Recommended predictions for the boosted-Higgs1

cross section2

Conveners of the gluon-fusion Working Group:3

K. Becker,a F. Caola,b A. Massironi,c B. Mistlberger,d P. F. Monni.e4

5

In collaboration with:6

X. Chen,f S. Frixione,g T. Gehrmann,f N. Glover,h K. Hamilton,i A. Y. Huss,e S. P. Jones,e7

A. Karlberg,f M. Kerner,f K. Kudashkin,j J. M. Lindert,h G. Luisoni,k M. L. Mangano,e S.8

Pozzorini,f E. Re,e G. P. Salam,b,l E. Vryonidou,e C. Wever.k9

aAlbert Ludwigs Universität Freiburg, Germany10

bRudolf Peierls Centre for Theoretical Physics,Oxford University, OX1 3PU, UK11

hInstitute for Particle Physics Phenomenology, Department of Physics, University of Durham, Durham,12

DH1 3LE, UK13

cCERN, Experimental Physics Department, and INFN, Sezione di Milano-Bicocca14

dCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA15

eCERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland16

fDepartment of Physics, University of Zürich, CH-8057 Zürich, Switzerland17

gINFN, Sezione di Genova, Via Dodecaneso 33, I-16146, Genoa, Italy18

iDepartment of Physics and Astronomy, University College London, London, WC1E 6BT, UK19

jInstitute for Theoretical Particle Physics (TTP), KIT, Karlsruhe, Germany20

kMax-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany21

lAll Souls College, Oxford OX1 4AL, UK22

Abstract: In this note we study the inclusive production of a Higgs boson with large transverse23

momentum. We provide a recommendation for the inclusive cross section based on a combination of24

state of the art QCD predictions for the gluon-fusion and vector-boson-fusion channels. Moreover,25

we compare such predictions to those obtained with commonly used event generators. We observe26

that the description of the considered kinematic regime provided by these tools is in good agreement27

with state of the art QCD predictions.28

Contents29

1 Introduction 130

2 Predictions for the gluon-fusion channel 231

2.1 Fixed-order 232

2.2 Event generators 533

3 Predictions from Vector Boson Fusion 834

4 Summary and conclusions 935

A Contribution from other production modes 1036

1 Introduction37

The continuously increasing amount of data recorded at the LHC opens the possibility to explore38

properties of the Higgs boson in a multitude of kinematic regimes. Of particular interest is the39

transverse momentum distribution of the Higgs boson for very large transverse momenta. Measure-40

ments of this observable allow for unique insights into the microscopic structure of the interactions41

of the Higgs boson with strongly interacting particles and might shed light on physics beyond the42

Standard Model. The observation of the Higgs boson in this kinematic regime is however extremely43

challenging.44

The inclusive search for the standard model Higgs boson produced at large transverse momen-45

tum (p⊥), and decaying to a bottom quark-antiquark pair has been performed using data collected46

in pp collisions at√s = 13 TeV by the CMS experiment [1]. A highly boosted Higgs boson decay-47

ing to bb̄ is reconstructed as a single, large radius jet (using the anti-kT algorithm with a distance48

parameter of R = 0.8) and is identified using jet substructure and dedicated b-tagging techniques.49

The method is validated with Z→ bb̄ decays. The cross section has been measured in a phase space50

defined by reconstructed level variables, namely Higgs p⊥ > 450 GeV and pseudorapidity |η| < 2.5.51

Future experimental endeavour will build on this first analysis and allow the exploration of this52

particular observable in depth.53

It is the objective of this note to briefly summarise currently available theoretical predictions for54

the transverse momentum distribution with p⊥ > 400 GeV. We analyse state of the art predictions55

based on perturbative QCD computations. In particular we perform a rough combination of next-to-56

next-to leading order (NNLO) calculations in the heavy top quark effective theory [2–4] with next-57

to-leading order (NLO) predictions in full QCD [5, 6]. Subsequently, we compare these predictions58

with state of the art parton shower Monte-Carlo event generators [7–11]. We find that indeed the59

most advanced MC event generators describe the cross sections of interest within uncertainties.60

