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Towards NNLO QCD predictions for off-shell top-quark pair production and decays

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper delivers the first NNLO QCD prediction for off-shell top-quark pair production with leptonic decays, estimating the missing two-loop virtual corrections through the double-pole approximation anchored to the on-shell cross…

desk verdict First NNLO number for off-shell W+W-bbbar with massive bottom quarks, but the fitted non-factorisable two-loop piece is load-bearing and only indirectly validated. read the letter →

arxiv 2507.11410 v1 pith:3SJG6TDD submitted 2025-07-15 hep-ph hep-ex

Luca Buonocore , Massimiliano Grazzini , Stefan Kallweit , Jonas M. Lindert , Chiara Savoini This is my paper · ORCID
classification hep-phhep-ex PACS 12.38.Bx14.65.Ha
keywords NNLOQCDtop-quarkpairproductionoff-shelleffectsdouble-poleapproximationqTsubtractionnon-factorisablecorrectionsW+W-bbbarLHCphysics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is trying to establish that NNLO QCD accuracy is attainable for the full off-shell process $pp \to e^+\nu_e b\,\mu^-\bar\nu_\mu\bar b$ — the clean "golden channel" of top-quark pair production — even though the exact two-loop amplitudes for such a $2\to 6$ process are beyond current techniques. The strategy is to compute every other ingredient exactly and to supply only the genuine two-loop virtual piece through the double-pole approximation: factorisable corrections built from polarised two-loop on-shell $t\bar t$ production amplitudes and two-loop top-decay form factors, plus non-factorisable corrections fixed by matching the off-shell result to the known on-shell $t\bar t$ NNLO cross section as the top-quark width goes to zero. The paper claims an inclusive NNLO cross section of $\sigma = 10623(55)\pm152$ fb at 13 TeV, an 11% increase over NLO, with numerical uncertainty below 2%, smaller than the roughly 5% residual perturbative uncertainty. A sympathetic reader would care because this is the first NNLO-level prediction for the dilepton $t\bar t$ final state that keeps top-quark off-shell effects, $tW$ interference, and finite-width effects consistently, and it lays out a path for approximating two-loop amplitudes in similarly complex processes.

What carries the argument

The load-bearing device is the double-pole approximation (DPA) for the two-loop virtual amplitude: the off-shell amplitude is expanded keeping only its double-resonant $t\bar t$ topology, with the residue evaluated on projected on-shell momenta, so production and decay factorise. The factorisable part multiplies polarised tree-level production and decay amplitudes by their two-loop corrections; the non-factorisable part is assumed to have the one-log-per-loop form $\Delta\sigma_{\rm NNLO,H}^{\rm non-fact} = A^{(2)}_{\rm nf}\log^2(\Gamma_t/Q_h) + B^{(2)}_{\rm nf}\log(\Gamma_t/Q_h) + C^{(2)}_{\rm nf} + O(\Gamma_t/Q_h)$, whose unknown coefficients are extracted by computing the cross section at several artificially small top-quark widths, fitting the residual width dependence, and matching the $\Gamma_t\to 0$ limit to the known on-shell $t\bar t$ NNLO cross section. Two supporting mechanisms carry the accuracy: massification, which restores bottom-quark mass logarithms in the two-loop production and decay amplitudes obtained in the massless limit, and $q_T$-subtraction with an $r_{\rm cut}\to 0$ extrapolation, which handles the infrared singularities of the $2\to 6$, $2\to 7$ and $2\to 8$ real and real-virtual contributions.

What would settle it

A direct calculation of the genuine two-loop non-factorisable contributions — soft-gluon exchanges linking the $t\bar t$ production stage to the top or anti-top decays, i.e., two-loop six-point integrals with internal masses — in the gluon-fusion channel would settle the claim: if the resulting constant differs from the fitted $C^{(2)}_{\rm nf}$ by more than the quoted ~1.5% systematic, the on-shell matching has fixed the wrong value. A cheaper immediate check is to rerun the $\Gamma_t \to 0$ fit restricted to the smallest widths ($\Gamma_t/\Gamma_t^{\rm phys} \le 0.1$), where the logarithmic ansatz should dominate; a significant shift of the extracted constant would signal contamination by power corrections beyond the fitted linear term.

