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NNLO QCD corrections to the Higgs-strahlung processes at hadron colliders

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Oliver Brein, Abdelhak Djouadi, Robert Harlander

High Energy Physics - Phenomenology · Submitted 16 Jul 2003 · 31 references

Pith's verdict: The standard NNLO reference for Higgs-strahlung, and it holds up — just don't quote the 2–3% theory error as if it were derived rather than inferred from scale variation.

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arXiv IDhep-ph/0307206 (version 1)
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CitationAbdelhak Djouadi, Oliver Brein, Robert Harlander. "NNLO QCD corrections to the Higgs-strahlung processes at hadron colliders." arXiv:hep-ph/0307206 (2003). https://pith.science/paper/7SJZROZR
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The paper

The paper claims

The Higgs-strahlung channel is now a few-percent prediction, not a tens-of-percent one.

The reading

The paper asks how precisely the Standard Model predicts Higgs boson production in association with a $W$ or $Z$ boson at the Tevatron and the LHC, once the second-order QCD corrections are included. The authors' route is to note that $qbar q to HV$ is just Drell–Yan production of an off-shell vector boson followed by the splitting $V^* to HV$, so the full next-to-next-to-leading-order (NNLO) machinery already built for Drell–Yan can be carried over, with one extra ingredient for $Z$ final states: gluon-initiated loops, especially $gg to HZ$. The reported outcome is that the two-loop corrections are modest, raising the cross section by under about 5% at the LHC and 10% at the Tevatron, while cutting the residual dependence on unphysical renormalization and factorization scales to roughly 2–3%. That matters because it turns Higgs-strahlung into a channel whose QCD theory error is smaller than the parton-distribution and luminosity uncertainties, which is exactly the regime where a measured rate can be read as a measurement of the Higgs coupling to vector bosons.

What carries the argument

The load-bearing device is the factorization identity of Eq. (5), $dhatsigma/dk^2(pp to HV + X) = sigma(pp to V^* + X) times dGamma/dk^2(V^* to HV)$, which splits the process into a piece governed purely by QCD (off-shell vector-boson production, known to NNLO) and a piece governed by electroweak kinematics (the splitting of the virtual boson into a real boson plus the Higgs). This is what lets the authors import the two-loop Drell–Yan calculation rather than perform a new multi-scale NNLO computation. The second ingredient is the set of quark-loop amplitudes for $gg to HZ$ — a triangle with $Z^* to HZ$ and box diagrams with both $H$ and $Z$ emitted from the internal quark lines — which interfere destructively and are evaluated with the full $m_t$, $m_b$ dependence. The third is the use of NNLO-evolved parton densities, which at the time required approximate three-loop splitting functions, supplied by the MRST sets.

Extended reading notes

Core claim

The central claim is that the QCD corrections to $pp to HV$ are known through $O(alpha_s^2)$ and are small, so the channel becomes one of the theoretically cleanest Higgs production modes at hadron colliders. The authors obtain this by factorizing the hadronic cross section as the Drell–Yan cross section for $pp to V^*$ times the energy distribution $dGamma/dk^2(V^* to HV)$, so that the known two-loop Drell–Yan result transfers directly. For $Z$ final states they add the heavy-quark loop processes, principally $gg to HZ$, computed with full top and bottom mass dependence, and they argue that the $Zgg$-induced corrections to $qbar q to Z^*$ are negligible (below the percent level at the Tevatron and smaller at the LHC). Numerically, with scales fixed at $mu_F = mu_R = M_{HV}$ and matched-order MRST parton densities, the NLO $K$-factor is about 1.27–1.29 at the LHC and 1.30–1.35 at the Tevatron, and the NNLO corrections add only about 1–3.5% at the LHC and about 10% at the Tevatron. The scale-variation band shrinks from roughly 10% (LO) to 5% (NLO) to 2% (NNLO) at the LHC for $M_H = 120$ GeV, and from 20% to 7% to 3% at the Tevatron. The $gg to HZ$ subprocess is found to contribute around 10% of the total $HZ$ rate at the LHC for low Higgs masses and to be negligible at the Tevatron.

Load-bearing premise

The whole NNLO result is inherited through the factorization of Eq. (5), which the authors themselves note in footnote 2 holds only at first order in the electroweak coupling, so QCD corrections to the $V^* to HV$ splitting and final state are neglected.

