Electronic structure and optical properties of prominent phases of $$\hbox {TiO}_{2}$$: First-principles study

We present the properties of the C-parameter as an event-shape variable. We calculate the coupling constants in the perturbative and also in the non-perturbative parts of the QCD theory, using the dispersive as well as the shape function models. By fitting the corresponding theoretical predictions to our data, we find \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.117 ± 0.014 and α ₀(μ _I ) = 0.491 ± 0.043 for dispersive model and \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.124 ± 0.015 and λ ₁ = 1.234 ± 0.052 for the shape function model. Our results are consistent with the world average value of \(\alpha _{\mathrm {s}} (M_{Z^{0}})\) = 0.118 ± 0.002. All these features are explained in the main text.