Phase space veto method for next-to-leading order event generators in hadronic collisions

TL;DR

The paper tackles the challenge of turning NLO QCD corrections into realistic, exclusive event samples suitable for detector simulation by introducing the phase-space veto ($\Phi$-space Veto) method. This approach uses event-by-event dynamic determination of the $s_{\mathrm{min}}$ boundary and a projection-based test to keep event weights positive and avoid double counting, while preserving the reduced scale dependence characteristic of NLO calculations. A concrete implementation for $pp\to Z^0/\gamma^*+X\to l^+l^-+X$ demonstrates that NLO cross sections and key distributions can be matched to traditional $s_{\mathrm{min}}$-slicing results, with the added advantage of direct interfacing to a parton shower (PYTHIA) and hadronization, yielding detector-ready events. The method provides NLO normalization with positive weights and offers a practical, generalizable framework for combining NLO calculations with showering and hadronization in hadronic collision phenomenology.

Abstract

A method for organizing next-to-leading order QCD calculations using a veto which enforces the cancellations between virtual and real emission diagrams is applied to hadronic collisions. The method employs phase space slicing with the slicing parameter determined dynamically event-by-event. It allows for the generation of unweighted events and can be consistently merged with a parton shower. The end product is more intuitive for the end user, as it is probabilistic, and can be easily interfaced to general purpose showering and hadronization programs to obtain a complete event description suitable for experimental analyses. As an example an event generator for the process pp --> Z + X at NLO is presented and interfaced consistently to the PYTHIA shower and hadronization package.

Figures (15)

• Figure 1: Feynman graphs contributing to $pp\!\!\!\!^{^{(-)}}\rightarrow Z^0+X$ at NLO. The wavy line represents either a $Z^0$ or $\gamma^\star$, and the vector-boson decay products are not shown.
• Figure 2: A projection of the $pp\!\!\!\!^{^{(-)}}\rightarrow Z^0 j$ phase space onto the $\hat{u}$ vs. $\hat{t}$ plane is shown, where $\hat{u}=(p_2-p_j)^2=-Q^2_{2j}$ and $\hat{t}=(p_1-p_j)^2=-Q^2_{1j}$, and $p_1,~p_2,~p_j$ are the momenta of the forward colliding parton, backward colliding parton, and real emission. The area above (below) the $\mathrm{s}_{\mathrm{min}}$ boundary is the region of resolved (unresolved) real emissions. When $\mathrm{s}_{\mathrm{min}}=\mathrm{s}_{\mathrm{zero}}$, it denotes the boundary defining the region inside of which the the n-body and (n+1)-body contributions sum to zero (i.e. the cross section integrated over the unresolved region is zero).
• Figure 3: The two roots of the quadratic n-body differential cross section presented in Eq. \ref{['e_xsec_nbody']} are plotted as a function of the lepton-pair rapidity, evaluated at parton center-of-mass energy equal to the $Z^0$ mass for $p\bar{p}$ collisions at 2 TeV (Tevatron, top) and for $pp$ collisions at 14 TeV (LHC, bottom). The smaller solution is the $\mathrm{s}_{\mathrm{zero}}$ function of interest, the larger solution should not be interpreted physically.
• Figure 4: The dependence of $\mathrm{s}_{\mathrm{zero}}$ as a function of lepton-pair rapidity at several choices of parton the center-of-mass energy $Q$ is shown for the $p\bar{p}$ collisions at 2 TeV (Tevatron, top) and for $pp$ collisions at 14 TeV (LHC, bottom). The $\mathrm{s}_{\mathrm{zero}}$ function does not depend strongly on the vector-boson decay angles.
• Figure 5: The scale variation of the $\mathrm{s}_{\mathrm{zero}}$ function evaluated at parton center-of-mass energy equal to the $Z^0$ mass is shown for $p\bar{p}$ collisions at 2 TeV (Tevatron, top) and for $pp$ collisions at 14 TeV (LHC, bottom). The $\mathrm{s}_{\mathrm{zero}}$ function encodes information about the factorization and renormalization scale choices into the $\Phi$-space Veto method, preserving the NLO calculation's reduced scale dependence.