Furthermore, we show contributions from the vector boson fusion production mode [12, 13] for the61

observable under consideration. We provide a recommendation for the theoretical prediction to be62

used by the ATLAS and CMS collaborations.63

– 1 –

2 Predictions for the gluon-fusion channel64

We start by summarising the predictions for the gluon-fusion (ggF) channel, and by giving an65

approximate NNLO result, which we quote as our recommendation for the cross section in the66

boosted regime. This is obtained by combining the NNLO prediction in the large-top mass limit67

with the NLO prediction in the full theory.68

The setup used for the NNLO results in the large-top-mass limit is as follows69

• mH = 125 GeV, mt = 173.2 GeV,70

• PDF4LHC15_nnlo_mc,71

• central scales (unless stated otherwise) µF = µR =√m2H + p2

⊥.72

The NLO predictions in the full theory used in Section 2.1 to estimate mass effects, on the other73

hand, have a slightly different scale choice. However, we argue that this does not change the final74

conclusions.75

In Section 2.2, we also consider the predictions from common event generators. Such predictions76

come with their own scale setting, as reported in the discussion below. We, of course, stress that77

the above scale choice is not unique, and different choices might lead to sizeable differences in the78

final predictions. However, the goal of this note is to benchmark the comparison between different79

theory predictions for the observable under study. Therefore, we limit ourselves to our choice for80

the discussion that follows.81

2.1 Fixed-order82

In this section we review the current state of the art of predictions for the transverse momentum83

(p⊥) spectrum of the Higgs boson in the boosted regime. The transverse momentum distribution84

was computed at NNLO in perturbative QCD in the heavy top quark effective theory (EFT) in85

refs. [2–4]. Specifically, refs. [2–4] compute NNLO corrections to the Born level production of a86

Higgs boson and a jet. In the EFT approximation the top quark is treated as infinitely heavy and87

its degrees of freedom are integrated out. It is however well known that the pure EFT computation88

fails to describe the p⊥ spectrum for transverse momenta larger than ∼ 200 GeV.89

One possibility to improve on the pure EFT computation is to create the so-called Born-90

improved EFT approximation. To this end the EFT cross section is simply rescaled by the exact91

leading order QCD cross section [14, 15].92

dσEFT-improved (0), NNLO

dp⊥=

dσQCD, LO

dp⊥dσEFT, LO

dp⊥

dσEFT, NNLO

dp⊥. (2.1)

The numerical implications of this Born-improved NNLO predictions were first studied in ref. [4]93

and show deviations from the pure EFT computation at the level of 50% for transverse momenta94

of 400 GeV. Since this modification is performed at leading order, a considerable perturbative95

uncertainty has to be associated with this procedure and higher order corrections are desirable.96

In order to further improve the result several approximations were considered including exact real97

matrix elements at NLO in QCD and approximations for virtual matrix elements in refs. [10, 16–18].98

Finally, the two-loop virtual matrix elements were included through an asymptotic expansion in99

refs. [5, 19], and exactly in ref. [6], hence allowing for the computation of the full NLO corrections.100

For the inclusive (cumulative) cross section, defined as101

Σ(pcut⊥ ) =

∫ ∞pcut⊥

dσ

dp′⊥dp′⊥ , (2.2)

– 2 –

pcut⊥ LO [fb] NLO[6] [fb] K

400 GeV 11.9+43.7%−28.9% 25.5+6.4%

−17.0% 2.14

430 GeV 8.2+44%−29.1% 17.6+6.2%

−17.0% 2.14

450 GeV 6.5+44%−29% 13.9+6.4%

−17.1% 2.14

500 GeV 3.6+44.2%−29.4% 7.7+6.2%

−17.2% 2.12

550 GeV 2.1+44.7%−29.1% 4.4+6.2%

−17.0% 2.12

600 GeV 1.2+44.9%−29.5% 2.6+6.7%

−17.5% 2.10

650 GeV 0.74+45.1%−29.9% 1.6+6.5%

−17.5% 2.09

700 GeV 0.45+45.1%−29.6% 0.93+6.4%

−17.5% 2.07

750 GeV 0.27+45.9%−29.7% 0.56+5.6%

−17.5% 2.05

800 GeV 0.16+45.0%−29.9% 0.33+6.1%

−17.5% 2.02

850 GeV 0.09+45.8%−29.9% 0.19+6.4%

−18.7% 2.00

Table 1: Inclusive cross sections and K-factors for pp → H+jet in the SM for the relevant pcut⊥

values as computed in ref. [6]. The exact two-loop virtual corrections are included. The resultsare obtained with the parton densities set PDF4LHC−30−pdfas (used both for LO and NLO) andcentral scales µR = µF = 1/2