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Extended reading notes

Core claim

The paper's central claim is that the inclusive cross section for $W^+W^-b\bar b$ production with leptonic decays and massive bottom quarks can be computed at NNLO accuracy by treating only the genuine two-loop virtual contribution in the double-pole approximation. The factorisable two-loop corrections are constructed from available polarised two-loop on-shell $t\bar t$ production amplitudes combined with two-loop top-quark decay amplitudes obtained from heavy-to-light form factors, with bottom-quark mass effects restored by massification. The non-factorisable corrections, which are not directly computable, are fixed by requiring that the off-shell cross section reproduces the on-shell $t\bar t$ NNLO cross section times branching ratios in the $\Gamma_t \to 0$ limit, using the known functional form $A^{(2)}_{\rm nf}\log^2(\Gamma_t/Q_h) + B^{(2)}_{\rm nf}\log(\Gamma_t/Q_h) + C^{(2)}_{\rm nf}$ for their width dependence. With this construction the paper obtains $\sigma_{\rm NNLO} = 10623(55)\pm 152$ fb at 13 TeV, an 11% upward shift of the NLO result, a numerical uncertainty below 2%, and residual perturbative uncertainties of about $+3.2\%/-4.6\%$. Along the way it establishes that the double-pole approximation reproduces the exact NLO result at the per-mille to few-percent level across fiducial and differential observables, and that the non-factorisable two-loop corrections, though roughly 20% of the factorisable ones, shift the NNLO correction at order one because other contributions largely cancel.

Load-bearing premise

The whole result rests on the assumption that the missing two-loop non-factorisable corrections have exactly the assumed dependence on the top-quark width — a squared logarithm, a single logarithm, and a constant term — and that the constant fixed by matching to the known on-shell $t\bar t$ cross section in the zero-width limit is the correct one, a check performed so far only at lower orders and in the subdominant channels.

Editorial extensions

If this is right

  • The inclusive $e^+\nu_e \mu^-\bar\nu_\mu b\bar b$ cross section at 13 TeV is predicted at $\sigma_{\rm NNLO} = 10623(55)\pm 152$ fb, an 11% upward correction over NLO with numerical uncertainty below 2%.
  • The double-pole approximation, validated at NLO to per-mille accuracy in fiducial cross sections and a few percent in distribution tails, supplies a stand-in for two-loop amplitudes in processes where exact multi-loop results are out of reach.
  • Non-factorisable two-loop corrections matter at order one for the NNLO shift because other contributions largely cancel, so a DPA-based NNLO prediction that omits them would be qualitatively incomplete.
  • After removing spurious finite-width terms, the matched prediction gives $\sigma^{\Delta\rm trunc}_{\rm NNLO} = 10278(55)\pm152$ fb, allowing consistent future comparisons with narrow-width treatments of $t\bar t$ plus $tW$ production.
  • The exact NLO computation of the process with a resolved jet in the 4-flavour scheme with massive bottom quarks is the first of its kind and provides a new handle for jet-based background modelling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • My inference: the same construction — exact real and real-virtual pieces, DPA for the genuine two-loop virtual, on-shell matching in the zero-width limit — is portable to other unstable-particle processes (such as $t\bar t H$ or single-top channels) for which exact two-loop amplitudes are unavailable but an on-shell NNLO anchor exists.
  • My inference: the headline number is only as strong as the fitted constant $C^{(2)}_{\rm nf}$; a future direct computation of the non-factorisable two-loop diagrams that shifts that constant would move the predicted cross section by about the quoted 1.5% systematic — the method's conclusion, not its architecture, is the vulnerable piece.
  • My inference: because the fit leaves the single-log coefficient $B^{(2)}_{\rm nf}$ weakly constrained, an analytic computation of the double- and single-log coefficients from soft-gluon exponentiation on the resonant propagators would tighten the prediction more efficiently than additional Monte Carlo statistics.
  • My inference: extending the $\Gamma_t \to 0$ fit bin-by-bin would test whether the non-factorisable two-loop corrections change sign or grow in kinematic tails, as the one-loop study in the bottom-jet-tagged setup suggests; differential NNLO predictions for this process would then be the natural next target.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents an NLO QCD calculation of pp -> W+W- b bbar production with leptonic decays and massive bottom quarks in the 4FS, implemented in the qT-subtraction framework and validated against an independent dipole-subtraction calculation. It then constructs the double-pole approximation (DPA) for the virtual corrections at NLO and shows that the DPA reproduces exact NLO results at the per-mille-to-percent level. The main new result is an approximate NNLO prediction for the inclusive cross section: the exact double-real and real-virtual contributions and the one-loop squared terms are combined with factorisable two-loop corrections built from polarised on-shell t-tbar amplitudes and massified top-decay form factors, while the non-factorisable two-loop contribution is inferred from the Gamma_t -> 0 limit through an on-shell matching procedure. The quoted NNLO cross section is 10623(55) +- 152 fb, corresponding to an NNLO/NLO K-factor of 1.108.

