Flavor symmetry

Flavor symmetry rotates quark species whose strong interactions are identical. Quark masses break it explicitly, so it is a useful approximate symmetry only among quarks light compared with the hadronic scale.

For a colour singlet hadron built entirely from up quarks, down quarks, strange quarks, antiquarks and gluons, the net number of quarks is a multiple of three. The central SU(3) colour element multiplies the state by ${\omega }^{n_q-n_{\bar{q}}}$, while a singlet must be unchanged. The central flavour element supplies the same phase because all three light flavours form the fundamental representation. Every resulting flavour multiplet therefore has zero SU(3) triality. Heavy quarks are singlets of light-flavour SU(3), so hadrons containing them do not obey this argument. The zero-triality restriction allows higher light-quark multiplets as well as the familiar singlet, octet and decuplet.

The $SU(2)$ flavor symmetry rotating down and strange quarks, with the up quark a singlet. Since down and strange quarks have equal electric charge, the charge operator commutes with all U-spin generators, and every irreducible U-spin multiplet has a common charge. Unequal down- and strange-quark masses break this symmetry.

For flavor symmetry acting on $(u,d,s)$, trace-orthonormal Hermitian diagonal generators are $h_1=\mathrm{diag}(1,-1,0)/\sqrt{2}$ and $h_2=\mathrm{diag}(1,1,-2)/\sqrt{6}$. The isospin and flavor hypercharge operators are $I_3=h_1/\sqrt{2}$ and $Y=\sqrt{2/3}{h}_2$. Their multiples by $i$ lie in the compact SU(3) Lie algebra.

An additive flavor symmetry quantum number: a strange quark has $S=-1$, its antiquark has $S=+1$, and up and down quarks have $S=0$. The strong interaction conserves net strangeness. Together with baryon number, it determines flavor hypercharge $Y=B+S$.

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.

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.

The baryon octet has rows $Y=1/2,0,-1/2$ with isospins $1/2$, $1\oplus 0$, $1/2$. The central weight has two states, ${\Sigma }^0$ and ${\Lambda }^0$. The baryon decuplet has rows $Y=1/2,0,-1/2,-1$ with isospins $3/2,1,1/2,0$. The three-quark flavour singlet has one state at the origin. Both abstract octet copies in $3^{\otimes 3}$ share the same weights.

For the light quark flavors, electric charge is $Q=I_3+Y/2$, with flavor hypercharge $Y=B+S$. On $(u,d,s)$ it gives $(2/3,-1/3,-1/3)$, and additivity extends it to the baryon octet and meson octet.

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.

A nonzero vacuum value of the scalar quark density breaks the massless nonsinglet chiral symmetry to its diagonal vector subgroup. Indeed, the nonsinglet axial charge algebra of quark densities gives $⟨[Q_{5,a},P_b]⟩=-i{\delta }_{ab}⟨\bar{q}q⟩$, whereas invariance of the vacuum under every conserved axial charge would make these expectations vanish. The three broken generators produce three massless Goldstone bosons in the exact chiral limit. Small equal quark masses explicitly break that symmetry and turn the pions into pseudo-Goldstone bosons, with squared masses linear in the quark mass at leading order.

Let $S=\bar{q}q$, $P_a=i\bar{q}{\gamma }_5{\tau }_{a}q$ and $Q_{5,a}=\int q^{†}{\gamma }_5{\tau }_{a}q/2d^{3}x$. The canonical anticommutation relations imply $[Q_{5,a},q]=-{\gamma }_5{\tau }_{a}q/2$ and $[Q_{5,a},\bar{q}]=-\bar{q}{\gamma }_5{\tau }_a/2$. The Leibniz rule for the commutator, together with $\{{\tau }_a,{\tau }_b\}=2{\delta }_{ab}$ and ${\gamma }_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero quark masses remove conservation of the charges without changing their canonical action on the fields.

For the Dirac equation with two equal quark masses $m$, the equations for $q$ and $\bar{q}$ give the divergence of the nonsinglet axial current as $im\bar{q}{\gamma }_5{\tau }_{a}q=m{P}_a$. The color chiral anomaly is absent in this nonsinglet identity because the quark flavor generator is traceless. Equivalently, the mass term in the Hamiltonian is $m\int S{d}^{3}x$, and the nonsinglet axial charge algebra of quark densities gives ${\dot{Q}}_{5,a}=m\int P_a{d}^{3}x$.

Isospin is the approximate $SU(2)$ flavor symmetry rotating the up and down quarks. It is accurate because their masses are close and both are small compared with the QCD scale.

Two pions, each of isospin one, have a nine-dimensional charge-state space. Clebsch-Gordan coefficients decompose it into an isospin-two quintet, an isospin-one triplet and an isospin-zero singlet. Under exchange of the two particle slots, the $I=2,0$ states are symmetric and the $I=1$ states are antisymmetric. Because pions have zero spin and obey bosonic exchange symmetry, the $I=1$ two-pion channel requires odd relative orbital angular momentum, while $I=0,2$ require even orbital angular momentum.

Nuclear quantum states in different nuclear isobars belonging to the same isospin multiplet, with equal total isospin and approximately the same spatial and spin structure. An isospin raising or lowering operator connects the states in the limit of exact isospin symmetry. Nuclear Coulomb energy and other isospin-breaking effects shift their energies and modify this ideal relationship.

Suppose a transition operator obeys $[I_3,H]=H/2$ and $[I_{+},H]=0$, and acts on the lower component of an isospin-half kaon. Its output has third component zero, and $I_{+}^{2}H|K⟩=H{I}_{+}^2|K⟩=0$. This excludes an isospin-two output. A spinless parent decaying to two spinless pions has zero relative orbital angular momentum, so Bose-Einstein statistics excludes the antisymmetric isospin-one sector. The output is therefore the two-isovector singlet. In the equal-mass isospin limit, charged and neutral two-pion decay widths have ratio two, including the identical-particle counting.