(√m2H + p2

⊥ +∑i |pt,i|

). Uncertainties are estimated by varying µF

and µR separately by factors of 0.5 and 2 excluding opposite variations.

the results for some relevant p⊥ cuts from refs. [6] are reported in Table 1.102

The exact NLO QCD corrections computed in ref. [6] modify the exact leading order prediction103

significantly but in a uniform way, as it can be appreciated from Fig. 1, from which one can extract104

KQCD ∼ 2.14 (2.3)

with a very mild p⊥ dependence.105

An analogous behaviour is observed in predictions obtained within the EFT. As a consequence,106

the modifications of the shape of the p⊥ distribution of the Higgs boson due to finite top quark107

mass effects is already accounted for in Eq. (2.1) by the inclusion of exact leading order matrix108

elements. The EFT K-factor is of the size of KEFT ∼ 1.93.109

Ideally, we want to combine the NNLO predictions computed in the EFT with the exact NLO110

prediction. Under the assumption that the exact NNLO QCD corrections follow the pattern of111

the NNLO EFT corrections, i.e. they would lead to a a uniform K-factor, this can be achieved by112

rescaling EFT NNLO predictions in the following way:113

dσEFT-improved (1), NNLO

dp⊥=

dσQCD, NLO

dp⊥dσEFT, NLO

dp⊥

dσEFT, NNLO

dp⊥. (2.4)

We combine the above K factors [6] with the NNLO prediction of ref. [4], which uses the setup114

reported at the beginning of Section 2. The prediction of ref. [6] is obtained with a different scale115

choice, namely116

µR = µF =1

2

(√m2H + p2

⊥ +∑i

|pt,i|

), (2.5)

– 3 –

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

dσ/dp t

,H[pb/G

eV]

ratio

NLO/L

OLHC 13 TeVPDF4LHC15 NLO

µ = HT2

10−1

100

ratio to LO HEFT

1.0

2.0

0 200 400 600 800 1000

LO HEFTNLO HEFT

LO FullNLO Full

10−1

100

pt,H [GeV]

1.0

2.0

0 200 400 600 800 1000

Figure 1: Transverse momentum distribution of the Higgs boson at the LHC with√s = 13 TeV

computed in ref. [6]. The upper panel shows absolute predictions at LO and NLO in the full SMand in the infinite top-mass approximation (HEFT). The lower panel shows respective NLO/LOcorrection factors. The bands indicate theoretical errors of the full SM result due to scale variation.

and parton densities set PDF4LHC−30−pdfas. While a study with a consistent setup should be117

performed to have an accurate predictions, we hereby assume that the estimated K factor is not118

much affected by these changes, which is reasonable within the current experimental precision for119

the cross section under study. This assumption is supported by the fact that in the limit of very120

large Higgs p⊥ the two scale choices become equivalent.121

We quote the prediction obtained with Eq. (2.4) as the current best prediction. To estimate122

the theory uncertainty in the resulting cross section we proceed as follows:123

• We perform a correlated variation of µR and µF by a factor of two around their central value124

by keeping µR = µF in the fixed order NNLO cross section.125

• We assume that the uncertainty due to mass effects in the NNLO EFT correction is obtained126

by rescaling the latter by the relative mass correction at NLO. Thus, we assess the uncertainty127

δNNLO,mtas128

δNNLO,mt=δΣQCD, NLO − δΣimproved(0), NLO

δΣimproved(0), NLO × δΣimproved(0), NNLO

∼ KQCD −KEFT

KEFT − 1× δΣimproved(0), NNLO

∼ 0.2× δΣimproved(0), NNLO. (2.6)

Here, δΣ refers to the perturbative correction at a given order in QCD perturbation theory.129

