Problems in resumming interjet energy flows with k_t clustering

TL;DR

The paper analyzes energy flow into gaps between hard jets and the impact of $k_t$ clustering on resummation. It shows that while clustering reduces non-global logarithms, it invalidates straightforward single-gluon Sudakov exponentiation, introducing a previously neglected ${\mathcal{O}}(\alpha_s^2)$ piece $C_2^{\mathrm{primary}}$ that cannot be obtained from naive exponentiation. By comparing with fixed-order results from EVENT2, the authors demonstrate that correct agreement requires including $C_2^{\mathrm{primary}}$ in addition to the non-global term, especially for $e^{+}e^{-}$ and DIS topologies; the situation is more delicate for dijet photoproduction and hadron-hadron processes due to intricate color structures. They propose a practical all-orders approach using large-$N_c$ dipole evolution with clustering for simple processes, while acknowledging limitations and the need for further work in more complex hadronic environments.

Abstract

We consider the energy flow into gaps between hard jets. It was previously believed that the accuracy of resummed predictions for such observables can be improved by employing the $k_t$ clustering procedure to define the gap energy in terms of a sum of energies of soft jets (rather than individual hadrons) in the gap. This significantly reduces the sensitivity to correlated soft large-angle radiation (non-global leading logs), numerically calculable only in the large $N_c$ limit. While this is the case, as we demonstrate here, the use of $k_t$ clustering spoils the straightforward single-gluon Sudakov exponentiation that multiplies the non-global resummation. We carry out an ${\mathcal{O}}(α_s^2)$ calculation of the leading single-logarithmic terms and identify the piece that is omitted by straightforward exponentiation. We compare our results with the full ${\mathcal{O}} (α_s^2)$ result from the program EVENT2 to confirm our conclusions. For $e^{+}e^{-} \to 2$ jets and DIS (1+1) jets one can numerically resum these additional contributions as we show, but for dijet photoproduction and hadron-hadron processes further studies are needed.

Figures (3)

• Figure 1: Comparison of the $C_F^2 \alpha_s^2 L$ term produced by EVENT2 and the analytical calculation (referd to as resummed since it is derived by expanding the naive Sudakov resummation to NLO) with and without $C_2^{\mathrm{primary}}$. The figures are for $R=1$ and $\Delta \eta=1.0$ (above) and $\Delta \eta=0.5$ (below). The agreement for the unclustered case is also shown for comparison.
• Figure 2: Comparison of the $C_F C_A$ (above) and $C_Fn_f$$\alpha_s^2 L$ term (below) produced by EVENT2 and the expanded Sudakov result, supplemented with non-global logs for the $C_F C_A$ term. The figures are for $R=1$ and $\Delta \eta=1.0$ and and as we expect the difference between the exact and resummed result expanded to NLO is a constant at large $L$.
• Figure 3: The results for the primary emission resummation with and without $k_t$ clustering for $R=1 \, , \Delta \eta=1$, using an adaptation of the program used for Ref. AS1. As can be seen, the clustering affects the primary emission term and the effect for $t=0.25$ is an increment of over 30 %.