Mass and the E1 decay rate of the singlet P state of charmonium.

Abstract

We calculate the hyperfine splitting of the [ital P] states of charmonium using the perturbative QCD hyperfine interaction to order [alpha][sub [ital s]][sup 2], in an improved quasistatic approximation whereby the quark-antiquark scattering amplitude is expanded in powers of [ital p][sup 2]/[ital p][sub 0][sup 2] instead of [ital p][sup 2]/[ital m][sup 2] and terms of up to first order in [ital p][sup 2]/[ital p][sub 0][sup 2] are kept. We evaluated the hyperfine splitting using the wave functions obtained from the unperturbed Hamiltonian of Gupta, Radford, and Suchyta. We find the splitting [Delta][ital M][sub [ital P]]=[ital M][sub c.o.g.][sup 3][ital P][sub [ital J]][minus][ital M][sup 1][ital P][sub 1] to be [minus]0.63 MeV. Our result is very similar to the result of Halzen, Olson, Olsson, and Stong who find [Delta][ital M][sub [ital P]]=[minus]0.7[plus minus]0.2 MeV, using various potential models. It also confirms the recent published experimental result on [Delta][ital M][sub [ital P]]. We also note that if we had used the improved quasistatic approximation of Gupta to extract the hyperfine interaction from the [ital q[bar q]] scattering amplitude to order [alpha][sub [ital s]][sup 2] in QCD, we would have obtained entirely different results for the [ital P]-wave hyperfine splitting in charmonium. We also calculatemore » the electric dipole decay rate of the process 1 [sup 1][ital P][sub 1][r arrow]1 [sup 1][ital S][sub 0]+[gamma] and find it to be about 630 keV for charmonium.« less