Significance. If the matching procedure is reliable, this is the first NNLO-quality prediction for off-shell top-quark pair production with decays in the 4FS with massive bottom quarks, and it would be a significant step towards a complete NNLO calculation for this process. The paper contains several genuine strengths: the exact NLO calculation is cross-checked against an independent implementation at the 0.04% (CKMP) and 0.2% (CMP) level; the NLO DPA is validated against exact NLO in fiducial cross sections and differential distributions; the rcut extrapolation is studied with a replica method and with alternative fit models; and the small-width extrapolation is validated at LO, at NLO, and in the off-diagonal NNLO channels that are computed exactly. The paper is also unusually transparent about the limitations of the non-factorisable two-loop inference. The main caveat is that the constant term of the non-factorisable two-loop correction is not computed but is fixed by the on-shell matching, so the final NNLO number inherits the model assumptions of that matching.

major comments (3)
  1. [Sec. 5.3.3, Eq. (76)] The central number is controlled by the constant C^(2)_nf of the non-factorisable two-loop corrections, but this constant is not extracted from data at the physical width. It is defined through Eq. (77) as the value that makes the Gamma_t -> 0 extrapolation of the off-shell NNLO_fact result coincide with the on-shell NNLO cross section. The fitted quantity is therefore the full off-shell-minus-on-shell difference, which also includes single-top/non-resonant contributions, 4FS/5FS effects, bottom-mass effects, and truncation subtleties. The ansatz in Eq. (73) assumes that all of these are described by A^(2)_nf log^2 + B^(2)_nf log + C^(2)_nf + D^(2)(Gamma_t/mt) plus power-suppressed terms. Any Gamma_t-dependent contamination not of this form is absorbed into C^(2)_nf and shifts sigma_NNLO at the physical width by an amount that is not covered by the quoted +-152 fb fit error. The NLO validation in this section checks the procedure at one loop, where the non-factorisable correction is relatively small, but at NNLO the non-factorisable contribution is O(1) of the NNLO correction, so the one-loop cross-check cannot bound the two-loop model error. I request a concrete sensitivity test: repeat the fit with additional power-correction terms, e.g. E(2)(Gamma_t/mt)^2 or (Gamma_t/mt) log(Gamma_t/mt), and report the resulting shift in C^(2)_nf and in sigma_NNLO. Without such a test, the statement that the numerical uncertainty is below 2% is not fully supported.
  2. [Sec. 5.4, Table 3] The main result does not follow from its stated components. In Table 3 the all-channel values are sigma_NNLO_qT,DPA' = 10623(55) +- 152 fb and sigma_NNLO_fact = 9546(55) fb, while the last row gives Delta_sigma_NNLO,H|non-fact = 1107 +- 152 fb. The text in Sec. 5.4 says that the total NNLO cross section is obtained as the sum of sigma_NNLO_fact and Delta_sigma_NNLO,H|non-fact, but 9546 + 1107 = 10653, not 10623. This is a 30 fb (about 0.3%) discrepancy in the headline cross section. Please correct the table or the definition of the sum, and explain where the difference comes from.
  3. [Appendix A, Sec. 3.1] The modified on-shell projection relaxes conservation of the invariant mass Q of the event, rescales the top-quark energies, and adjusts the initial-state momenta, and it is used over the entire phase space rather than only near threshold. This projection is part of the construction of the factorisable two-loop corrections and of the reweighting in Eq. (63), so any bias introduced by the projection can propagate into the fitted non-factorisable constant through the matching in Sec. 5.1. The NLO DPA validation is reassuring, but it does not directly bound the effect on the two-loop factorisable contribution, which is much larger in absolute terms. I ask for an explicit estimate of the projection dependence, for example by comparing the modified projection with the standard Q-conserving projection in the phase-space region where both are defined, and by reporting the change in Delta_sigma_NNLO,H|non-fact when the projection choice is varied.
minor comments (4)
  1. [Sec. 2.2.1] The word 'perfomed' appears in the consistency-check paragraph; it should be 'performed'.
  2. [Appendix A] The phrase 'off-shell skaddones momenta' appears to contain a stray word; it should presumably read 'off-shell momenta'.
  3. [Sec. 5.3.3, Eq. (85)] The formula for the replica standard deviation appears garbled in the printed version (the sum over sqrt terms is not a standard deviation). Please clarify the exact expression used.
  4. [Sec. 5.3.3, Fig. 16] The fitted coefficients A^(2)_nf, B^(2)_nf and D^(2) are shown only through correlation plots; reporting their central values and uncertainties in a table would make the model-dependence discussion more quantitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the on-shell matching is an externally validated boundary condition, not a tautology.