Two spin-one isospin multiplets have a unique normalized singlet. In the displayed formula each ket on the right records the two third components, not their individual total isospins. The total raising operator annihilates this combination because the coefficients of $|1,0⟩$ and $|0,1⟩$ cancel. Its exchange symmetry is positive; the five-dimensional total-isospin-two sector is also symmetric and the three-dimensional total-isospin-one sector antisymmetric.

An isospin multiplet is an irreducible finite-dimensional SU(2) representation with labels $I,M$, where $I\ge 0$, $2I$ is an integer and $M=I,I-1,\dots ,-I$. With Hermitian generators, the lowering coefficient is $\sqrt{(I+M)(I-M+1)}$. The quadratic Casimir operator has value $I(I+1)$, fixing all ladder norms and the termination at $M=-I$.

An isospin-invariant decay amplitude is an intertwining map from the initial isospin multiplet into the final tensor product. The Clebsch-Gordan decomposition for SU2 contains each allowed final isospin once. Commutation with the third generator and the quadratic Casimir operator places the image in the matching coupled state. Commutation with the ladder operators makes its scalar coefficient independent of the magnetic component. Orthogonality of the Clebsch-Gordan coefficients then gives a summed squared amplitude $|a{|}^2$. A physical decay rate additionally includes phase space and spin sums, which can be absorbed into a common reduced coefficient when exact isospin symmetry makes them identical between channels.

With positive ladder coefficients, the second-axis half-turn is $e^{-i\pi I_2}|I,M⟩=(-1{)}^{I-M}|I,-M⟩$. Conjugation sends $I_{\pm }$ to $-I_{\mp }$, making consecutive phases differ by a minus sign. The highest-state rotation formula, expanded through its last nonzero lowering term, fixes the highest-state phase to one.

Weak isospin labels representations of the SU(2) factor in the electroweak gauge group. A left-handed lepton doublet has $T=1/2$ with third-component eigenvalues $+1/2,-1/2$; a right-handed charged-lepton singlet has $T=0$. Along with electric charge, the third component determines the weak neutral current through $T^3-Q{\sin }^2{\theta }_W$.

The product of charge conjugation and a half-turn in isospin is G parity. Both conjugations act on the isospin generators by $(J_1,J_2,J_3)\mapsto (-J_1,J_2,-J_3)$, so their product commutes with every generator. The Schur lemma therefore gives a common G-parity value throughout each irreducible isospin multiplet on which the operator acts internally. It can classify a charged multiplet even though individual charged states are not eigenstates of charge conjugation.

For integer isospin $I$ and neutral-state charge conjugation value $\eta$, the entire isospin multiplet has $G=\eta (-1{)}^I$. Both charge conjugation and the second-axis half-turn reverse the first and third generators, so their product commutes with the full isospin algebra. Evaluate it on the $M=0$ state and use ladder connectivity to obtain the value for every $M$.

Each pion has G parity minus one, so an $n$-pion state has $G=(-1{)}^n$. In the isospin-symmetric strong interaction, a parent with positive G parity can decay only to an even pion multiplicity, and a negative parent only to an odd multiplicity. The rule does not forbid small isospin-breaking or electromagnetic amplitudes.

The neutral pion has positive charge conjugation eigenvalue and its spin-one isospin half-turn has eigenvalue minus one. Hence the G parity eigenvalue is minus one. Since G parity commutes with isospin, both charged members have the same value. Charged-state phase changes alter individual charge-conjugation and rotation formulas together, leaving this eigenvalue unchanged.

An isorotation is a global transformation generated by isospin. In the pion Skyrme model it acts as $U\mapsto AU{A}^{†}$, with constant $A\in SU(2)$, rotating the three pion components while preserving the scalar component. The matrices $A$ and $-A$ induce the same classical rotation, so this action factors through $SO(3)$; quantum states may carry half-integer isospin on its $SU(2)$ cover. This global internal rotation is distinct from a spatial rotation or a local gauge transformation.

The eightfold way organizes light hadrons into approximate $SU(3)$ flavor multiplets built from the up, down, and strange quarks.

The antisymmetric flavour representation ${\bigwedge }^{3}3$ has dimension one and consists of the alternating $uds$ combination. Its sole weight is $(I_3,Y)=(0,0)$, distinct as a representation from the octet $\Lambda$ at the same weight. The Pauli constraint on three-quark flavour multiplets requires orbital excitation for a three-quark colour-singlet realization.

The completely symmetric three-quark representation of flavour symmetry has dimension ten. Its weight diagram has rows $(Y,I)=(1,3/2),(0,1),(-1,1/2),(-2,0)$. It includes $uuu$ at the ${\Delta }^{++}$ weight and $sss$ at the ${\Omega }^{-}$ weight. A spatially symmetric three-quark colour singlet realizes this multiplet with spin $3/2$.

The pseudoscalar octet of flavor symmetry contains three pions, four kaons and the Eta octet state. In its weight diagram, the neutral pion and ${\eta }_8$ are separate states at the origin.

The zero-isospin, zero-flavor hypercharge state ${\eta }_8=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ in the pseudoscalar meson octet. It is a pure octet basis state; the physical eta may mix it with the flavor-singlet state.

The spin-one-half octet of flavor symmetry contains the nucleons, Sigma baryons, Lambda baryon, and Xi baryons. Its weight diagram has six peripheral states and two states at $(I_3,Y)=(0,0)$: ${\Sigma }^0$ and ${\Lambda }^0$, with different total isospin.