Matching Z $\to$ Hadrons at NNLO with Sector Showers

TL;DR

This work develops a fully differential NNLO matching framework for sectorised VINCIA showers applied to hadronic $Z$ decays, enabling precise control of 2-, 3-, and 4-jet final states through multi-differential jet rates. The authors derive explicit fixed-order and shower-matching coefficients, most notably the NLO 3-jet density ${\rm A}_{2\to3}^{\rm NLO}$ and its accompanying Sudakov-based factor ${K}^{\rm NLO}$, while implementing a subtraction-based procedure in sector subtraction to handle the intricate singular structure in the unordered phase-space regions. They incorporate interference effects between different Born states (via an interference factor ${k_I}$) and address $g\to q\bar q$ splittings with a modified sector-resolution variable, along with renormalisation-scale and scheme choices that ensure the correct fixed-order expansion. The result yields a practical route to NNLO accuracy within a shower algorithm, with explicit formulas and numerical strategies designed for efficient implementation in VINCIA, and sets the stage for phenomenological studies and public code release. The methodology has potential to generalise to more complex final states and to subleading colour effects, initial-state radiation, and higher-multiplicity Born configurations.

Abstract

We present a detailed technical derivation of matching conditions at next-to-next-to-leading order in the sectorised VINCIA parton shower, by considering leading-colour 2-, 3- and 4-jet rates in hadronic Z-boson decays. In particular, we introduce a full subtraction-based calculation of the matching coefficient required to obtain the NLO 3-jet rate. This is achieved through a judicious choice of the counter-terms, which optimises the numerical evaluation of the subtracted double-real matrix element. We additionally give a consistent prescription for incorporating interference effects due to higher-order mixing between Born states with different flavour contents. Finally, we briefly comment on higher-order uncertainty estimates.

Figures (7)

• Figure 1: The Dalitz plane for the evaluation of eq. \ref{['eq:IAsector']}. All singularities persist in the red-filled polygon, while the empty triangle is singularity-free by virtue of the sector constraint.
• Figure 2: The numerical evaluation of the $w_{\rm{RS}}^A$ (left) and $\mathcal{I}^A$ (right) contributions for various Born+1 configurations, as a function of the number of points used in the MC evaluation.
• Figure 3: The numerical contribution from eq. \ref{['eq:wRS-I-tot']} as a function of $(\hat{y}_{qg}, \hat{y}_{g\bar{q}})$ (left) and in logarithmic space (right).
• Figure 4: The contribution $v^{\mathrm{NLO}}_{N_C}$ as a function of $(\hat{y}_{qg}, \hat{y}_{g\bar{q}})$ (left) and in logarithmic space (right). Here, we subtract the expected logarithmic terms $\mathcal{L}(\tau_3) = \frac{\alpha_s N_C}{2\pi} \left(-\frac{1}{2} \ln^2 \tau_3 - \frac{10}{3} \ln \tau_3\right)$, with $\tau_3 = \hat{y}_{qg} \hat{y}_{g\bar{q}}$ in eq. \ref{['eq:wNC-expand']} from the full correction.
• Figure 5: Illustration of shower histories starting from $Z\to u\bar{u}$ (blue) and $Z\to d\bar{d}$ (red). Both histories are allowed to contribute to the same phase-space points for $Z\to u\bar{u}d\bar{d}$ (purple).