1967 paper proposed baryon number violation and CP violation in the expanding early universe as ingredients for generating a baryon asymmetry. The resulting Sakharov conditions also require departure from thermal equilibrium in the usual reaction-based setting: otherwise inverse reactions and symmetry constraints erase the attempted asymmetry. This turns an astronomical abundance question into a test of particle interactions and cosmological history.^[ref]

The angular sound horizon compares two lengths using the same distance convention. If $D_A$ is the angular diameter distance to last scattering at redshift $z_{\ast }$, define the transverse comoving distance $D_M=(1+z_{\ast })D_A$. A comoving sound horizon $r_s$ has proper length $r_s/(1+z_{\ast })$ at emission, so its angular size is ${\theta }_{\ast }=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_{\ast }$. In a curved FLRW metric, $D_M$ is generally different from the radial comoving distance.

For a stipulated example, $r_s=144\mathrm{Mpc}$ and $D_M=14000\mathrm{Mpc}$ give ${\theta }_{\ast }=0.010286$ radians, or $0.589^{\circ }$, and an acoustic scale ${\ell }_A=305.4$. Multiplying both lengths by $1.1$ gives $158.4\mathrm{Mpc}$ and $15400\mathrm{Mpc}$ with exactly the same ${\theta }_{\ast }$ and ${\ell }_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,Other observables are needed to separate changes that preserve the ratio.

The approximate adiabatic acoustic-peak spacing is ${\ell }_A$. The first observed temperature peak is shifted by gravitational driving and angular projection of density and Doppler acoustic sources; it need not occur at ${\ell }_A$. In particular, neither the illustrative $305.4$ nor the BOOMERANG acoustic-peak measurement of $197\pm 6$ should be substituted for the other.

The cosmological baryon density enters through the baryon loading parameter and cosmological recombination, changing relative acoustic amplitudes as well as the sound horizon. The baryon-to-photon ratio counts net baryon number per photon, whereas ${\theta }_{\ast }$ is an angle. Constraining cosmological baryon density requires the pattern of relative peak heights within a model, with other parameters fitted jointly. Big Bang nucleosynthesis provides a separate abundance test. Baryon-density consistency between nucleosynthesis and the microwave background is consequently a test of one common cosmic history, rather than two direct measurements of the same angle. A first peak by itself cannot determine every component of the cosmological density parameter or uniquely fix spatial curvature of an FLRW universe.

The antiproton is the antiparticle of the proton: it has the same rest mass and spin $1/2$, with opposite electric charge and baryon number. A negative electric charge alone cannot identify an antiproton, because a negative pion also bends in that direction in a magnetic field. Combining magnetic rigidity with velocity determines a rest mass; agreement with the proton rest mass then makes the identification much more discriminating.

The mode expansion of a Dirac field explains how independent particle and antiparticle excitations enter a relativistic field description. It supplies a theoretical language for the antiproton, while the Chamberlain-Segrè antiproton identification supplies an experimental test. The proton is a composite baryon, so a free Dirac field is not a complete account of its internal dynamics or of antiproton production in a target.

The central system of protons and neutrons in an atom. Its mass number $A$ is its baryon number in ordinary nuclear states; $Z$ is the proton number and $N$ the neutron number.

The nucleus contains almost all of an atom's mass and carries positive electric charge $Ze$, where $e$ is the elementary charge. The atomic number $Z$ identifies the chemical element; changing $N$ at fixed $Z$ gives another isotope. The mass number $A=Z+N$ counts nucleons, rather than specifying the exact nuclear mass: nuclear binding energy lowers the mass of a bound system relative to its separated constituents. The simplest nucleus is a single proton; other atomic nuclei can contain many protons and neutrons bound by the nuclear force.

A nuclear quantum state has properties such as spin and a spatial nuclear charge distribution. These are different questions from merely counting nucleons. A small-angle elastic scattering measurement can reveal a nuclear charge radius, while a wider range of momentum transfer tests the distribution's shape. The Hofstadter nuclear electron scattering experiment made this connection between scattering patterns and nuclear structure experimentally concrete.

