Precision QCD: Matched NNLO+NNLL Results

• The paper demonstrates that matched NNLO+NNLL methods significantly reduce theoretical uncertainties in collider observables compared to lower-order predictions.
• The methodology integrates fixed-order NNLO calculations with NNLL resummation by subtracting overlapping contributions to maintain accuracy across soft and hard regimes.
• The framework utilizes profile scales and rigorous uncertainty assessments, ensuring reliable precision for a wide range of QCD analyses in modern collider experiments.

Matched NNLO+NNLL (next-to-next-to-leading order plus next-to-next-to-leading logarithmic) results are the state-of-the-art in precision QCD calculations for collider observables sensitive to both large invariant-mass scales and regions dominated by soft and/or collinear radiation. The matched approach accurately incorporates both the fixed-order perturbative expansion through NNLO and the all-order resummation of logarithmically enhanced contributions up to NNLL by systematically subtracting double-counted terms. This ensures reliable results across the full observable spectrum, with quantified theoretical uncertainties appropriate for modern precision phenomenology.

1. Fundamentals of NNLO+NNLL Matching

For collider observables subject to large logarithmic corrections—such as event shapes, resummation variables, or cross sections with jet vetoes—fixed-order QCD expansions become unreliable in certain regions (e.g., $p_T \to 0$, small thrust, or tight jet-veto cuts) due to terms of the form $\alpha_s^n L^k$, with $L$ a large logarithm. All-order resummation of these enhanced contributions is essential. NNLL resummation systematically captures the leading, next-to-leading, and next-to-next-to-leading towers of logarithms using factorization theorems based on soft-collinear effective theory (SCET) or equivalent frameworks.

However, resummation is only formally accurate in the singular (small observable) regime, while the fixed-order result is reliable in the hard region. To retain the accuracy of both, NNLO+NNLL matching constructs a composite prediction by adding the full NNLL-resummed result to the NNLO fixed-order calculation, and subtracting their common terms as given by the expansion of the resummed formula to $\mathcal{O}(\alpha_s^2)$. Explicitly, the additive master matching formula reads

$$\sigma_{\rm NNLO+NNLL} = \sigma_{\rm NNLL\;resum} + [\sigma_{\rm NNLO}^{\rm FO} - (\sigma_{\rm NNLL\;resum})_{\rm expanded\ to\ NNLO}],$$

ensuring fixed-order accuracy at large observable values and all-order logarithmic accuracy as the observable tends to zero. This structure is implemented, with appropriate generalizations, in all current high-precision event-shape, jet-veto, and resummation-sensitive analyses.

2. Formalism and Key Ingredients

The factorization underlying NNLL resummation generally separates the observable into hard, jet/beam, and soft functions, each evaluated at their canonical scale and evolved to a common scale using renormalization group evolution (RGE) kernels. For example, the SCET 0-jet cross section with a rapidity-dependent veto (as in Drell-Yan) is factorized as

$$\frac{d\sigma_0^{\rm resum}}{dQ^2\,dY} = \sigma_B \sum_{ij} H_{ij}(Q^2,\mu_H)\,U_H(Q,\mu_H,\mu)\,B_i\,B_j\,U_B^2\,S_f\,U_S + d\sigma^{R\rm sub},$$

with the hard matching $H_{ij}$, beam functions $B_i$, and soft function $S_f$, each expanded through two loops for NNLL$^\prime$. The evolution factors $U_X$ encode the solution to the RGE, driven by the cusp and non-cusp anomalous dimensions.

The fixed-order expansion of all ingredients is required through $\mathcal{O}(\alpha_s^2)$. The non-singular piece, ensuring correct matching as the veto scale increases, is computed as the difference between the NNLO fixed-order result and the expansion of the resummed formula evaluated with all scales at the fixed-order point: $$\sigma_0^{\rm nons,NNLO} = \sigma_0^{\rm FO,NNLO} - \sigma_0^{\rm resum,NNLL'}\Big|_{\mu_H=\mu_B=\mu_S=\mu_{\rm FO}}.$$

The matched NNLO+NNLL cross section is then

$$\sigma_{\rm NNLL'+NNLO}(Q,\mu,\{\mu_i\}) = \sigma_{\rm resum}(Q,\mu,\{\mu_i\}) + [\sigma_{\rm NNLO}(Q,\mu) - (\sigma_{\rm resum}(Q,\mu,\{\mu_i\}))_{\rm expanded\ to\ NNLO}],$$

as demonstrated explicitly in (Clark et al., 8 Apr 2025).

3. Profile Scales, Uncertainties, and Matching Schemes

To smoothly transition between regimes where resummation is essential ($\mathcal{T}^{\rm cut}\ll Q$ or small observable) and fixed-order dominance (large observable), profile scales are introduced. These interpolating functions for $\mu_H$, $\mu_B$, and $\mu_S$ (or resummation and factorization/renormalization scales in $b$-space or Mellin space) enforce canonical scaling in the asymptotic regime and turn off resummation at large values: $$\mu_H=-i\mu_{\rm FO},\quad \mu_S=\mu_{\rm FO}f_{\rm run}(x),\quad \mu_B=\mu_{\rm FO}\sqrt{f_{\rm run}(x)},\quad x\equiv\frac{\mathcal{T}^{\rm cut}}{M_Z},$$ with detailed parameterizations of $f_{\rm run}(x)$ optimized for each process and observable.