full rationale

The only step that could look circular is the determination of the non-factorisable two-loop constant C^(2)_nf through the Gamma_t-to-0 matching (Secs. 5.1 and 5.3.3). The paper explicitly states that this constant "cannot be extracted solely through this approach" and is fixed by requiring the off-shell cross section to reproduce the independent on-shell t-tbar NNLO cross section of Ref. [2] in the zero-width limit (Eqs. 76-77). The final physical-width prediction (Eq. 88) is not forced to equal the on-shell value: it is obtained by adding the off-shell and fitted logarithmic contributions evaluated at Gamma_t = Gamma_phys, and it differs from the on-shell-matched sigma_Delta_trunc by about 3% (10623 vs 10278 fb), showing that the physical-width result carries independent off-shell content. The NLO version of the same reconstruction is validated against the exact off-shell NLO calculation (Fig. 13), and the off-diagonal NNLO channels are computed exactly. The functional ansatz Eq. (73) is a modelling assumption and a genuine accuracy limitation, acknowledged by the authors, but it is not a case of defining the prediction in terms of its own output. All other ingredients, such as the two-loop factorisable amplitudes, real contributions, and massification, are external or independently checked. No circular reduction is exhibited.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new particles or forces are introduced. The central numerical claim rests on fitted coefficients for the non-factorisable two-loop corrections and on a set of physical assumptions about the small-width limit, the DPA, and the massification procedure.

