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Soft-gluon resummation for Higgs boson production at hadron colliders

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S. Catani, D. de Florian, M. Grazzini, P. Nason

High Energy Physics - Phenomenology · Submitted 23 Jun 2003 · 55 references

Pith's verdict: The canonical NNLO+NNLL gluon-fusion Higgs paper, and it holds up; the soft-gluon extrapolation is stated as an assumption rather than smuggled in, and the residual worries are minor.

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arXiv IDhep-ph/0306211 (version 1)
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CitationD. de Florian, M. Grazzini, P. Nason, S. Catani. "Soft-gluon resummation for Higgs boson production at hadron colliders." arXiv:hep-ph/0306211 (2003). https://pith.science/paper/H2TFZCKG
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The paper

The paper claims

Soft-gluon resummation pushes the gluon-fusion Higgs prediction to next-to-next-to-leading logarithmic accuracy, matched to exact NNLO.

The reading

The paper asks how precisely the production rate of a Standard Model Higgs boson at a hadron collider can be predicted from QCD, given that the process runs through gluon fusion and is therefore a pure strong-interaction quantity. Its answer is to resum the threshold logarithms of soft-gluon emission to next-to-next-to-leading logarithmic (NNLL) accuracy in Mellin-moment space, then add the exactly known fixed-order terms up to NNLO so that nothing already computed is lost. The authors argue that this is the first QCD cross section computed at nominal NNLO+NNLL accuracy, and they use it to quote residual perturbative uncertainties of roughly $8$ per cent at both the Tevatron and the LHC for $M_H < 200$ GeV. Along the way they show that the soft approximation reproduces the exact NLO and NNLO results well, and that the Mellin-space version of the approximation is numerically superior to the more common $x$-space version. The practical stake is whether a measured Higgs rate can be used to test for deviations in the Yukawa couplings, which requires a QCD prediction whose error bar is smaller than the effect being looked for.

What carries the argument

The load-bearing object is the Sudakov radiative factor $1+O(1/N)$ in Mellin space, an exponential whose argument is a double integral over the soft-gluon momentum fraction and the running coupling, controlled by two perturbative functions $A(1+O(1/N))$ and $D(1+O(1/N))$. $1+O(1/N)$ is fixed at any order by the large-$N$ limit of the gluon anomalous dimension in the $1+O(1/N)$ scheme, which is why the resummation can be pushed past NLL at all: the same function that governs the evolution of the parton densities governs the leading threshold logarithms. The companion object is the matching prescription $1+O(1/N)$, which subtracts the fixed-order truncation of the resummed series from the exact fixed-order cross section so that the sum reproduces the known NNLO result identically and adds only the beyond-NNLO soft terms. A third piece of machinery is the Minimal Prescription for the inverse Mellin contour, which keeps the Landau singularity at $1+O(1/N)$ to the right of the integration path so that no unjustified non-perturbative power corrections are introduced. Appendix A supplies the all-order generalization of the $1+O(1/N)$ replacement, expressed as a formal differential operator $1+O(1/N)$ acting on a step function, which is what makes the logarithmic accuracy tunable order by order.

Extended reading notes

Core claim

The central claim is that the gluon-fusion Higgs cross section can be computed with all soft-gluon logarithms resummed at NNLL and consistently matched to the exact NNLO result, and that the resulting prediction is under good perturbative control. The resummed coefficient function is written as $G^{res}_{gg,N} = C_{gg}(1+O(1/N))$ times a Sudakov radiative factor $1+O(1/N)$, with the factor given either as a $z$-integral over $(z^{N-1}-1)/(1-z)$ of the collinear and large-angle soft anomalous coefficients, or equivalently as a $y$-integral over the running coupling between $y = N_0/N$ and $1$. The authors supply the explicit NNLL ingredients: $A^{(1)} = C_a$, $A^{(2)} = C_a K/2$, the estimate $A^{(3)} = C_a[13.81 - 2.1467 N_f - N_f^2/108]$, $D^{(2)}$ from earlier work, and the constants $C^{(1)}_{gg}$, $C^{(2)}_{gg}$ extracted from the known one- and two-loop coefficient functions. Truncating the resummed formula reproduces the exact NLO and NNLO coefficient functions to within the scale-variation bands, which the authors present as the evidence that the soft approximation is trustworthy. Numerically, at the LHC the NNLO $K$-factor is about $2.1$-$2.2$ for $M_H < 200$ GeV and NNLL adds about $6$ per cent; at the Tevatron the NNLO $K$-factor is $3$-$3.2$ and NNLL adds $12$-$15$ per cent. The residual scale uncertainty is about $8$ per cent at NNLL in both cases, versus about $10$ per cent (LHC) and $13$ per cent (Tevatron) at NNLO.