Perturbative uncertainty bands are constructed by conventional scale variations. It was found (Clark et al., 8 Apr 2025) that standard profile-scale variations can underestimate uncertainty due to cancellations among partonic channels at NLL$^\prime$+NLO. A "MaxDev" prescription, applying independent variations in each channel and taking the largest resulting deviation, provides a more robust uncertainty estimate. Combined with variation of the fixed-order scale, this yields total uncertainties at the percent level in NNLO+NNLL predictions.

4. Canonical Examples Across Collider Observables

Matched NNLO+NNLL methodology is now the standard for a wide range of precision collider observables:

• Drell-Yan and Jet Vetoes: The master formula and the entire framework, including profile scales and MaxDev uncertainties, is implemented for rapidity-dependent jet vetoes in the Drell-Yan process (Clark et al., 8 Apr 2025).
• Transverse-Momentum and Event Shapes: For $pp \to H(b\bar b)$, the $p_T$ distribution is computed using impact-parameter ($b$-space) factorization, with matching yielding scale reduction from $\sim$45-50% at NLO+NLL to 22-25% at NNLO+NNLL in the low $p_T$ region (Harlander et al., 2014, Harlander et al., 2014).
• Energy-Energy Correlation in $e^+e^-$: The back-to-back region is analyzed with detailed NNLL+NNLO matching, additive or log-R schemes, and rigorous uncertainty estimation, producing competitive extractions of $\alpha_s(M_Z)$ (Tulipánt et al., 2017, Kardos et al., 2018).
• Heavy-Flavored Fragmentation: The Mellin-moment space approach with additive matching, complete soft resummation, and explicit discussion of Landau pole prescription impacts (Bonino et al., 2023).
• Event Shapes in Higgs Decays: Thrust, heavy-jet mass, and $C$-parameter in $H\to q\bar q$ and $H\to gg$ decays have been calculated at NNLO+NNLL, demonstrating substantial reductions in uncertainty and sensitivity to higher-logarithmic orders in gluonic channels (Fox et al., 13 Oct 2025, Fox et al., 2 Dec 2025).
• Diboson and Electroweak Processes: $W^+W^-$ and $ZZ$ pair invariant-mass distributions exploit threshold resummation to NNLO+NNLL, yielding a several-percent increase in the high-mass regime and a reduction of scale uncertainties (from 6.8% to 4.1% at $Q=2.5$ TeV for $WW$) (Banerjee et al., 12 Jun 2025, Banerjee et al., 24 Sep 2024).
• Top Quark Production: NNLO+NNLL' matching for $t\bar t$ distributions (both massive and small-mass boosted soft resummation) stabilizes systematic error and produces nearly perfect agreement with exact NNLO when optimal scales are chosen (Czakon et al., 2018, Kidonakis, 2012).

Representative numerical outcomes demonstrate a pattern: at hadronic scales $\mathcal{T}^{\rm cut}=\mathcal{O}(10\ \text{GeV})$, uncertainties fall from 5-6% (NLL$^\prime$+NLO) to 1-3% (NNLL$^\prime$+NNLO) in veto observables (Clark et al., 8 Apr 2025); for $p_T$ distributions, from 45% to 22% at the peak (Harlander et al., 2014); for diboson invariant masses, from $3.4\%$ to $2.6\%$ in the TeV regime (Banerjee et al., 24 Sep 2024). These improvements are consistent across collider observables.

5. Implementation and Cross-Validation

Matching schemes used in the literature are primarily additive, though log-R or multiplicative forms are also present for event shapes and fragmentation. All implementations enforce unitarity by construction—integrated matched results reproduce the total NNLO cross section. Two-loop hard, beam, and soft function boundary conditions are employed in SCET-based approaches (Alioli et al., 2015, Alioli et al., 2019). For impact-parameter or Mellin-space resummation, results depend on the handling of Landau pole effects, with minimal prescription (Catani-Trentadue) favored for stability (Bonino et al., 2023). Profile scale design and the adoption of robust uncertainty estimates are essential for reliable predictions.

Numerical codes are typically validated by comparison with established fixed-order tools (e.g., DYNNLO, NNLOJET) and by expansion of the resummed formula; careful crosschecks confirm that the matching procedure reproduces the singular fixed-order structure and smoothly transitions to non-singular regimes (Fox et al., 13 Oct 2025, Fox et al., 2 Dec 2025).

6. Phenomenological Impact and Outlook

Matched NNLO+NNLL predictions underlie current precision SM and BSM analyses at the LHC and anticipated Higgs and lepton colliders. They provide cross sections with percent-level accuracy for both total rates and resummation-sensitive differential distributions, stabilizing theoretical uncertainties. For jet-vetoed or exclusive observables, the matched results are particularly indispensable, as pure NNLO fixed-order series are unreliable at small veto scales or in the Sudakov region.

The systematic reduction of perturbative uncertainties by a factor two or more across all key observables enables high-precision SM tests, differential extractions of QCD parameters, and improved sensitivity to new phenomena. Further progress, such as extension to N$^3$LL matching and full two-dimensional resummation, and continued development of optimal uncertainty prescriptions and Monte Carlo integration, is ongoing in the field. For detailed formulations, implementations, and benchmark results, see (Clark et al., 8 Apr 2025, Harlander et al., 2014, Alioli et al., 2015, Fox et al., 13 Oct 2025, Fox et al., 2 Dec 2025, Banerjee et al., 24 Sep 2024), and (Banerjee et al., 12 Jun 2025).