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

TL;DR

The paper investigates the dead-cone phenomenon in top-quark jets within perturbative QCD by separating radiation from the $\widehat{t\bar t}$ dipole (top) from decay-related $\widehat{tb}$ dipole radiation, using Pythia 8.3 to analyze $e^+e^-\to t\bar t$ at $1$ TeV and exploring both angular and momentum-space observables. It demonstrates how angular ordering and jet-strategy selections can reveal a central dead-cone suppression and quantifies the tb-dipole's role in filling the dead cone, while testing MLLA predictions for heavy-quark fragmentation via $\xi$-spectra and Limiting Spectrum fits. Extrapolations that subtract tb-dipole contributions yield parton spectra in good agreement with stable-top expectations to within about $5-12\%$ in the relevant region, and the hadron spectra show similar trends with strong suppression of high-momentum fragmentation for the top quark. Finite-width effects are found to be small (order $\lesssim5\%$) and the results lend support to applying these methods in $pp$ collisions at the LHC with carefully designed fat-jet analyses, promising a new QCD test in this high-mass regime.

Abstract

The gluon emission from an energetic heavy quark is suppressed in the forward direction below the angle $Θ\lesssim m_Q/E$ 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 $t\to b\ellν$, where the b-quark radiates gluons as well and partially obscures the dead cone. Guided by the decay amplitude in leading order in $α_s$ 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 $Θ_b$ and extrapolated into forward direction $Θ_b=0$ 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.

Figures (20)

• Figure 1: Feynman diagrams for gluon emission in $e^+e^-\to t \bar{t} \to bW^+\bar{b} W^-$ relevant to the top quark hemisphere according to the decomposition Eq. \ref{['FKhozeOtt']}. Heavy lines denote off-shell quarks. Panel (a) shows the correction to top production (amplitude $A$), while panel (b) shows the corresponding corrections to top decay (amplitude $B_1$), including gluon emission from the $b$-quark in the decay (according to Orr:1992uv).
• Figure 2: Scatter plots for three representative events from $e^+e^-\to t \bar{t}+X$ with stable top-quark at $\sqrt{s}=1$ TeV (upper panels), showing the distribution of partons from top-quark fragmentation over the angles $X$ and $Y$ in Eq. \ref{['XYdef']} in units of the dead cone angle $\Theta_0$ with the top-quark direction in the centre (0,0) (shown as cross): the primary gluon as red point at angle $R_X=\Theta/\Theta_0$ to the centre surrounded by the emitted partons (black points) limited by the same angle $R_X$ (red circle) as required by "angular ordering"; lower panels: Scatter plots as above but for events with top quark decay $t\to b\ell \nu + g$ for primary gluons triggered with $X_g<0$ shown as red square and the b-quark as blue dot rotated into negative $X$-direction ($Y=0$) and with $X_b<-0.5$. The emitted partons come mainly from the b-quark and are found inside the blue circle around the b-quark position as required by angular ordering (lower panels); here $\Theta_0=20^\circ$ at 1 TeV (from Pythia 8.3).
• Figure 3: Angular distribution of the reconstructed primary gluon in $e^+e^- \to t\bar{t} \ +$ partons at 1 TeV around the top-quark direction for stable top quarks: the left panel shows the distribution in the rescaled angles $(X,Y)$, defined in Equ. \ref{['XYdef']}, with the dead cone boundary $R_X^2=X^2+Y^2 \simeq 1$ and the suppression in the centre, the right panel shows the corresponding gluon density. At this energy the dead cone angle $\Theta_0 \simeq 20^\circ$ and the forward hemisphere ends at $R_X\simeq 4.5$. Data generated with Pythia 8.3.
• Figure 4: Distribution of the reconstructed primary gluon as in Fig. \ref{['fig:ring']} in the rescaled opening angle $\Theta/\Theta_0$ for stable top-quark production without and with gluon energy cut-off $E_g>10$ GeV and distribution in momentum fraction $x_p$, both in comparison with the approximate analytic form Equ. \ref{['emission']}.
• Figure 5: The invariant masses M($b\ell \nu+ g$) vs. M($b\ell \nu$) in $e^+e^- \to t \bar{t} +X$ with $b$ and $g$ representing jets reconstructed from the final partons (left panel) and M($B\ell \nu+ g$) vs. M($B\ell \nu$) from the final charged and neutral hadrons (right panel). The narrow stripes form near the top mass $m_t=172.5$ GeV. The vertical stripe corresponds to the processes of hard gluon radiation from the top-quark $t^*\to tg$, the horizontal stripe to the top-quark decay with gluon emission $t\to b \ell \nu +g$.