A Simulation of QCD Radiation in Top Quark Decays

TL;DR

The paper develops a covariant, coherent parton shower for top-quark decays implemented in Herwig++, combining all-orders resummation of soft/collinear QCD radiation with unenhanced first-order matrix-element corrections for hard emissions. It introduces detailed shower variables, phase-space factorization, and angular ordering to accurately model radiation from both the top and bottom quarks, and then applies soft and hard matrix-element corrections to ensure correct soft limits and fill dead regions of phase space. The authors demonstrate improved agreement with prior HERWIG results for key observables and provide a generalized quasi-collinear splitting function to mitigate spuriously enhanced emissions, while reconstructing kinematics with momentum reshuffling to preserve jet structure. The approach lays groundwork for extending accurate QCD radiation modeling to decays of other heavy particles, including supersymmetric states, in future Herwig++ versions.

Abstract

In this paper we describe a theoretical framework and algorithms for implementing QCD corrections to top quark decays in the Herwig++ event generator. The dominant corrections, due to soft and collinear emissions, are summed to all orders through the coherent parton branching formalism. In addition, unenhanced first-order matrix-element corrections are included to account for large transverse momentum emissions.

Figures (9)

• Figure 1: An example of the decay $t\rightarrow bW+\left(n\right)g$.
• Figure 2: In this figure we show the azimuthally averaged radiation pattern $\left\langle W_{tb}^{t}\right\rangle$, for $v_{t}=0.65$, $v_{b}=1$. In this plot, the ratio of the integral of $\left\langle W_{tb}^{t}\right\rangle$ outside the cone, to the same integral inside the cone is around $-0.33$.
• Figure 3: In this figure we show the phase space boundaries for the symmetric (left) and maximal (right) choices of phase space partitioning, in the $x_{g},x_{W}$ plane, where $x_{g}$ and $x_{W}$ are equal to two times the energy fraction of the gluon and W boson in the top quark rest frame. The regions T1 and T2 are populated by gluon emissions from the top quark while the region labelled B is populated by emissions from the b-quark. The region labelled D is the dead region. For the symmetric choice the phase space volume is divided more or less evenly between that accessible to emissions from the top and bottom quarks, while the maximal choice maximises the volume to be populated by emissions from the b-quark.
• Figure 4: In this figure we sketch the 'momentum reshuffling' procedure. Initially the top quark decay to a b-quark and a W boson is simulated, this momentum configuration is shown first on the far left. Afterwards some additional radiation is produced from the parton shower, represented by the spiral in the central configuration above, clearly this configuration does not conserve momentum. The initial b-jet momenta are rescaled and boosted to give the configuration shown on the right hand side: the three-momenta of the W boson and b-jet are scaled down and the momentum of the additional radiation, transverse to the W-boson direction, is absorbed by the b-jet .
• Figure 5: Dalitz plot for gluonic radiation in top decay. In both plots the soft and hard matrix element corrections have been applied, but only one emission has been allowed. a) shows the radiation for the symmetric choice of Gieseke:2003rz for emission from the top and bottom while b) shows the radiation with the scales chosen to give the maximum amount of radiation from the bottom quark. The blue (innermost) line gives the limit for radiation from the bottom, the green (middle) line from the top and the red (outer) line the boundary of the phase space region.