Standard Model fermion

A Standard Model generation contains left-handed quark and lepton doublets and right-handed up-quark, down-quark and charged-lepton singlets. A right-handed neutrino may be added as a gauge singlet.

A generation groups corresponding quarks and leptons with the same gauge quantum numbers as their counterparts in other generations. The Standard Model has three such families. Yukawa matrices assign their different masses, and the mismatch of up-type and down-type mass bases produces the CKM matrix.

The three quark generations pair $(u,d)$, $(c,s)$ and $(t,b)$, each containing one charge-$2/3$ and one charge-$-1/3$ quark. The number of generations controls the number of physical mixing angles and CP-violating phases in the CKM matrix.

A lepton is a Standard Model fermion without color charge. The charged leptons have electric charge $-1$ and their corresponding neutrinos have charge zero. There are three generations; their antiparticles carry opposite electric charge and lepton number. Charged leptons and neutrinos have lepton number $+1$, with the corresponding antiparticles carrying $-1$.

The tau is the charged lepton of the third fermion generation. In the massless approximation its charged-current W boson partial decay width equals those of the Electron and muon, reflecting lepton universality.

The muon is the negatively charged second-generation lepton, with electric charge $-1$ and lepton number $+1$. Its antiparticle is the antimuon.

A weak charged current decay described at energies much below the W boson mass by a four-fermion interaction with coefficient given by the Fermi constant. At leading order with negligible final-state masses, its width is $G_F^2{m}_{\mu }^5/(192{\pi }^3)$.

For a negative muon with rest-frame spin vector $\mathbf{s}$, the leading massless-electron distribution is $d\Gamma /(dxd\Omega )=G_F^2{m}_{\mu }^5{x}^2[(3-2x)+(1-2x)\hat{\mathbf{k}}\cdot \mathbf{s}]/(384{\pi }^4)$, where $x=2E_e/m_{\mu }\in [0,1]$. Its final-spin sum is $64G_F^2(k\cdot q^{\prime })(q\cdot (p-m_{\mu }s))$. The spin vector is axial whereas the electron direction is polar, so their scalar product reverses under parity. Its occurrence demonstrates parity violation by the left-chiral weak charged current.

The antimuon is the antiparticle of the muon. It has electric charge $+1$ and lepton number $-1$.

A quark mass is the coefficient of its Dirac mass term. This term couples left- and right-handed quarks, so nonzero quark masses explicitly break independent rotations of the two chiralities. For two equal masses, the vector flavor symmetry survives, while the nonsinglet axial current obeys the flavor-nonsinglet axial Ward identity with equal quark masses. In the renormalized theory the mass parameter depends on the subtraction scale and scheme, while measurable hadron masses are physical observables.

A quark field is a Dirac field or a chiral Weyl spinor carrying the colour fundamental representation of $SU(3)$. Its three colour components transform as ${\psi }^i\mapsto U^i{}_j{\psi }^j$ under local gauge transformations. The colour index is independent of the Lorentz spinor index, and the weak representations of its left and right chiral components differ in the Standard Model.

A quark flavor is one of the six Standard Model quark species: up, down, strange, charm, bottom and top. Each flavor occurs in three colour states and is counted as one Dirac species when both chiralities are included. The active flavor number in a low-energy QCD description excludes quarks heavier than the relevant scale, with matching of the running coupling when a heavy threshold is crossed.

The top quark is the up-type member of the third quark generation. Its mass exceeds the W boson mass, so a real W boson cannot decay into an on-shell top quark. A massless approximation for accessible decay products must not silently remove this channel threshold.

A constituent quark that does not participate in the elementary weak vertex of a meson decay. It combines with a produced quark or antiquark in the final hadron.

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

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

The antiparticle of a quark.

The Standard Model representations under $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ are $Q_L=(3,2{)}_{1/6}$, $u_R=(3,1{)}_{2/3}$, $d_R=(3,1{)}_{-1/3}$, $L_L=(1,2{)}_{-1/2}$, $e_R=(1,1{)}_{-1}$, and $H=(1,2{)}_{1/2}$.

