Heavy quark jet production near threshold

In this paper, we study the fragmentation of a heavy quark into a jet near threshold, meaning that final state jet carries most of the energy of the fragmenting heavy quark. Using the heavy quark fragmentation function, we simultaneously resum large logarithms of the jet radius $R$ and $1-z$, where $z$ is the ratio of the jet energy to the initiating heavy quark energy. There are numerically significant corrections to the leading order rate due to this resummation. We also investigate the heavy quark fragmentation to a groomed jet, using the soft drop grooming algorithm as an example. In order to do so, we introduce a collinear-ultrasoft mode sensitive to the grooming region determined by the algorithm's $z_{\mathrm{cut}}$ parameter. This allows us to resum large logarithms of $z_{\mathrm{cut}}/(1-z)$, again leading to large numerical corrections near the endpoint. A nice feature of the analysis of the heavy quark fragmenting to a groomed jet is the heavy quark mass $m$ renders the algorithm infrared finite, allowing a perturbative calculation. We analyze this for $E_JR \sim m$ and $E_JR\gg m$, where $E_J$ is the jet energy. To do the latter case, we introduce an ultracollinear-soft mode, allowing us to resum large logarithms of $E_JR/m$. Finally, as an application we calculate the rate for $e^+e^-$ collisions to produce a heavy quark jet in the endpoint region, where we show that grooming effects have a sizable contribution near the endpoint.