Load-bearing premise

The whole extrapolation rests on the premise the authors state in the introduction: that because the soft-gluon approximation reproduces the exact NLO and NNLO corrections, it will keep the same reliability for the still higher orders that no exact calculation can yet check.

Editorial extensions

If this is right

  • For a light Higgs at the LHC, the NNLO+NNLL prediction carries roughly $8$ per cent residual perturbative uncertainty, small enough that a measured rate differing by more than that would be a serious candidate for non-Standard-Model Yukawa couplings.
  • At the Tevatron the resummation matters more (about $12$-$15$ per cent on top of NNLO), so the two colliders' predictions are brought to a comparable relative precision despite the Tevatron sitting closer to threshold.
  • The soft-gluon expansion truncated at N3LO, given explicitly in Appendix E, provides a ready-made check value for any future exact three-loop calculation of the gluon coefficient function.
  • The comparison of N-space and $x$-space approximations, with N-space tracking the exact NLO and NNLO results more closely, argues that threshold resummation should be formulated in Mellin space rather than in $x$-space.
  • Differences between the MRST2002 and Alekhin A02 gluon luminosities, $8$ per cent at the LHC and up to $14$ per cent at the Tevatron, exceed the quoted parton-uncertainty envelopes, so parton determination rather than QCD radiation becomes the dominant theory error.

Reading between the lines

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

  • Because the paper's soft-gluon coefficients $A^{(n)}$ are process-independent up to the colour factor $C_a$, the same NNLL machinery transfers directly to Drell-Yan and to vector-boson production, so the Higgs calculation functions as a template for a whole class of threshold-resummed hadron-collider predictions.
  • The authors note that a future exact N3LO calculation would settle the reliability question directly: if the exact N3LO coefficient function agrees with Eq. (97) in its known logarithmic terms and in the size of the unknown constant, the extrapolation premise is vindicated; if the constant term is large, the quoted $8$ per cent band is optimistic.
  • The finding that parton-density differences exceed the perturbative error bar suggests that further work on the QCD side will yield diminishing returns for Higgs coupling measurements until the gluon luminosity at $x \sim 0.01$-$0.1$ is pinned down better.
  • The Minimal Prescription's role here, keeping the Landau singularity on the far side of the contour while still reproducing the truncated series rapidly (Appendix D), is a concrete demonstration that resummed perturbation theory can be used without introducing uncontrolled power corrections.
Original abstract

We consider QCD corrections to Higgs boson production through gluon--gluon fusion in hadron collisions. We compute the cross section, performing the all-order resummation of multiple soft-gluon emission at next-to-next-to-leading logarithmic level. Known fixed-order results (up to next-to-next-to-leading order) are consistently included in our calculation. We give phenomenological predictions for Higgs boson production at the Tevatron and at the LHC. We estimate the residual theoretical uncertainty from perturbative QCD contributions. We also quantify the differences obtained by using the presently available sets of parton distributions.

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.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.63 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.72 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We consider QCD corrections to Higgs boson production through gluon--gluon fusion in hadron collisions. We compute the cross section, performing the all-order resummation of multiple soft-gluon emission at next-to-next-to-leading logarithmic level. Known fixed-order results (up to next-to-next-to-leading order) are consistently included in our calculation. We give phenomenological predictions for Higgs boson production at the Tevatron and at the LHC. We estimate the residual theoretical uncertainty from perturbative QCD contributions. We also quantify the differences obtained by using the presently available sets of parton distributions.

