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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 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.

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.

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 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.

Some formulas write $Q=I_3+Y$ rather than $Q=I_3+Y_{\mathrm{usual}}/2$. Then $Y=(B+S)/2$, where $B$ is baryon number and $S$ is strangeness. The light-quark weights are $u:(1/2,1/6)$, $d:(-1/2,1/6)$, $s:(0,-1/3)$. Baryon weight rows therefore have half the usual vertical coordinates. This is a flavour convention, distinct from weak hypercharge.

Consider equal initial abundances of a heavy boson $X$ and its antiparticle. Two decay channels with different baryon numbers, for example $X\to uu$ and $X\to \bar{d}{e}^{+}$, have branching fractions $r$ and $1-r$. The conjugate channels have branching fractions $\bar{r}$ and $1-\bar{r}$. Their mean baryon numbers are $r-1/3$ and $-\bar{r}+1/3$, so each particle-antiparticle pair produces $r-\bar{r}$, or $ϵ$ per decaying particle. Unequal branching fractions require CP violation; equal total decay widths are consistent with CPT symmetry. Baryon number cannot be assigned so that both channels conserve it. An initial abundance decaying outside thermal equilibrium, after inverse production becomes ineffective, can therefore generate a nonzero baryon asymmetry. At the microscopic level a decay-rate difference caused by CP violation requires interference of decay amplitudes with both a relative weak phase and an absorptive phase; complex couplings alone do not guarantee it. Subsequent baryon-number washout must be ineffective for the produced asymmetry to survive.

The gauge-covariant kinetic term for two massless quarks separates into left- and right-handed parts. Independent constant unitary quark flavor rotations preserve these terms, giving classical $U(2{)}_L\times U(2{)}_R$ internal symmetry in addition to local color $SU(3)$. The singlet axial $U(1{)}_A$ has a chiral anomaly, while the nonsinglet chiral symmetry and vector baryon number remain exact quantum symmetries, up to their shared discrete identifications. Equal nonzero quark masses reduce the continuous flavor symmetry to $SU(2{)}_V\times U(1{)}_V$.

Matter parity assigns odd parity to the quark and lepton chiral superfields and even parity to the Higgs chiral doublets and gauge superfields. It is related to component R-parity by $R_p=P_M(-1{)}^{2s}$. The Lorentz-invariant action contains an even number of fermions, so conserved matter parity is equivalent to conserved R-parity. Products of four matter superfields can be even while violating baryon number; matter parity alone does not exclude higher-dimensional proton-decay operators.

Matter parity assigns $-1$ to every matter chiral superfield, including a singlet $N$ describing a conjugate right-handed neutrino, and $+1$ to Higgs chiral superfields. Thus the quadratic superpotential term $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even, although it changes lepton number by two. The higher-degree operators $QQQL$ and $(L{H}_u{)}^2$ are also even and violate baryon number or lepton number. Exact R-parity therefore forbids the renormalizable trilinear violations $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$, but does not imply conservation of these continuous charges in every allowed interaction.

Scattering with one meson and one baryon in the initial and final states. The strong interaction conserves electric charge, baryon number, net strangeness, four-momentum, total angular momentum, and parity. Approximate isospin symmetry imposes additional relations among charge-channel amplitudes.

With hypercharge normalized by $Q_{\mathrm{em}}=T_3+Y$, the renormalizable R-parity-conserving MSSM superpotential is $W=(y_u{)}_{ij}{U}_i^c(Q_j\cdot H_u)+(y_d{)}_{ij}{D}_i^c(Q_j\cdot H_d)+(y_e{)}_{ij}{E}_i^c(L_j\cdot H_d)+\mu H_u\cdot H_d$, with $A\cdot B={ϵ}_{ab}{A}^a{B}^b$. Additional renormalizable R-parity violation allows $\frac{1}{2}{\lambda }_{ijk}(L_i\cdot L_j)E_k^c+{\lambda }_{ijk}^{\prime }(L_i\cdot Q_j)D_k^c+\frac{1}{2}{\lambda }_{ijk}^{\prime \prime }{ϵ}_{abc}{U}_i^{c,a}{D}_j^{c,b}{D}_k^{c,c}+{\kappa }_i{L}_i\cdot H_u$. The two antisymmetries are ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$. The first, second and fourth additional terms violate lepton number; the third violates baryon number.

After cosmological weak freeze-out, the surviving neutron fraction is $x_n=r_f(1+r_f{)}^{-1}{e}^{-\Delta t/{\tau }_n}$, where $r_f$ is the initial neutron-to-proton ratio and ${\tau }_n=t_{1/2}/\ln 2$ is the mean lifetime. Baryon number is conserved because a decaying neutron becomes a proton. If almost all surviving neutrons enter helium-4, neutron-limited helium synthesis gives the displayed primordial helium mass fraction. Increasing the half-life increases survival. If the longer lifetime also represents uniformly weaker weak interactions, it raises the temperature of cosmological weak freeze-out and the initial neutron fraction, reinforcing the helium increase. That last conclusion requires specifying how the weak rates change, rather than treating lifetime alone as a complete microscopic model.

The proton and neutron form the nucleon isospin doublet, with spin angular momentum $1/2$ and baryon number one. A unit Skyrmion with the fermionic Finkelstein-Rubinstein constraints gives these quantum numbers in its lowest rotational multiplet.