Mesons: J^PC and neutral-meson mixing

A meson is |ψ⟩ = |orbital⟩ ⊗ |spin⟩ ⊗ |flavour⟩ ⊗ |colour⟩, with the colour factor pinned to the singlet (rr̄ + gḡ + bb̄)/√3, which is symmetric under q ↔ q̄ and drops out of every sign count below. The orbital factor is a relative-coordinate wavefunction R(r) Y_L^m(r̂), the spin factor is one of the four states of ½ ⊗ ½ (singlet S = 0, triplet S = 1), and J comes from L ⊗ S by Clebsch–Gordan: J = L for S = 0, and J = L − 1, L, L + 1 for S = 1 and L ≥ 1, but only J = 1 when L = 0. J, P and C commute with the strong Hamiltonian; L and S do not: they are labels of the potential-model basis in which the state is written, and spin–orbit and tensor forces mix them, so the ρ is mostly ³S₁ with a ³D₁ admixture. That is why assignments use the spectroscopic name n^(2S+1)L_J for the dominant component and J^PC for what is actually conserved.

Parity sends r → −r, and on the relative wavefunction Y_L^m(−r̂) = (−1)^L Y_L^m(r̂): even L is even, odd L is odd. The second factor is intrinsic. Applying the parity operator γ⁰ to the plane-wave solutions of the Dirac equation returns the particle spinors u with one sign and the antiparticle spinors v with the opposite one, so a fermion and its antifermion have opposite intrinsic parity and the pair contributes −1 whatever convention fixes the quark's own sign. Together, P = (−1)^(L+1). Every L = 0 meson is therefore parity-odd — π, K, η, ρ, ω, φ all carry P = −1 — and the first positive-parity nonets, the a₀, a₁, a₂ and b₁ families near 1.2 to 1.5 GeV, cost one unit of orbital excitation. Read the sign off a decay and you have a check: ρ → ππ puts the two pions in an L = 1 wave, parity (−1)¹ for the wave times (−1)² for two pions, giving −1 and matching the ρ.

Ĉ turns every quark into its antiquark, so Ĉ|q(1) q̄(2)⟩ ∝ |q̄(1) q(2)⟩. To compare with the original, relabel the two particles: exchanging positions costs (−1)^L, exchanging spins costs +1 for the symmetric triplet and −1 for the antisymmetric singlet, that is (−1)^(S+1), and exchanging two fermion operators costs −1. The product is C = (−1)^(L+S). It is an eigenvalue only if the swapped state is the same state: uū, dd̄, ss̄, cc̄, bb̄ and their combinations. For K⁰ = ds̄ the operator returns K̄⁰ = sd̄, a different ket, and for π⁺ it returns π⁻. Charged members of an isospin multiplet get G = C(−1)^I instead, with C the neutral member's value: pions have G = −1, so n pions carry (−1)ⁿ, which is why the ρ (G = +1) goes to two pions and the ω (G = −1) to three. Tabulate L and S: L = 0 gives 0⁻⁺ and 1⁻⁻; L = 1 gives 1⁺⁻ and 0⁺⁺, 1⁺⁺, 2⁺⁺; L = 2 gives 2⁻⁺ and 1⁻⁻, 2⁻⁻, 3⁻⁻. Missing from every row are 0⁻⁻ and the series 0⁺⁻, 1⁻⁺, 2⁺⁻, … with P = (−1)^J and C = −P. Those are the exotics; π₁(1600) with 1⁻⁺ is evidence for glue or a second qq̄ pair inside the hadron.