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 consider QCD corrections to Higgs boson production through gluon--gluon fusion in hadron collisions. background 0.74 0.36 0.03 asserted 0.76
s1 We compute the cross section, performing the all-order resummation of multiple soft-gluon emission at next-to-next-to-leading logarithmic level. method 0.64 0.97 0.26 asserted 1.00
s2 Known fixed-order results (up to next-to-next-to-leading order) are consistently included in our calculation. method 0.87 0.65 0.08 asserted 1.00
s3 We give phenomenological predictions for Higgs boson production at the Tevatron and at the LHC. result 0.52 0.92 0.14 asserted 1.00
s4 We estimate the residual theoretical uncertainty from perturbative QCD contributions. result 0.43 0.94 0.07 asserted 0.93
s5 We also quantify the differences obtained by using the presently available sets of parton distributions. contribution 0.49 0.93 0.05 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from simulation or computation 0.85.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • 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. The SV (dotted lines) and SVC (dashed lines) approximations in N-space versus the exact results (solid lines) at the LHC (√ s = 14 TeV). The central curves are obtained by fixing µF = µR = MH . The bands are obtained by varying µF and µR simultaneously and independently in the range 0.5MH ≤ µF , µR ≤ 2MH with the constraint 0.5 ≤ µF /µR ≤ 2. Here and in the following, we use the MRST2002 [50] set of parton distributions, which includes (approximated [45, 46, 51]) NNLO parton distributions. The parton densities and QCD coupling 15
Figure 2. The SV (dotted lines) and SVC (dashed lines) approximations in N-space versus the exact results (solid lines) at the Tevatron (√ s = 1.96 TeV). are evaluated at each corresponding order, by using 1-loop αS at LO, 2-loop αS at NLO and 3-loop αS at NNLO. The corresponding values of αS(MZ) are 0.130, 0.1197, 0.1154, at 1-loop, 2-loop and 3-loop order, respectively. As can be observed from Figs. 1 and 2, the SV and SVC approximations in N-space agree very well with the exact NLO and NNLO calculations. Moreover, the differences between the approximated and exact results are substantially…
Figure 3. The SV-x (dotted lines) and SVC-x (dashed lines) approximations versus the exact results (solid lines) at the LHC (√ s = 14 TeV). At NLO, the x-space soft-virtual contribution (called SV-x approximation, here) to the gluon coefficient function G (1) gg in Eq. (8) is G (1) SV-x gg (z; M2 H /µ2 R; M2 H/µ2 F ) = δ(1 − z)  11 2 + 6ζ(2) + 33 − 2Nf 6 ln µ 2 R µ 2 F  + 6 D0(z) ln M2 H µ 2 F + 12 D1(z) . (57) The same approximation can be defined at NNLO, and the corresponding contribution to the gluon coefficient function G (2) gg (z) has the form G (2) SV-x
Figure 4. The SV-x (dotted lines) and SVC-x (dashed lines) approximations versus the exact results (solid lines) at the Tevatron (√ s = 1.96 TeV). G (2) SVC-x gg (z; M2 H /µ2 R; M2 H/µ2 F ) = G (2) SV-x gg (z; M2 H /µ2 R; M2 H/µ2 F ) − 72 ln3 (1 − z) . (60) Figures 3 and 4 are obtained in the same way as Figs. 1 and 2, except for the use of the SV-x and SVC-x approximations instead of the analogous N-space approximations. These figures show that the bulk of the fixed-order radiative corrections is given by the SV-x and SVC-x approximations, as observed in Refs. [8, 16]. However, c
Figure 5. N-dependence of resummed (left-hand side) and fixed-order (right-hand side) gluon coefficient functions for Higgs production (MH = 150 GeV) with fixed coupling constant αS = 0.1. 19
Figure 6. Scale dependence of the Higgs production cross section at the LHC for MH = 115 GeV at a) (upper) LO, NLO, NNLO and b) (lower) LL, NLL, NNLL accuracy. As expected from the QCD running of αS, the cross sections typically decrease when µR increases around the characteristic hard scale MH , at fixed µF = MH. In the case of variations of µF at fixed µR = MH , we observe the opposite behaviour. In fact, when MH = 115 GeV, the cross sections are mainly sensitive to partons with momentum fraction x ∼ 10−2 , and in this x-range scaling violation of the parton densities is (moderately)…
Figure 7. Scale dependence of the Higgs production cross section at the LHC for MH = 400 GeV at a) (upper) LO, NLO, NNLO and b) (lower) LL, NLL, NNLL accuracy. that the µF dependence is much weaker than the µR dependence at each fixed order (LO, NLO, NNLO). In
Figure 8. Fixed-order and resummed K-factors for Higgs production at the LHC. The impact of higher-order corrections is sometimes presented through the K-factors, defined as the ratio of the cross section evaluated at each corresponding order over the LO result. The K-factors are shown in
