Strong Coupling Constant from the Photon Structure Function

We extract the value of the strong coupling constant ${\alpha }_s$ from a single-parameter pointlike fit to the photon structure function $F_2^{\gamma }$ at large $x$ and $Q^2$ and from a first five-parameter full (pointlike and hadronic) fit to the complete $F_2^{\gamma }$ data set taken at PETRA, TRISTAN, and LEP. In next-to-leading order and the $\overline{\mathrm{M}\mathrm{S}}$ renormalization and factorization schemes, we obtain ${\alpha }_s(m_Z)=0.1183\pm 0.0050(\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}{)}_{-0.0028}^{+0.0029}(\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r})$ (pointlike) and ${\alpha }_s(m_Z)=0.1198\pm 0.0028(\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{t}{)}_{-0.0046}^{+0.0034}(\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r})$ (pointlike and hadronic). We demonstrate that the data taken at LEP have reduced the experimental error by about a factor of 2, so that a competitive determination of ${\alpha }_s$ from $F_2^{\gamma }$ is now possible.