Relativistic quantum mechanics and the quark-pair creation model

In the context of a 3P₀ quark-pair creation model for the process ρ→ππ, a method is developed for taking relativity into account in the calculation of hadron decays. Following a brief review of relativistic quantum mechanics, an expression is derived for the general relation between a momentum-space, two-particle, instant form wave function in an arbitrary frame and the wave function associated with the c.m. frame. This relation is used to develop relativistic wave functions for the π and ρ mesons. Second quantized state vectors for ππ and ρ states are constructed with the help of these relativistic wave functions. The ρ→ππ transition amplitude is obtained by using these state vectors to calculate matrix elements of a second quantized 3P₀ quark-pair creation operator derived from a scalar Lagrangian density. The amplitude differs from the one obtained using nonrelativistic wave functions in the appearance of Wigner rotations. In spite of the complications arising from these rotations the calculation of the relativistic amplitude is reduced to carrying out a two-dimensional integral. The amplitude is of the same form as one derived from an effective ρππ Lagrangian except for the presence of a form factor that depends on the magnitude of the three-momentum of a final-state pion. The shape of the form factor is determined by the relativistic π and ρ wave functions. Using the ρ→ππ transition amplitude as a vertex interaction in a relativistic model of ππ scattering, the p-wave, ππ scattering amplitude is calculated and fit to data by adjusting the interaction strength and the ρ bare mass. This leads to a mass shift and decay width for the ρ meson. Using nonrelativistic wave functions to calculate the form factor leads to a negligible mass shift, whereas using the relativistic wave functions leads to a bare ρ mass of 855.7 MeV, corresponding to a physical ρ mass of 775.5 MeV. The quark-pair creation operator strength parameter for the relativistic case is roughly a factor of 2 larger than that for the nonrelativistic case.