Figure 9. NNLL and NNLO cross sections at the LHC, using MRST2002 parton densities. resummed results. We use MH = 150 GeV. As in the LHC case, the cross sections typically decrease when µR increases around the characteristic hard scale MH . Figure 10a shows that the fixed-order cross section decreases when µF increases at fixed µR. This is not unexpected: at the Tevatron, the cross section is mainly sensitive to partons with x ∼ 0.05–0.1, where the scaling violation is slightly negative. As in the LHC case, the µF dependence of the resummed results appears to be stronger than the µF…
Figure 10. Scale dependence of the Higgs production cross section at the Tevatron for MH = 150 GeV at a) (upper) LO, NLO, NNLO and b) (lower) LL, NLL, NNLL accuracy. knowledge of the parton distributions. Uncalculated higher-order QCD radiative corrections are the most important source of un￾certainty on the coefficients Gab. A method, which is customarily used in perturbative QCD calculations, to estimate their size is to vary the renormalization and factorization scales around the hard scale MH . In general, this procedure can only give a lower limit on the ‘true’ uncertainty. This is well…
Figure 11. Fixed order and resummed K-factors for Higgs production at the Tevatron Run II ( √ s = 1.96 TeV). ±8% when MH ∼< 200 GeV. At the Tevatron, when MH ∼< 200 GeV, the NNLO scale dependence is about ±13%, whereas the NNLL scale dependence is about ±8%. Another method to estimate the size of higher-order corrections is to compare the results at the highest available order with those at the previous order. Considering the differences between the NNLO and NNLL cross sections, we obtain results that are consistent with the uncertainty estimated from scale variations. We have also considered
Figure 12. NNLL and NNLO cross sections at the Tevatron (√ s = 1.96 TeV), using MRST2002 parton densities. heavy quark in the loop at the Born level. In other words, as for the size of the QCD radiative corrections, what matters is that the heavy-quark mass, Mt is actually larger than the soft-gluon scale, MH/N, rather than the Higgs boson scale MH. This feature, i.e. the dominance of soft-gluon effects, persists at NNLO (see Sect. 4) and it is thus natural to assume that, having normalized our cross sections with the exact Born result, the uncertainty ensuing from the large-Mt approximation…
Figure 13. Comparison of A02 and MRST2002 gg luminosities at the LHC. that, once problems with parton density determinations are overcome, a theoretical precision of about 10% in the predicted cross section can be attained. The impact of the NNLL corrections included in this work was found to be modest in the case of light-Higgs production at the LHC. The NNLL corrections are more significant at the Tevatron, and allow us to reduce the theoretical error band at the same precision level as at the LHC. We believe that our results add much confidence in the predicted cross sections for the…
Figure 14. Comparison of A02 and MRST2002 gg luminosities at the Tevatron. We consider the N-moments In(N) of the singular distributions Dn(z) defined in Eq. (11): In(N) ≡ Z 1 0 dz zN−1Dn(z) = Z 1 0 dz z N−1 − 1 1 − z lnn (1 − z) . (63) These N-moments were computed in Ref. [26]. It was shown that, in the large-N limit, the leading and next-to-leading logarithmic terms (lnn+1 N and lnn N) arising from the integration over z can straightforwardly be obtained by a very simple prescription. It is indeed sufficient [26] to replace the integrand weight (z N−1 − 1) by the approximation z N−1

Figures reproduced from arXiv:hep-ph/0306211 by the authors.

References
How later papers used it

216 later papers cite it

Evidence: substantial. 207 of the citing papers share no author with it, and 104 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 300 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 73 papers with this label
Uses its data or values 1 of 24 papers with this label
How the literature uses it

208 arXiv papers cite this in 289 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 · 137 sentences without a confident label

Contests it 1 paper
  • At NLO, in the large-$m_t$ limit, we have [ref] [Actually there is a missing term in these references: the correct result can be found for example in Refs. [ref] .] [equation] where [equation] and P_gg^(0)(z)=P̃_gg^(0)(z) + β_0δ(1-z), P_gq^(0)(z)=C_F/2π 1+(1-z)^2/z with P̃_gg^(0)(z) = C_A/π/1-z + P_gg^reg(z), P_gg^reg(z) = C_A/π[1/z-2+z(1-z)].

    Resummation of soft and hard gluon radiation in perturbative QCD · Higgs production at fixed perturbative order · contest 0.88

Builds on it 2 papers
Uses its method or tool 72 papers
Uses its data or values 20 papers
Compares with it 10 papers
Cites it as support 2 papers
Cites it as background 33 papers

Discussion

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