free parameters (2)
  • Gamma_t-extrapolation fit coefficients (A(2)_nf, B(2)_nf, C(2)_nf, D(2)) = not individually tabulated; the combined non-factorisable corrections are Delta_sigma_NNLO,H|non-fact = 1107 +/- 152…
    These coefficients parameterise the assumed functional form of the non-factorisable two-loop virtual contribution (Eq. 73). They are fitted to the rcut-extrapolated factorisable-only NNLO results at nine Gamma_t values and matched to the on-shell NNLO ttbar cross section. The final NNLO cross section in Eq. (88) depends on this fit.
  • rcut-extrapolation fit parameters per partonic channel = not quoted; results are given for extrapolated cross sections
    The rcut -> 0 limit is obtained by fitting each partonic channel with a specific polynomial/log-enhanced ansatz (Sec. 5.3.2). The extrapolated values depend on the chosen fit model, though linquad and improved fits agree within errors.
assumptions (5)
  • domain assumption The off-shell cross section factorises as sigma_ttbar times branching ratios in the Gamma_t -> 0 limit (Eq. 77)
    Invoked in Sec. 5.1 to match to the on-shell ttbar cross section; expected from formal all-order arguments cited in Refs. [21,22], validated here at NLO and in off-diagonal NNLO channels, but not at NNLO for diagonal channels.
  • domain assumption The non-factorisable two-loop virtual corrections have the functional form A(2)_nf log^2(Gamma_t/Qh) + B(2)_nf log(Gamma_t/Qh) + C(2)_nf + O(Gamma_t/Qh) (Eq. 73)
    Based on the soft-gluon origin and at most one log per loop; no explicit two-loop derivation is given (Sec. 2.2 and Sec. 5.3.3).
  • domain assumption The double-pole approximation captures the two-loop virtual amplitude with sufficient accuracy for the inclusive cross section
    Validated at NLO in Sec. 4.4, but the two-loop exact amplitude is unavailable; the extension is an assumption of the paper.
  • domain assumption Massification restores bottom-quark mass effects in the two-loop production/decay amplitudes up to power corrections O(mb^2/mt^2)
    Applied in Secs. 2.2.2 and 2.2.4 using the mass factorisation formula of Refs. [65-69]; pointwise one-loop check at the 0.3% level.
  • ad hoc to paper The modified on-shell projection (relaxing Q conservation) does not introduce a significant bias
    Appendix A: the projection is used in the entire phase space to avoid discarding below-threshold events; no systematic study of this distortion is provided.

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Pith. "Pith review of Towards NNLO QCD predictions for off-shell top-quark pair production and decays." pith.science (2026). https://pith.science/paper/3SJG6TDD

@misc{pith2026250711410,
  author       = {Pith},
  title        = {Pith review of: Towards NNLO QCD predictions for off-shell top-quark pair production and decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SJG6TDD}},
  note         = {Machine review of arXiv:2507.11410}
}
abstract

We consider QCD radiative corrections to $W^+W^-b {\bar b}$ production with leptonic decays and massive bottom quarks at the LHC. We perform an exact next-to-leading order (NLO) calculation within the $q_T$-subtraction formalism and validate it against an independent computation in the dipole subtraction scheme. Non-resonant and off-shell effects related to the top quarks and the leptonic decays of the $W^\pm$ bosons are consistently included. We also consider the approximation in which the real-emission contribution is computed exactly while the virtual is evaluated in the double-pole approximation (DPA), which formally requires the inclusion of both factorisable and non-factorisable corrections. We evaluate such contributions and show that the DPA performs remarkably well at both the inclusive and differential levels. We then extend our calculation to the next-to-next-to-leading order (NNLO). All tree-level and one-loop amplitudes are evaluated exactly, while the missing two-loop virtual contribution is estimated using the DPA. The factorisable two-loop corrections are explicitly computed by relying on available results for the polarised two-loop on-shell top-quark pair production amplitudes and the corresponding top-quark decays. The non-factorisable contributions are inferred by exploiting the cancellation of logarithmic singularities in the $\Gamma_t\to 0$ limit through an on-shell matching procedure. The NNLO corrections for the inclusive cross section are found to increase the NLO prediction by approximately $11\%$, with a numerical uncertainty that is conservatively estimated to be below the $2\%$ level $\unicode{x2013}$ significantly smaller than the $5\%$ residual perturbative uncertainties.

Figures

Figures reproduced from arXiv: 2507.11410 by the authors.

