Production of double P-wave heavy quarkonia at a super Z factory

The production of double P -wave heavy quarkonia is studied systematically within the framework of non-relativistic quantum chromodynamics at the exclusive process of the electron–positron collision annihilation at the CM energy √s=91.1876 s = 91.1876 GeV of the Z⁰ Z 0 pole, i.e., double P -wave charmonia, double P -wave Bc B c mesons, and double P -wave bottomonia in e⁺e⁻→ γ ^*/Z⁰ → |(QQ'̄)[n] +|(Q'Q̄)[n^ ] e + e - → γ ∗ / Z 0 → | ( Q Q ′ ¯ ) [ n ] ⟩ + | ( Q ′ Q ¯ ) [ n ′ ] ⟩ ( $$Q/Q'=c$$ Q / Q ′ = c - or b -quarks) at a future super Z factory, where [ n ] or [n^ ] [ n ′ ] stands for the color-singlet [¹P₁] [ 1 P 1 ] and [³PJ] [ 3 P J ] ( $$J=0,1,2$$ J = 0 , 1 , 2 ) Fock states. As an improved trace technology is useful for calculating the complicated double P -wave channels, the analytical result can be obtained at the amplitude level. According to our study, the generation rates for the double P -wave heavy quarkonia at the future super Z factory are considerable. The values obtained for the total cross sections of the double P -wave charmonium for σ (|(cc̄)[1] +|(cc̄)[1] )=4.191+0.489-0.498× 10⁻³~fb σ ( | ( c c ¯ ) [ 1 ] ⟩ + | ( c c ¯ ) [ 1 ] ⟩ ) = 4 . 191 - 0.498 + 0.489 × 10 - 3 f b , the total cross sections of the double P -wave Bc B c mesons for σ (|(cb̄)[1] +|(bc̄)[1] )=0.2045+0.0084-0.0080~fb σ ( | ( c b ¯ ) [ 1 ] ⟩ + | ( b c ¯ ) [ 1 ] ⟩ ) = 0 . 2045 - 0.0080 + 0.0084 f b , and the total cross sections of the double P -wave bottomonium for σ (|(bb̄)[1] +|(bb̄)[1] )=5.244+0.134-0.136× 10⁻³~fb σ ( | ( b b ¯ ) [ 1 ] ⟩ + | ( b b ¯ ) [ 1 ] ⟩ ) = 5 . 244 - 0.136 + 0.134 × 10 - 3 f b . The main uncertainties come from the mass of heavy quarkonia and the corresponding first-derivative radial wave functions at the origin under the Buchmüller–Tye (BT) potential. The number of events in the production of double P -wave charmonium, Bc B c -meson, and bottomonium states via e⁻e⁺→ γ ^*/Z⁰ → |(QQ̄')[n] + |(Q'Q̄)[n^ ] e - e + → γ ∗ / Z 0 → | ( Q Q ¯ ′ ) [ n ] ⟩ + | ( Q ′ Q ¯ ) [ n ′ ] ⟩ at the integrated luminosity L≈ 10³⁴cm⁻²s⁻¹ L ≈ 10 34 c m - 2 s - 1 at the super Z factory is almost unobservable.