Logarithmically-accurate and positive-definite NLO shower matching

TL;DR

This work constructs and validates a framework for logarithmically accurate NLL+NLO shower matching across processes with two coloured legs, including neutral- and charged-current Drell–Yan, Higgs production, DIS, and e^+e^- to jets. It introduces ESME, a positive-definite matching scheme that unifies real-radiation handling with NLO normalisation through exponentiated subtraction, and complements it with dBNLO and Projection-to-Born approaches for broader applicability. The authors perform extensive NLO and NNDL tests across $e^+e^-$, $pp$, and DIS, including $oldsymbol{\alpha_s \to 0}$ extrapolations to isolate pure NLO coefficients, demonstrating consistency with fixed-order results and event-shape resummation predictions. Phenomenological comparisons to data and performance assessments show competitive generation speeds and reasonable data agreement, underscoring the practical potential of these methods for showers with higher logarithmic accuracy. Overall, the work represents a significant step toward NNLL shower accuracy for hadronic processes and provides a transferable, positive-definite NLO matching framework within PanScales.

Abstract

We present methods to achieve NLL+NLO accurate parton showering for processes with two coloured legs: neutral- and charged-current Drell-Yan, and Higgs production in $pp$ collisions, as well as DIS and $e^+e^-$ to jets. The methods include adaptations of existing approaches, as well as a new NLO matching scheme, ESME, that is positive-definite by construction. Our implementations of the methods within the PanScales framework yield highly competitive NLO event generation speeds. We validate the fixed-order and combined resummation accuracy with tests in the limit of small QCD coupling and briefly touch on phenomenological comparisons to standard NLO results and to Drell-Yan data. The progress reported here is an essential step towards showers with logarithmic accuracy beyond NLL for processes with incoming hadrons.

Figures (12)

• Figure 1: Simple illustration of the different possible actions in the two streams of the ESME algorithm with joint reals and subtractions. The actions are shown separately for the cases $R(\Phi)<C(\Phi)$ (left) and $R(\Phi)>C(\Phi)$ (right). In each case, when summing the two streams, one sees that the "accept evt" action occurs with total weight $R/M$. One can also verify that the contribution to the total event rate change relative to the ${\bar{B}}_C$ normalisation is $(R-C)/M$. Recall that the default action in stream 1 (2) is to accept (reject) the event if the shower scale reaches $v_{\min}$ --- only when the action is different from the stream's default is the total event rate affected.
• Figure 2: Tests of NLO-matched showers, showing the oriented thrust axis distribution in $e^+e^- \to \gamma^* \to q\bar{q}$ collisions. The top panel shows the ratio to the total Born cross section for a phenomenological setup with $\sqrt{s} = 91.1876\;\mathrm{GeV}$, $\alpha_s(\sqrt{s}) = 0.118$ and a showering cutoff of $0.5\;\mathrm{GeV}$. The middle panel shows the ratio to the differential NLO cross section with the same settings for $\sqrt{s}$ and $\alpha_s$. The bottom panel shows the ratio of the pure NLO coefficient in the matched shower to the known exact NLO coefficient, i.e. the ratio in Eq. (\ref{['eq:alphas-nlo-coeff']}).
• Figure 3: NLO tests for the $pp\to Z/\gamma^* \to e^+ e^-$ process with cuts on the lepton transverse momentum and rapidity. Left (right): the invariant mass (rapidity) of the colour singlet. The top and middle panels show results with phenomenological settings, compared to NLO predictions from MCFM. Bands correspond to 7-scale uncertainty, $m_{\ell\ell}/2\le \mu_R,\mu_F \le 2m_{\ell\ell}$ with $1/2\le \mu_R/\mu_F \le2$. The bottom panel shows the ratio of the shower NLO coefficient (extracted in an $\alpha_s \to 0$ limit) to the NLO coefficient from MCFM. The bands represent the combined statistical uncertainty on the ratio.
• Figure 4: Analogue of Fig. \ref{['fig:pp2Zdec-NLO']}, showing the $W$ transverse mass and the charged lepton rapidity, without lepton or missing momentum cuts.
• Figure 5: Analogue of Fig. \ref{['fig:pp2Zdec-NLO']} for Higgs production, showing the Higgs rapidity distribution (left) and the transverse momentum distribution (right).