Matching Fully Differential NNLO Calculations and Parton Showers

TL;DR

This work develops a general, fully differential framework to combine NNLO fixed-order calculations with LL resummation in jet-resolution variables and to interface the resulting NNLO+LL cross sections with parton showers to produce hadron-level events. It constructs Monte Carlo cross sections for N, N+1, and N+2 jet multiplicities that simultaneously satisfy fixed-order accuracy and LL resummation, using two jet-resolution scales to separate exclusive and inclusive regions. The authors detail two NNLO+LL construction strategies (Case 1 and Case 2), show how to couple these cross sections to showers while preserving perturbative accuracy, and relate the approach to existing methods such as GENEVA, HJ-MiNLO, and UNLOPS. The framework provides a path toward practical NNLO+PS event generation across multiple jet bins, with explicit guidance on consistency conditions and shower interfacing, enabling higher-precision collider phenomenology. Overall, this work offers a comprehensive, theory-grounded blueprint for achieving NNLO accuracy in fully differential event generation with realistic showered final states.

Abstract

We present a general method to match fully differential next-to-next-to-leading (NNLO) calculations to parton shower programs. We discuss in detail the perturbative accuracy criteria a complete NNLO+PS matching has to satisfy. Our method is based on consistently improving a given NNLO calculation with the leading-logarithmic (LL) resummation in a chosen jet resolution variable. The resulting NNLO$+$LL calculation is cast in the form of an event generator for physical events that can be directly interfaced with a parton shower routine, and we give an explicit construction of the input "Monte Carlo cross sections" satisfying all required criteria. We also show how other proposed approaches naturally arise as special cases in our method.

Figures (3)

• Figure 1: Illustration of the $N$-jet, $(N+1)$-jet, and $(N+2)$-jet regions in eq. \ref{['eq:NNLOevents']} for resolution variables that satisfy $\mathcal{T}_{N+1} < \mathcal{T}_N$ (e.g., the $p_T$ of the leading and subleading jet or $N$-jettiness Stewart:2010tn). The $N$-jet bin has $\mathcal{T}_N < \mathcal{T}_N^\mathrm{cut}$ and is represented by $N$-parton events with $\mathcal{T}_N = \mathcal{T}_{N+1} = 0$ (shown by the black dot at the origin). The $(N+1)$-jet bin has $\mathcal{T}_N > \mathcal{T}_N^\mathrm{cut}$ and $\mathcal{T}_{N+1} < \mathcal{T}_{N+1}^\mathrm{cut}$ and is represented by $(N+1)$-parton events with $\mathcal{T}_{N+1} = 0$ (shown by the black line on the $\mathcal{T}_N$ axis). The inclusive $(N+2)$-jet bin has $\mathcal{T}_N > \mathcal{T}_N^\mathrm{cut}$ and $\mathcal{T}_{N+1} > \mathcal{T}_{N+1}^\mathrm{cut}$ and is represented by $(N+2)$-parton events.
• Figure 2: Illustration of the perturbative accuracy of the cross section in different regions of the jet resolution variable $\mathcal{T}_N$. On the left we show the differential spectrum in $\mathcal{T}_N$, and on the right we show the cumulant as a function of $\mathcal{T}_N^c$, which approaches the total $N$-jet cross section (blue dashed line) for large $\mathcal{T}_N^c$. For large $\mathcal{T}_N^{(c)}$, the FO contributions (blue) determine the perturbative accuracy. As $\mathcal{T}_N^{(c)}$ decreases into the transition region, the resummed terms become increasingly important. At small $\mathcal{T}_N^{(c)}$ the resummation order determines the perturbative accuracy. The LL accuracy (green) that determines the shape at small $\mathcal{T}_N^{(c)}$ can be improved by higher-order resummation (orange). In the LL cumulant, we show that two different $\mathcal{T}_N^\mathrm{cut}$ values should produce the same cumulant cross section above $\mathcal{T}_N^\mathrm{cut}$.
• Figure 3: Illustration of the issues in defining an IR-safe phase space separation at NNLO using single-parton variables in case of vector boson production. Limiting each emission to be below $p_T^\mathrm{cut}$ (dashed lines) results in a miscancellation of IR divergences between the tree-level contribution on the left, which would contribute to $\mathrm{d}\sigma^\textsc{mc}_{0}(p_T^\mathrm{cut})$, and the corresponding one-loop contribution on the right, which would contribute to $\mathrm{d}\sigma^\textsc{mc}_{\ge 1}(p_T>p_T^\mathrm{cut})$.