CERN Accelerating science

Lund plane~\cite{Andersson:1988gp} representation of the phase space for soft and/or collinear emission, and illustration of the interplay between a hardest emission generator (HEG), a parton shower and a constraint on an observable (black line). Dimensionful quantities ($k_t$, $v$) are taken normalised to the centre-of-mass energy $Q$.

Schematic illustration of the issue associated with gluon asymmetrisation. (a) Contours on the Lund plane, in the PanLocal family of showers, highlighting the fact that a given physical point $X$ in the Lund plane (highlighted with a red cross) can come from two different values of $v$. The shading of the green curves represents the variation in radiation intensity along the contour. (b) Density plot, at each point in the Lund plane, representing schematically the fraction of the emission intensity at that point that has been excluded once the HEG has reached a given $v$ value ($v_\Phi$) without emitting, and an illustration that as the shower continues there may still be phase-space points (such as that marked with a cross) where the Sudakov has only been partially accounted for. The implications are discussed in the text.

Schematic illustration of the issue associated with gluon asymmetrisation. (a) Contours on the Lund plane, in the PanLocal family of showers, highlighting the fact that a given physical point $X$ in the Lund plane (highlighted with a red cross) can come from two different values of $v$. The shading of the green curves represents the variation in radiation intensity along the contour. (b) Density plot, at each point in the Lund plane, representing schematically the fraction of the emission intensity at that point that has been excluded once the HEG has reached a given $v$ value ($v_\Phi$) without emitting, and an illustration that as the shower continues there may still be phase-space points (such as that marked with a cross) where the Sudakov has only been partially accounted for. The implications are discussed in the text.

An illustration of the spin-sensitive observable $\Delta \Psi_{12}$ as the difference of the azimuthal angles of the branching plane of the matched emission $\mathcal{P}_1$ and the branching plane of a subsequent collinear branching $\mathcal{P}_2$.

Second-order validation of the implementation of spin correlations in the hard region for $e^+ e^- \to \gamma^* \to q \bar{q}$. Following Eq.~\eqref{eq:hard-spin-xsec-form}, the ratio $a_2 / a_0$ is computed for the shower with matching enabled (blue curve). Below, the ratio of the shower $a_2 / a_0$ to the exact matrix element $a_2 / a_0$ is shown.

Same as Figure~\ref{fig:spin-qqbar}, but for all possible configurations of $H \to gg$. Note that in the lower plots, it is the gluon of the underlying 3-jet system that undergoes collinear splitting, rather than the hard emission from the 2-jet system (which is a quark or anti-quark, and so does not mediate azimuthal correlation). This is also the reason for the lack of symmetry in the matched result between positive and negative $\eta_1$ values.

Same as Figure~\ref{fig:spin-qqbar}, but for all possible configurations of $H \to gg$. Note that in the lower plots, it is the gluon of the underlying 3-jet system that undergoes collinear splitting, rather than the hard emission from the 2-jet system (which is a quark or anti-quark, and so does not mediate azimuthal correlation). This is also the reason for the lack of symmetry in the matched result between positive and negative $\eta_1$ values.