The Lund $b$-jet plane

TL;DR

The paper develops a single-logarithmic framework to compute the primary Lund plane density for jets initiated by a massive quark, incorporating quasi-collinear factorisation, a variable-flavour running of $\alpha_s$, and both collinear and soft resummation implemented via a large-$N_c$ dipole shower with full mass dependence. It derives massive-quark Lund plane densities at ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha_s^2)$, highlighting dead-cone suppression and mass effects through the dead-cone angle $\Delta_d=m/Q$ and dead-cone time $t_d$, and combines these ingredients into an all-orders master formula with running-coupling and flavour-evolution effects. The phenomenological application to $e^+e^-$ collisions at $\sqrt{s}=M_Z$ demonstrates clear dead-cone signatures in the $\!b$-jet Lund plane and includes matched predictions to tree-level matrix elements; comparisons to Pythia show qualitative agreement, with better concordance when flavoured-branch declustering is used. The results provide a theoretically solid basis for studying heavy-quark fragmentation in the Lund plane and open avenues for extending to LHC processes and multi-pronged jets, including $c$-jets and top/boson-initiated jets.

Abstract

We compute the primary Lund plane density for jets initiated by a massive ($b$) quark to single logarithmic accuracy in Quantum Chromodynamics (QCD). In order to capture mass effects, we consider quasi-collinear factorisation and we include contributions from the running of the QCD coupling and from collinear evolution, in a variable flavour-number scheme. Furthermore, the resummation of soft logarithms, including clustering effects, is performed numerically, keeping the full dependence on the $b$-quark mass. While our all-order results can be applied to both hadron and lepton colliders, we present, as first phenomenological application, the resummed calculation of the Lund plane density in $e^+e^-$ collisions at $\sqrt{s}=M_Z$, matched to tree-level matrix elements.

Figures (10)

• Figure 1: On the left, we show the average partonic fraction $f_{i|i_0}(t)$ assuming the collinear evolution starts from a light quark (dashed lines) or from a heavy quark (solid lines). On the right, we show instead the average momentum fraction $\braket{x(t)}_{i|i_0}$ assuming the collinear evolution starts from a light quark (dashed lines) or from a heavy quark (solid lines). Collinear evolution is performed by following the hard branch, which corresponds to the standard Lund plane declustering.
• Figure 2: Plots of the distributions $p_{i|b}(x,t)$ at four different snapshots of the evolution time. In each plot, the different solid lines correspond to a final parton $i$ of a given flavour or any flavour, using standard Lund declustering. The dashed curve instead shows the distribution that corresponds to flavour declustering.
• Figure 3: On the left, comparison between the Abelian and non-Abelian contributions to $\rho_{\mathcal{Q}}^{(\text{soft})}$. The Abelian contribution leads to a stronger suppression than the non-Abelian one, as discussed in the text. On the right, all-order resummation of soft contributions for a $b$-quark dipole ($\mathcal{Q}=b$) obtained with the numerical method described in Sec. \ref{['sec:num_soft_res']}, as well as its first and second order expansions, plotted as a function of the Lund-plane pseudorapidity $\eta=-\log \tan \frac{\theta_{ij}}{2}$, see Eq. (\ref{['eq:lund-kinematic-variables-ee']}). The second order expansion is compared to the analytic calculation performed in the collinear limit.
• Figure 4: Soft contribution to the primary density for an initial $b \bar{b}$ dipole (at LEP energies), as a function of the evolution time, for two different slices in pseudorapidity $\eta$. In each plot, we show the all-order result (in black), and its ${\cal O}\left(\alpha_\text{s}\right)$ (red) and ${\cal O}\left(\alpha_\text{s}^2\right)$ (blue) expansions.
• Figure 5: Pictorial representations of the different contributions that enter the resummation of the primary Lund plane density. The collinear evolution of the originating parton $i_0$ from the initial angular scale $\Delta_0$ to parton $i$ at an angle $\Delta$ is represented by a green arrow. During this evolution the transverse momentum is reduced from $Q \Delta_0$ to $x Q \Delta$. The red arrow represents instead soft evolution, at fixed angular scale, from $z Q \Delta$ to $k_t$. Finally, the black blob represents the splitting that is recorded.