The measured value of the forward-backward asymmetry in the process $e^{+} e^{-} \to Z \to b \bar{b}$ at the electron-positron collider LEP significantly deviated from the theoretical predictions given by the Standard Model of elementary particles and interactions. According to the theoretical calculation, this asymmetry is proportional to the difference of the squared coupling constants of the boson $Z$ with left- and right-handed $b$ quarks ($g^{2}_{Lb}-g^{2}_{Db}$). In this master thesis, we were searching for observables at hadron colliders, such as the Large Hadron Collider (LHC), which are sensitive to the same difference of the coupling constants. We have investigated the $Z$ boson production and decay into a $b \bar{b}$ pair. At the leading order in quantum perturbation theory, we have calculated, at parton level, the total and differential cross section for the process $q \bar{q} \to Z \to b \bar{b}$ depending on the angle between the $q$ quark and $b$ quark momenta in the center-of-mass reference frame. In exclusive production of the $Z$ boson in (symmetric) collisions of protons, the forward-backward asymmetry is undefined. We have considered more inclusive processes, where additional hadron jets are formed. We were interested in the angular distributions of the $Z$ boson's decay products with respect to the momenta of the incoming partons and additional hadron jets at the parton level. We have then defined and calculated an observable at parton level sensitive to $g^{2}_{Lb}-g^{2}_{Db}$. In a similar way it would be also possible to obtain sensitivity to $g^{2}_{Lb}-g^{2}_{Db}$ at the hadron level.
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