Figure 1
Figure 1. Three diagrams contributing to the factorisable (left) and non-factorisable corrections (right), re￾spectively. The diagram in the centre contributes to both factorisable and non-factorisable corrections. separated into production and decay stages. It is worth stressing that the distinction between factorisable and non-factorisable corrections is not made at the level of individual Feynman diagrams, but at the level… view at source ↗
Figure 2
Figure 2. Manifestly non-factorisable diagrams contributing to off-shell tt¯ production. 2. non-manifestly non-factorisable corrections (see [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Non-manifestly non-factorisable diagrams contributing to off-shell tt¯ production. Explicit expressions for these ∆ functions, written in terms of scalar one-loop integrals, are given in Appendix B. In our Matrix implementation of the non-factorisable corrections according to Eq. (14), we rely on Collier [84] for the evaluation of the required one-loop scalar integrals. Considering only the contributions from IR-div… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: LO and NLO differential results for the CKMP setup at √ s = 8 TeV in the invariant mass (a) and the transverse momentum (b) of the b ¯b pair. The central scale is fixed at µ0 = mt. Absolute predictions at LO (grey) are compared against NLO predictions based on CS (red)…
Figure 5
Figure 5. Figure 5: LO and NLO differential results for the CKMP setup at √ s = 8 TeV for the invariant-mass distri￾bution me+µ− of the two charged leptons (a) and the azimuthal-angle separation ∆ϕe+µ− (b). The central scale is fixed at µ0 = mt. Curves and bands as in [PITH_FULL_IMAGE:fi…
Figure 6
Figure 6. Figure 6: Comparison between factorisable (magenta curve) and non-factorisable (blue curve) one-loop cor￾rections, computed at scale Qe within the CMP setup at √ s = 13 TeV and central scale µ0 = mt. 29 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]
Figure 7
Figure 7. Figure 7: NLO differential results for the CMP setup at √ s = 13 TeV and central scale µ0 = mt. We consider the gg partonic channel as a case study. We show a comparison between the NLO result in which only the virtual contribution at scale Q has been approximated in DPA (green …
Figure 8
Figure 8. Figure 8: NLO differential results for the CKMP setup at √ s = 8 TeV. The central scale is fixed at µ0 = mt. 32 [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]
Figure 9
Figure 9. Figure 9: NLO differential results for the CMP setup at √ s = 13 TeV. The central scale is fixed at µ0 = mt. 33 [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: Study of the rcut-dependence at NLO in QCD, for different partonic channels, gg, q¯q, gq, as well as the combination of all channels. The seven columns refer to the different values of the top-quark width Γt considered for the small-width extrapolation. The extrapolat…
Figure 11
Figure 11. Figure 11: Study of the rcut-dependence at NNLO in QCD, for different partonic channels: gg, q¯q, gq and qq + ¯q¯q + q′¯q from the first to the fourth row, respectively. The seven columns refer to the different values of the top-quark width Γt considered for the small-width extr…
Figure 12
Figure 12. Figure 12: Γt → 0 extrapolation at LO for the gg (upper plot) and qq¯ (lower plot) partonic channels. W boson. Without the quasi-collinear enhancement, linear power corrections remain small in the qq¯ channel, allowing quadratic corrections from non-resonant topologies to domina…
Figure 13
Figure 13. Figure 13: Γt → 0 extrapolation at NLO in qT subtraction. On the left, we show the comparison, for the diagonal partonic channels, between the exact NLO correction (orange markers) and the on-shell result (blue curve). We also display the NLO correction where the one-loop contri…
Figure 14
Figure 14. Figure 14: Γt → 0 extrapolation at NNLO in qT subtraction. On the left, we present the results for the diagonal channels where the “missing” non-factorisable two-loop corrections have been fitted according to the procedure described in the main text. On the right, we show the co…
Figure 15
Figure 15. Figure 15: Behaviour of the original ∆σNNLOfact qT,DPA results (purple markers) as a function of Γt/mt. We also display the ensemble of 1000 fitting replicas (pink band), used in the numerical extrapolation of the non￾factorisable corrections. Results for the physical top-quark …
Figure 16
Figure 16. Figure 16: Two-dimensional correlation plots between pairs of fit parameters A (2) nf , B (2) nf , C (2) nf , D(2) (fit-wPC model), corresponding to the 1σ variation of ∆σNNLO,H|non−fact. The first (second) row refers to gg (qq¯) channel. turn remain close to the result in the o…

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Reviewed August 6, 2026 · model on record in the stance chip above.