Electron-positron annihilation cross section near the new threshold

Abstract

Heavy-quark models of the new narrow resonances are presented. The relative heaviness of the assumed new quarks suggests applying the adiabatic approximation familiar from molecular physics here. This approach provides partial justification for the use of potential models to describe the new resonances. Vacuum polarization plays an important role in determining the shape of the adiabatic potential. Model calculations of the ${\psi }^{\prime \prime }\mathrm{}\left(4.1\right)\mathrm{}$ are presented. Agreement with experiment is good. A coupled-channel formulation of the narrow resonances and the new threshold is presented. The resulting coupled-channel Schrödinger equations can be solved iteratively. The lowest-order approximation reproduces the adiabatic picture and higher-order corrections are calculable. Several coupled-channel models are analyzed numerically and qualitatively. Although detailed numerical results are model-dependent, certain trends occur. We estimate that the radiative decays of the ${\psi }^{\prime }\mathrm{}\left(3.7\right)\mathrm{}$ should be less important than is indicated by naive-pure-quark-model calculations. In addition, the ${\psi }^{\prime }\mathrm{}\left(3.7\right)\mathrm{}$ is not a simple two-quark state but has considerable charm-anticharm-meson content. The leptonic widths of the $\psi \mathrm{}\left(3.1\right)\mathrm{}$ and ${\psi }^{\prime }\mathrm{}\left(3.7\right)\mathrm{}$ are computed and good agreement with experiment is found. Corrections to the adiabatic-potential-model approximation to $R\left(Q^2\right)=\frac{\sigma (e^{+}{e}^{-}\to \mathrm{hadrons})}{\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})}$ above threshold are computed. If the coupled-channel model uses the SU(4) charm scheme, then $D^{*}\text{'}\mathrm{s}$ are expected to outnumber $D\text{'}\mathrm{s}$ by a wide margin above threshold. The potential-model calculations of $R\left(Q^2\right)$ which accept the conventional SU(4) scheme are systematically below the data for $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}\gtrsim 4.6-4.8$ GeV. By adding the contribution of a heavy lepton with mass 2.3-2.4 GeV to the four-quark prediction, a curve of $R$ vs $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}$ which is in good agreement with the data over the entire energy range occurs.