In one Standard Model generation, the left-handed up and down quark fields form a weak $SU(2)$ doublet of hypercharge $1/6$. Each weak component has three colours. The same field therefore counts as three doublets for weak gauge-loop contributions and two colour fundamentals for strong gauge-loop contributions. Right-handed up and down quarks are separate weak singlets.

In normalization $Q=T_3+Y$, the left quark doublet has $Y=1/6$, right up/down singlets have $2/3,-1/3$, the left lepton doublet has $-1/2$, the right charged lepton has $-1$, and the Higgs doublet has $1/2$. The alternative convention $Q=T_3+\hat{Y}/2$ uses $\hat{Y}=2Y$. Up-type Yukawa interactions use the conjugate doublet of hypercharge $-1/2$.

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 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.

Proton decay is a hypothetical baryon-number-violating transition of the proton into lighter particles. Its nonobservation places strong lower bounds on the scale, or strong upper bounds on the couplings, of baryon-number-violating interactions.

A proton-decay operator is a local effective interaction that violates baryon number. Standard Model gauge invariance first permits such operators at dimension six, including $QQQL$ and $u^c{u}^c{d}^c{e}^c$.

Lepton number assigns $L=1$ to leptons, $L=-1$ to antileptons and zero to quarks. Neutrino masses show that separate family lepton numbers need not be exact.

A neutrino is an electrically neutral spin-one-half lepton. The Standard Model has three active neutrino flavours associated with the electron, muon and tau.

If all neutrinos are exactly massless, arbitrary unitary rotations among them preserve both their kinetic terms and their zero mass matrix. After diagonalizing the charged-lepton Yukawa matrix, choose the neutrino basis to coincide with that rotation. The resulting weak charged current has no physical lepton mixing matrix. With nondegenerate neutrino masses this extra basis freedom is lost, permitting observable neutrino oscillations and lepton mixing.

The muon neutrino is the neutral lepton coupled to a muon by the weak charged current. It carries lepton number $+1$; its antineutrino has the opposite lepton number.

An antineutrino is the antiparticle of a neutrino and carries the opposite additive lepton number.

Neutrino oscillation is the coherent conversion between neutrino flavors caused by different propagation phases of mixed mass eigenstates. It measures mass-squared differences and mixing parameters.

A right-handed neutrino transforms as $(1,1{)}_0$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, so it is neutral under every Standard Model gauge interaction.

A right-handed neutrino is a singlet of the entire Standard Model gauge group. Its same-chirality bilinear with the field obtained by charge conjugation therefore admits the displayed renormalizable Majorana mass term, in addition to a Higgs-induced Dirac mass term. It violates lepton number by two units. Analogous terms for electrically charged Standard Model fermions carry nonzero electric charge; quarks also have nontrivial color charge. The presence of a right-handed field permits mass but does not force a nonzero mass parameter.

The type-I seesaw mechanism combines a Dirac neutrino mass $m_D$ with a much larger right-handed-neutrino Majorana mass term $M$. Diagonalizing the mass matrix gives a light eigenvalue of order $m_D^2/M$ and a heavy eigenvalue of order $M$.

The Weinberg operator is the dimension-five Standard Model operator $(LH)(LH)/\Lambda$. After the Higgs field condenses, it gives a left-handed neutrino Majorana mass of order $v^2/\Lambda$.

For the gauge-singlet composite $\chi ={\tilde{\varphi }}^{†}L$, the interaction $G{\chi }^T{C}^{-1}\chi -G^{\ast }\bar{\chi }C{\bar{\chi }}^T$ is electroweak gauge-invariant. At the Higgs field vacuum, $\chi =v{\nu }_L/\sqrt{2}$, giving $\frac{1}{2}(G{v}^2{\nu }_L^T{C}^{-1}{\nu }_L-G^{\ast }{v}^2{\bar{\nu }}_{L}C{\bar{\nu }}_L^T)$. The Grassmann variation of a chiral Majorana mass term therefore identifies the mass parameter as $G{v}^2$, with physical mass $|G|v^2$. The dimension-five interaction has a coupling of mass dimension minus one. Under lepton number phase rotation, its two terms acquire factors $e^{\pm 2i\alpha }$, so it violates continuous lepton number by two units. It is an effective extension of the minimal renormalizable Standard Model.

