How to identify the dead cone in the top-quark jet

The gluon emission from an energetic heavy quark is suppressed in the forward direction below the angle $\Theta\lesssim m_Q/E$ for a quark of mass $m_Q$ and energy $E$ according to perturbative Quantum Chromodynamics (QCD) (``dead cone"). Another consequence is the suppression of energetic particles in the jet which has been observed already for c- and b-quark jets. The suppression of the forward particles can be explained by an application of the Modified Leading Logarithmic Approximation (MLLA) of perturbative QCD. In this paper we investigate whether this type of analysis can be carried out also for top-quark jets with the much higher heavy quark mass allowing for QCD tests in this new kinematic regime. The new aspect of this analysis is the finite lifetime of the top quark. We consider for simplicity the decay $t\to b\ell\nu$, where the b-quark radiates gluons as well and partially obscures the dead cone. Guided by the decay amplitude in leading order in $\alpha_s$ for $e^+e^- \to t \bar t$ we propose a method to separate the radiation by the $\widehat{tb}$ dipole in the decay process which is superimposed to the primary radiation from the $\widehat{t \bar t}$ dipole involving the top-quark dead cone effect. The momentum distributions of partons or hadrons are determined for finite decay angles of the b-quark $\Theta_b$ and extrapolated into forward direction $\Theta_b=0$ where the radiation from the decay process is expected to vanish. This method is successfully tested at the parton level and results obtained for hadrons are compatible with the MLLA relation within an accuracy of around 15\%. Our calculations are carried out with the Pythia 8.3 Monte Carlo Event Generator.