Editorial extensions

If this is right

  • The residual QCD uncertainty on $pp to HW$ and $pp to HZ$ falls to 2–3%, below the roughly 5% parton-distribution error quoted at NLO, so PDFs and luminosity become the dominant theoretical limitation.
  • Because the $K$-factor is large and nearly constant in $M_H$, a measured Higgs-strahlung rate can be converted into a determination of the $HVV$ coupling with a small and slowly varying QCD correction.
  • At the LHC, the $gg to HZ$ channel must be included for low Higgs masses at the 10% level of the total $HZ$ rate, but can be dropped at the Tevatron where it stays at the fraction-of-a-femtobarn level.
  • The NNLO corrections are of the same order as the electroweak $O(alpha)$ corrections but of opposite sign, so the two must be combined before quoting a final theory error.
  • The same factorization argument extends to supersymmetric Higgs-strahlung $pp to VPhi$ and to the associated $Phi A$ production channel, up to global coupling factors and an additional one-loop $gg to Phi A$ piece.

Reading between the lines

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

  • If the QCD error really sits at 2–3%, the practical bottleneck for using Higgs-strahlung as a coupling measurement shifts to the integrated luminosity, which the authors elsewhere estimate at about 5%; a luminosity calibration from $W$ and $Z$ Drell–Yan would then be the natural next step, since much of the $K$-factor cancels in the ratio.
  • The paper's own logic suggests a sharper test of the factorization ansatz than any cross-section measurement: a dedicated calculation of the $O(alpha_s^2)$ corrections to the $V^* to HV$ splitting itself, which would either confirm that the final-state QCD corrections are negligible or move the quoted precision.
  • The same Drell–Yan factorization could be pushed to the $b bar b to H$ channel or to charged-Higgs associated production, where the final-state QCD corrections are less obviously small and the ansatz would be under more stress.
  • Because the $gg to HZ$ box and triangle amplitudes interfere destructively and depend on the top mass, a measurement of the $HZ$ rate at high $M_H$ would be sensitive to the sign and size of that interference, giving a handle that the inclusive $qbar q$ channel does not provide.
Original abstract