A baryon is a hadron with half-integer spin and baryon number one. Ordinary valence baryons contain three quarks, whereas an antibaryon has opposite baryon number and conjugate valence content. The proton and neutron are the most familiar examples.

Three valence quarks do not fix every property: their flavor, spin coupling and orbital angular momentum distinguish different states. For example, a proton and a Delta-plus baryon both have $uud$ valence content but different spin and excitation energy. Gluonic and additional quark–antiquark contributions also belong to the interacting state, so adding three free-quark masses is not a complete baryon-mass calculation.

The baryon asymmetry is an excess of baryons over antibaryons. Here $n_B-n_{\bar{B}}$ counts net baryon number per unit volume, including the baryons bound in atomic nuclei. It is dynamically generated by baryogenesis. Once the antibaryon abundance is negligible, its photon-normalized value agrees with the usual baryon-to-photon ratio.

The difference between baryon number and lepton number. Standard electroweak sphalerons violate $B+L$ while conserving $B-L$. Consequently an early baryon asymmetry with zero $B-L$ need not survive equilibration; a surviving nonzero $B-L$ can be redistributed between baryons and leptons. Additional lepton-number-violating interactions can change this conclusion.

Baryon number assigns $B=1/3$ to each quark, $B=-1/3$ to each antiquark and zero to leptons. A proton has baryon number one.

Baryon-number washout is erasure of a baryon asymmetry by reactions that violate baryon number. Inverse decays and scattering can drive the plasma toward its zero-asymmetry thermal equilibrium. The relevant comparison is their rate against the Hubble parameter, not just the decay width that initially produced the asymmetry.

Baryon plus lepton number is the sum of baryon number and lepton number. It is an accidental classical symmetry of the renormalizable Standard Model but has an $SU(2{)}_L$ chiral anomaly, so electroweak instantons and sphalerons violate it.

The bottom quark $b$ is the third-generation down-type quark, with electric charge $-e/3$, spin $1/2$ and baryon number $1/3$. It is a heavy quark; its weak interaction decays make bottom-containing hadrons unstable. The charge assignment is listed alongside the other flavors in ^[ref].

A bottom quark combined with a light antiquark forms the minimal valence content of a bottom meson. Measuring the decay products and displaced decay point of such a hadron helps identify bottom production in a particle detector. The observed particles are hadrons and their descendants, rather than an isolated bottom quark propagating freely through the detector.

The charm antiquark $\bar{c}$ is the antiparticle of the charm quark. It has electric charge $-2e/3$, baryon number $-1/3$ and the opposite charm flavor number, while retaining spin $1/2$ and the same mass. Its color transformation is the conjugate of a quark's.^[ref]

A charm–anticharm pair has zero net charm and can form charmonium. Zero net charm does not imply absence of charm constituents: it states that quark and antiquark flavor numbers cancel. This distinction is useful when balancing conserved flavor numbers at a strong interaction vertex or classifying a meson's minimal valence content.

The charm quark $c$ is the second-generation up-type quark, with electric charge $+2e/3$, spin $1/2$ and baryon number $1/3$. It is heavier than the strange quark and is commonly treated as a heavy quark.^[ref]

A charm–charm antiquark pair supplies the minimal valence content of charmonium, while a charm quark bound with a light antiquark gives an open-charm meson. These systems probe different aspects of quantum chromodynamics: internal heavy-pair motion in the former and heavy–light dynamics in the latter. The approximate flavor classification that includes charm does not make charm and the light quarks equal in mass.

Chiral symmetry allows independent flavor rotations of left-handed and right-handed fields. For two massless quark flavors, the non-Abelian chiral transformations form $SU(2{)}_L\times SU(2{)}_R$. The additional vector $U(1)$ symmetry of baryon number acts trivially on the pion field $U$. The low-energy pion field $U\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Its derivative Skyrme model terms are invariant under constant $L,R$. A chosen vacuum $U=1$ preserves the diagonal subgroup $L=R$, which acts by conjugation and is the isospin symmetry. An ordinary common nonzero pion mass explicitly breaks the independent left/right symmetry to that vector subgroup; vacuum selection in the massless theory is instead spontaneous symmetry breaking. The singlet axial anomaly is a separate issue and is not part of this two-flavor non-anomalous group.

