Towards understanding $b\bar{b}$ production in $\gamma \gamma$ collisions

Understanding the data on the total cross section ${\sigma }_{tot}($e${}^{+}$e${}^{-}\to$e${}^{+}$e${}^{-}b\bar{b})$ measured at LEP2 represents a serious challenge for perturbative QCD. In order to unravel the origins of the discrepancy between data and theory, we investigate the dependence of four contributions to this cross section on $\gamma \gamma$ collision energy. As the reliability of the existing calculations of ${\sigma }_{tot}($e${}^{+}$e${}^{-}\to$e${}^{+}$e${}^{-}b\bar{b})$ depends, among other things, on the stability of calculations of the cross section ${\sigma }_{tot}(\gamma \gamma \to b\bar{b})$ with respect to variations of the renormalization and factorization scales, we investigate this aspect in detail. We show that in most of the region relevant for the LEP2 data the existing QCD calculations of ${\sigma }_{tot}(\gamma \gamma \to b\bar{b})$ do not exhibit a region of local stability and should thus be taken with caution. The source of this instability is suggested and its phenomenological implications for LEP2 data are discussed.