Flavour SU(3) gives 3 ⊗ 3̄ = 8 ⊕ 1, and the neutral I = 0 members of the pseudoscalar nonet are η₈ = (uū + dd̄ − 2ss̄)/√6 and η₁ = (uū + dd̄ + ss̄)/√3. If the symmetry were exact those would be the eigenstates. It is not: the quark-mass term with mₛ ≫ mᵤ, m_d transforms as the eighth component of an octet and has a non-zero matrix element between 8 and 1, so in the (η₈, η₁) basis the mass-squared matrix has an off-diagonal entry and the physical states are a rotation, |η⟩ = cos θ_P |η₈⟩ − sin θ_P |η₁⟩ and |η′⟩ = sin θ_P |η₈⟩ + cos θ_P |η₁⟩. The angle is fixed by the eigenvalues: with m_η₈² = (4m_K² − m_π²)/3 = (569 MeV)² from Gell-Mann–Okubo, sin² θ_P = (m_η₈² − m_η²)/(m_η′² − m_η²) = 0.039, so θ_P ≈ −11°. The vector nonet does the same with θ_V ≈ 39°, near the ideal 35.3° at which ω = (uū + dd̄)/√2 and φ = ss̄, which is why φ → KK̄ takes 83% of its decays while φ → 3π is suppressed. Pseudoscalars sit far from ideal because the U(1)_A anomaly lifts the singlet: η′ at 958 MeV is the cost of that, and the angle itself is fitted, never derived.

The strong interaction makes definite strangeness: π⁻p → ΛK⁰ produces |K⁰⟩ = ds̄, never K̄⁰ = sd̄. The weak interaction ignores that label: a two-W box diagram connects K⁰ to K̄⁰ (ΔS = 2), and both decay to the same ππ and πππ channels. Project onto the two-dimensional kaon subspace and second-order perturbation theory (Weisskopf–Wigner) gives H_eff = M − iΓ/2, where Mᵢⱼ holds masses and energy shifts and Γᵢⱼ = 2π Σ_f ⟨i|H_W|f⟩⟨f|H_W|j⟩ δ(E_f − m_K) holds decay rates. Γ is Hermitian but −iΓ/2 is anti-Hermitian, so H_eff is not: the norm of a state falls with time, which is what decay means. CPT sets M₁₁ = M₂₂ and Γ₁₁ = Γ₂₂. With CP conserved and the phase convention CP|K⁰⟩ = |K̄⁰⟩, the eigenvectors are K_S = (K⁰ + K̄⁰)/√2 with CP = +1 and K_L = (K⁰ − K̄⁰)/√2 with CP = −1. The eigenvalues differ in both parts: K_S → ππ is fast, τ_S = 0.895 × 10⁻¹⁰ s, while K_L, being CP-odd, cannot reach ππ and is left with three-body channels, πππ with under 100 MeV to spare and semileptonic πℓν, so τ_L = 5.12 × 10⁻⁸ s, 570 times longer; and Δm = m_L − m_S = 3.48 × 10⁻⁶ eV. Since 1964, K_L → ππ at 2 × 10⁻³ has shown CP is not exact, so this holds only to that accuracy.

Time evolution is diagonal in the eigenbasis, so write the initial state there: |K⁰⟩ = (|K_S⟩ + |K_L⟩)/√2. Each component picks up its own factor, e^{(−im_S t − Γ_S t/2)} and e^{(−im_L t − Γ_L t/2)}, and rotating back gives the K̄⁰ amplitude ½(e_S − e_L). Squaring, P(K⁰ → K̄⁰) = ¼[e^{(−Γ_S t)} + e^{(−Γ_L t)} − 2e^{(−(Γ_S+Γ_L)t/2)} cos(Δm t)], the standard two-level beat at angular frequency Δm/ħ with two different dampings. At t = 2τ_S the numbers are e⁻² = 0.135, e^(−0.0035) = 0.997 and 2e^(−1.002) cos(0.948) = 0.428, so P(K̄⁰) = 0.176 while P(K⁰) = 0.390: nearly a third of the surviving kaons have flipped strangeness. The curves cross when cos(Δm t) = 0, at t = π/(2Δm) = 3.3τ_S, and once the K_S component has died the beam is pure K_L, half K⁰ and half K̄⁰ from then on. Experiments tag the flavour through semileptonic decays, K⁰ → π⁻e⁺ν and K̄⁰ → π⁺e⁻ν̄, and fit the asymmetry to extract Δm. In NumPy it is four lines: build H = M − 1j*G/2, apply scipy.linalg.expm(−1j*H*t) to the flavour-basis vector, and watch its norm fall.