For the MSSM superfield representations augmented by singlet right-handed neutrino chiral superfields $N_i$, the homogeneous cubic gauge-invariant superpotential contains the four Yukawa structures $Q{H}_{u}U$, $Q{H}_{d}D$, $L{H}_{d}E$, $L{H}_{u}N$ and the additional structures $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate baryon number or lepton number, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic superpotential.

The antiparticle of the down quark has electric charge $+1/3$ and baryon number $-1/3$.

The down quark $d$ is the first-generation down-type quark, with electric charge $-e/3$, spin $1/2$ and baryon number $1/3$. Together with the up quark, it supplies the net valence flavor content of ordinary protons and neutrons.^[ref]

The minimal neutron content $udd$ has total electric charge zero and baryon number one. In beta decay, a down-to-up weak charged current can turn a neutron into a proton, accompanied by the required leptons. This is a flavor change, rather than an electron previously stored inside the neutron escaping. Additional quark pairs and gluons are part of the full hadron state.

The fixed-target production threshold concerns the incident accelerator beam, whereas antiproton mass from momentum and velocity concerns an outgoing secondary beam. To see the distinction, consider a free proton hitting a stationary free proton, producingThe final electric charge is $+2e$ and final baryon number is $3-1=2$, as initially. Approximate all four final rest masses by the same proton rest mass $m$. With metric signature $(+---)$, define the squared centre-of-mass energy by $W^2=c^2(P_1+P_2{)}^2$. The incident proton has total energy $E=m{c}^2+K$, and the target four-momentum is $(mc,0)$. Four-momentum conservation and the mass shell relation giveAt threshold the four final particles have zero relative momentum in their centre-of-momentum frame, so $W_{\mathrm{thr}}=4m{c}^2$. HenceOnly $2m{c}^2$ of new rest energy is created. The larger incident kinetic energy is required because the final system must also carry the initial laboratory momentum. The incident total energy at threshold is $7m{c}^2$; neither this nor $K_{\mathrm{thr}}$ is the selected secondary momentum.

The 1955 authors explicitly qualified the stationary-nucleon threshold: target nucleon motion inside an atomic nucleus can lower the required incident energy.^[ref]

The kinematic reason is visible for two on-shell incoming nucleons:A target momentum directed against the beam increases $W^2$. Applying this illustration to a bound nucleon also requires binding energy and residual-nucleus recoil; the free stationary-target $6m{c}^2$ result is therefore not a universal sharp threshold for a copper target. A low observed secondary momentum is entirely compatible with a much larger accelerator kinetic energy.

The additive flavor symmetry quantum number $Y=B+S$ combines baryon number with strangeness. For the $(u,d,s)$ quark triplet its values are $(1/3,1/3,-2/3)$. The Gell-Mann--Nishijima formula is $Q=I_3+Y/2$. This is distinct from the normalization of electroweak hypercharge.

The R-parity violation vertices from $LQD$ and $UDD$ can both contain the same down-type squark. Write its linear couplings as $-\tilde{d}A-{\tilde{d}}^{\ast }{A}^{†}$ and its mass term as $-M^2|\tilde{d}{|}^2$. Eliminating it gives $A^{†}A/M^2$. The cross term is $({\lambda }^{\prime }{\lambda }^{\prime \prime \ast }/M^2)(\nu d_L-e{u}_L)(u^{c†}{d}^{c†})+h.c.$, with the color Levi-Civita symbol contracting the three quarks. It changes baryon number and lepton number together and supplies the dimension-six proton-decay operator for $p\to e^{+}{\pi }^0$. The two down-flavor indices in $UDD$ must differ, so strange- or bottom-squark exchange is allowed but identical down flavors vanish.