The type-II seesaw mechanism couples two lepton doublets to an electroweak scalar triplet. A small neutral-triplet vacuum expectation value directly generates a left-handed neutrino Majorana mass term.

A Yukawa interaction couples left- and right-handed fermions through a scalar field. Replacing the Higgs field by its vacuum expectation value turns a gauge-invariant Standard Model Yukawa interaction into a fermion mass term.

For one four-component Dirac field with scalar interaction $-y\bar{\psi }\psi \varphi$ and the mostly-plus propagator convention, the fermion loop numerator is $4[M^2-k\cdot (k-p)]$. Reduce it to a bubble and tadpole: $-4y^2[(2M^2+p^2/2)J-I]$. Dimensional regularization gives pole parts $J=2/(\epsilon 16{\pi }^2)$ and $I=-2M^2/(\epsilon 16{\pi }^2)$. The local pole then requires both wavefunction renormalization and a scalar mass counterterm.

For a real scalar coupled through $-g\bar{\psi }\varphi \psi$, the scalar-fermion Feynman vertex is $-ig$. Two-fermion scattering has scalar exchange in the $t$ and $u$ channels with a relative minus from exchanging identical external fermions. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative fermionic minus for a fixed state ordering. Fermion-scalar scattering has the two orderings along one open fermion line, using internal momenta $p_1+p_2$ and $p_1-p_4$; these have the same sign. The Mandelstam variables specify the propagator denominators.

The pseudoscalar Yukawa interaction with coefficient $-g$ has the displayed Feynman vertex. Its two-fermion tree-level Feynman diagrams are the $t$ and $u$ exchanges; antisymmetry of the final identical fermions requires their relative minus. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative minus. With initial and final states ordered as $b_1^{†}{d}_2^{†}|0⟩$ and $b_3^{†}{d}_4^{†}|0⟩$, the normal-ordered current contributes $-d^{†}d\bar{v}{\gamma }^{5}v$ in the exchange channel but $db$ and $b^{†}{d}^{†}$ in the annihilation channel. These fermionic signs fix the overall sign as well as their interference; changing an external-state phase changes both terms together.

The Standard Model lepton doublet $L=({\nu }_L,e_L{)}^T$ has hypercharge $-1/2$, the electron singlet $e_R$ has $-1$, and the Higgs doublet has $1/2$. Therefore $-y_e\bar{L}\varphi e_R+h.c.$ is gauge-invariant even though a bare ${\bar{e}}_L{e}_R$ mass is not. The vacuum expectation value generates a Dirac mass $|y_e|v/\sqrt{2}$. In unitary gauge, the Higgs boson coupling is $-(m_e/v)h\bar{e}e$ after rephasing the electron.

A pseudoscalar Yukawa interaction couples a Dirac field bilinear containing the chirality matrix to a spin-zero field. In the standard mostly-minus gamma matrix convention with ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the usual Dirac adjoint, $ig\bar{\psi }{\gamma }^5\psi \varphi$ is Hermitian for real $g$ and real $\varphi$. If $\varphi$ is a pseudoscalar, the interaction also preserves parity symmetry in quantum field theory; if $\varphi$ is parity even, it breaks that symmetry. Moving the factor $i$ into the definition of ${\gamma }^5$ changes the displayed vertex coefficient but not the interaction. Each vertex has one bosonic leg and two oppositely oriented fermion legs, and preserves Dirac fermion number conservation.

With the usual Dirac adjoint, Hermitian ${\gamma }^5$ and $\{{\gamma }^0,{\gamma }^5\}=0$, taking the adjoint gives ${\psi }^{†}{\gamma }^5{\gamma }^0\psi =-{\psi }^{†}{\gamma }^0{\gamma }^5\psi$. Consequently a real pseudoscalar with coupling $-g\varphi \bar{\psi }{\gamma }^5\psi$ has a Hermitian interaction only if $g^{\ast }=-g$. Writing $g=i{g}_P$ with real $g_P$ gives the familiar interaction $-i{g}_P\varphi \bar{\psi }{\gamma }^5\psi$. Formal Euler-Lagrange field equations obtained with independent spinor and adjoint variables can still be computed for the original symbolic coupling; it must not silently be identified with a real physical coupling.