We implement, at next-to-next-to-leading order, the QCD corrections to Standard Model Higgs boson production in association with vector bosons at hadron colliders, $q\bar{q} \to HV$ with $V=W,Z$. They consist of the two--loop corrections to the Drell--Yan process for the production of off-shell vector bosons, $q\bar{q} \to V^*$, and in the case of $Z$ final states, of the additional contribution from heavy-quark loop mediated processes, in particular $gg \to HZ$. For the Higgs boson masses relevant at the Tevatron and the LHC, $M_H \lsim 200$--300 GeV, the two-loop corrections are small, increasing the production cross sections by less than 5% and 10%, respectively; the scale dependence is reduced to a level of less than a few per cent. This places these processes among the most theoretically clean Higgs boson production channels at hadron colliders.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,895 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.40 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.30 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We implement, at next-to-next-to-leading order, the QCD corrections to Standard Model Higgs boson production in association with vector bosons at hadron colliders, $q\bar{q} \to HV$ with $V=W,Z$. They consist of the two--loop corrections to the Drell--Yan process for the production of off-shell vector bosons, $q\bar{q} \to V^*$, and in the case of $Z$ final states, of the additional contribution from heavy-quark loop mediated processes, in particular $gg \to HZ$. For the Higgs boson masses relevant at the Tevatron and the LHC, $M_H \lsim 200$--300 GeV, the two-loop corrections are small, increasing the production cross sections by less than 5% and 10%, respectively; the scale dependence is reduced to a level of less than a few per cent. This places these processes among the most theoretically clean Higgs boson production channels at hadron colliders.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We implement, at next-to-next-to-leading order, the QCD corrections to Standard Model Higgs boson production in association with vector bosons at hadron colliders, $q\bar{q} \to HV$ with $V=W,Z$. contribution 1.00 0.95 0.08 asserted 0.99
s1 They consist of the two--loop corrections to the Drell--Yan process for the production of off-shell vector bosons, $q\bar{q} \to V^*$, and in the case of $Z$ final states, of the additional contribution from heavy-quark loop mediated processes, in particular $gg \to HZ$. contribution 0.65 0.62 0.05 asserted 1.00
s2 For the Higgs boson masses relevant at the Tevatron and the LHC, $M_H \lsim 200$--300 GeV, the two-loop corrections are small, increasing the production cross sections by less than 5% and 10%, respectively; the scale dependence is reduced to a level of less than a few per cent. result 1.00 0.90 0.50 asserted 1.00
s3 This places these processes among the most theoretically clean Higgs boson production channels at hadron colliders. interpretation 1.00 0.64 0.51 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from simulation or computation 0.77.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Diagram for associated Higgs and vector boson production in hadronic collisions. The total cross section for the subprocess is obtained by integrating over k 2 : σˆLO(qq¯ → V H) = G2 FM4 V 288πsˆ (v 2 q + a 2 q )λ 1/2 (M2 V , M2 H; ˆs) λ(M2 V , M2 H; ˆs) + 12M2 V /sˆ (1 − M2 V /sˆ) 2 , (3) where the reduced quark couplings to gauge bosons are given in terms of the electric charge and the weak isospin of the fermion as: aq = 2I 3 q , vq = 2I 3 q − 4Qqs 2 W for V = Z and vq = −aq = √ 2 for V = W, with s 2 W = 1 − c 2 W ≡ sin2 θW . 3
Figure 2. NLO QCD corrections to the vector boson–quark–antiquark vertex. Including these contributions, and taking into account the virtuality of the vector boson, the LO cross section is modified in the following way σNLO = σLO + ∆σqq¯ + ∆σqg , (6) with ∆σqq¯ = αs(µR) π Z 1 τ0 dτ X q dL qq¯ dτ Z 1 τ0/τ dz σˆLO(τzs) ωqq¯(z) , ∆σqg = αs(µR) π Z 1 τ0 dτ X q,q¯ dL qg dτ Z 1 τ0/τ dz σˆLO(τzs) ωqg(z) , (7) 2This is only valid at first order in the electroweak coupling; at two-loop order in GF , QCD corrections to the final state should also be taken into account. In addition,
Figure 3. Diagrams for the NNLO QCD corrections to the process qq¯ → W∗ . These corrections have been calculated a decade ago in Ref. [17] and recently updated [19]. 5
Figure 4. Diagrams for the QCD corrections to qq¯ → Z ∗ not present in qq¯ → W∗ . Because gluons have only vector couplings to quarks and the effective Zgg coupling must be a colour singlet, only the axial–vector part of the Zqq¯ coupling will contribute as a consequence of Furry’s theorem. The axial–vector of the Z coupling to quarks, aq = 2I 3 q , differs only by a sign for isospin up-type and down-type quarks, so that their contribution should vanish in the case of quarks that are degenerate in mass. Thus, in the SM, only the top and bottom quarks will contribute to these topologies3
Figure 5. Diagrams for the gg → HZ process, which contributes to O(α 2 s ). At the LHC, the contribution of this gluon–gluon fusion mechanism to the pp → HZ total production cross section can be substantial. This is due to the fact that the suppression of the cross section by a power (αs/π) 2 is partly compensated by the increased gluon luminosity at high-energies. In addition, the tree-level cross section for qq¯ → HZ drops for increasing c.m. energy and/or MH values, since it is mediated by s-channel gauge-boson exchange. We have recalculated the cross section for the process gg → HZ at th
Figure 6. The K-factors for pp → HW at the LHC as a function of MH at LO, NLO and NNLO (solid black lines). The bands represent the spread of the cross section when the renormalization and factorization scales are varied in the range 1 3MHV ≤ µR (µF ) ≤ 3MHV , the other scale being fixed at µF (µR) = MHV . 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 100 120 140 160 180 200 220 240 260 280 300 MH KWH(Tevatron) LO NLO NNLO
Figure 7. The same as in
Figure 8. The total production cross sections at NNLO for

Figures reproduced from arXiv:hep-ph/0307206 by the authors.

References
How the literature uses it

130 arXiv papers cite this in 148 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 67 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 7 papers
Uses its data or values 48 papers
Compares with it 3 papers
  • At this order we may compare with the results of Ref. [ref] for most processes and for the remaining $W^±H$ and $ZH$ processes with the code vh@nnlo [ref] .

    Non-local slicing approaches for NNLO QCD in MCFM · NNLO · 0.99

  • As a test of our implementation, we have validated our calculation in the absence of any cuts on the final state particles by comparison with the public code vh@nnlo [ref] .

    Associated production of a Higgs boson at NNLO · LHC Phenomenology · 0.99

  • In this case a sizable destructive interference between the triangle and the box diagrams takes place [ref] , while this does not happen for VBF (see also the plots in fig:hqxsection).

    Vector boson fusion at NNLO in QCD: SM Higgs and beyond · Quark-gluon initiated contributions via the square of one-loop diagrams · 0.94

Cites it as background 19 papers

Discussion

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