After a scalar vacuum expectation value, a Yukawa matrix produces a bilinear $-{\bar{f}}_L{M}_f{f}_R+h.c.$. Independent unitary left/right rotations diagonalize $M_f$ with nonnegative masses. Extra scalar-dependent operators can make the physical scalar coupling matrix fail to diagonalize with the mass matrix.

A Yukawa matrix contains the flavor-space couplings between left-handed fermions, right-handed fermions, and a scalar field. Its singular values determine fermion masses after symmetry breaking, while mismatches between its diagonalizing transformations produce flavor mixing.

A singular value decomposition of the charged-lepton Yukawa matrix uses independent unitary rotations of left- and right-handed leptons. Their gauge-covariant kinetic terms are unchanged because each rotation mixes identical Standard Model representations. After electroweak symmetry breaking, the nonnegative singular values give Dirac masses $m_i=y_{i}v/\sqrt{2}$ and diagonal Higgs boson couplings $m_i/v$. When the neutrinos are massless, rotating their left-handed fields with the charged partners preserves the diagonal weak charged current.

The Cabibbo-Kobayashi-Maskawa matrix is the unitary mismatch between the left-handed transformations that diagonalize the up- and down-quark Yukawa matrices. It appears in charged weak currents.

Orthogonality of two CKM rows or columns makes the three complex products form a closed triangle. Its area is $|J|/2$, where $J$ is the Jarlskog invariant. Normalizing the displayed column-orthogonality relation gives vertices $0,1,\bar{\rho }+i\bar{\eta }$, with $\bar{\rho }+i\bar{\eta }=-V_{ud}{V}_{ub}^{\ast }/(V_{cd}{V}_{cb}^{\ast })$. A collapsed triangle has zero weak rephasing-invariant CP violation for nondegenerate quark masses.

The Wolfenstein parametrization expands the three-generation CKM matrix in its small $1$-$2$ mixing parameter $\lambda =s_{12}$, defining $s_{23}=A{\lambda }^2$ and $s_{13}{e}^{i\delta }=A{\lambda }^3(\rho +i\eta )$. To cubic order its rows are $(1-{\lambda }^2/2,\lambda ,A{\lambda }^3(\rho -i\eta ))$, $(-\lambda ,1-{\lambda }^2/2,A{\lambda }^2)$, and $(A{\lambda }^3(1-\rho -i\eta ),-A{\lambda }^2,1)$. Higher orders restore exact unitarity. Rephasing-invariant barred parameters are defined from ratios of CKM entries rather than this truncated expansion.

For nondegenerate quark masses, independent phase changes of the up- and down-quark mass eigenfields alter the numerical CKM matrix by row and column phases without changing physics. Thus CP symmetry requires that some allowed phase convention makes the matrix real, rather than requiring real entries in every convention. With $N$ generic families, removing $2N-1$ phases from a unitary matrix leaves $(N-1)(N-2)/2$ physical CP-violating phases and $N(N-1)/2$ mixing angles.

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.

An $N\times N$ CKM matrix starts with $N^2$ parameters of a unitary matrix. Independent quark rephasings remove $2N-1$ phases because the common phase has no effect. The remaining $(N-1{)}^2$ parameters consist of $N(N-1)/2$ angles and $(N-1)(N-2)/2$ CP-violating phases. Extra mass degeneracies may reduce this generic count.

The angle of the real two-generation quark mixing rotation, with $V_{ud}=V_{cs}=\cos {\theta }_C$, $V_{us}=\sin {\theta }_C$, and $V_{cd}=-\sin {\theta }_C$.

A small weak charged current amplitude factor associated with $\sin {\theta }_C$. A doubly suppressed amplitude contains two such factors, giving a rate suppression of order ${\tan }^4{\theta }_C$ relative to a favoured process when other matrix elements are comparable.

The Jarlskog invariant is a rephasing-invariant measure of CP violation in a three-generation quark mixing matrix. It is the common magnitude of imaginary quartic products such as $\mathrm{Im}(V_{ij}{V}_{k\ell }{V}_{i\ell }^{\ast }{V}_{kj}^{\ast })$ for distinct rows and columns.