• The final uncertainty is obtained by combining the two uncertainties defined in the previous130

two items. In Table 2 we report the results for the cross sections, where the uncertainties are131

– 4 –

either combined in quadrature (NNLOapproximatequad.unc. ) or summed linearly (NNLOapproximate

lin.unc. ). In132

the following, we will consider the combination in quadrature as our central prescription.133

• An additional source of uncertainty is given by the top-mass scheme, for which we adopt134

the on-shell scheme used in the calculation of ref. [6]. The difference between the on-shell135

and the MS scheme can be substantial at LO for typical renormalisation scales in boosted136

Higgs production. This difference introduces an additional source of uncertainty that will be137

considerably reduced at NLO, although at present this dependence has not yet been studied.138

For this reason, we do not consider an estimate of the corresponding uncertainty in this note.139

We stress, however, that future precise determinations of the boosted Higgs cross section must140

include a careful assessment of the top mass scheme dependence.141

pcut⊥ NNLOapproximatequad.unc. [fb] NNLOapproximate

lin.unc. [fb]

400 GeV 32.0+9.1%−11.6% 32.0+9.4%

−11.9%

430 GeV 22.1+9%−11.4% 22.1+9.3%

−11.8%

450 GeV 17.4+8.9%−11.5% 17.4+9.3%

−11.9%

Table 2: Best prediction for the inclusive cross sections at different p⊥ cuts of phenomelogicalinterest.

In Fig. 2 we show the cumulative cross section as a function of the p⊥ cut. The figure compares142

the EFT result at NNLO to the NNLO EFT result rescaled by the LO spectrum in the full theory143

(labelled EFT-improved(0), NNLO), and to our best prediction obtained with Eq. (2.4) (labelled144

EFT-improved(1), NNLO). Fig. 3 shows the ratio of the latter two predictions to the central value145

of the EFT-improved(1) prediction. The uncertainties in the EFT-improved(0) band has been ob-146

tained by pure scale variation, while the uncertainty in the EFT-improved(1) prediction is estimated147

as outlined above.148

2.2 Event generators149

In this section we report the predictions obtained with different event generators for the boosted-150

Higgs scenario.151

We compare the following Monte-Carlo tools:152

• POWHEG gg−h [7]: NLO accurate for inclusive gluon fusion and LO in the p⊥ spectrum. The153

default POWHEG µR and µF scales are used. hfact = 104 GeV as in the CMS analysis note [1]154

(this only impacts the predictions matched to a parton shower below).155

• POWHEG HJ [8]: NLO accurate in the Higgs p⊥ spectrum. µR and µF are set to HT /2 =156

1/2(√

m2H + p2

⊥ +∑ni=1 |pt,i|

), where pt,i is the transverse momentum of the i-th radiated157

parton (n = 1 for Born/Virtual events, n = 2 for real events).158

• HJ-MiNLO [9]: NLO for inclusive gluon fusion and NLO in the p⊥ spectrum. µR and µF are159

always set to p⊥. Born events with one jet terms are proportional to α2s(mH)αs(p⊥), while160

NLO corrections are proportional to α2s(mH)α2

s(p⊥).161

• MG5−MC@NLO [10]: predictions obtained by merging samples of 0,1, and 2 jets. The scale is set162

following the FxFx [20] prescription and the merging scale is set to 30 GeV163

– 5 –

EFT, NNLO

EFT-improved(0), NNLO

EFT-improved(1), NNLO

500 600 700 800 900

1

10

1

10

100

p⟂cut [GeV]

Σ(p

⟂cut )[fb]

Figure 2: Cumulative cross section as a function of the p⊥ cut, see the text for description.

EFT-improved(0), NNLO

EFT-improved(1), NNLO

500 600 700 800 900

0.8

0.9

1.

1.1

p⟂cut [GeV]

Σp

⟂cut

ΣEFT-improved(1)

Figure 3: Ratio of the cumulative cross section to the central value of the EFT-improved(1)

prediction as a function of the p⊥ cut, see the text for description.

The results for the POWHEG/MiNLO generators are reported both at fixed order and matched to164

the Pythia 6 parton shower [21], in Table 3 and 4, respectively. Table 3 shows the predictions from165

the POWHEG/MiNLO generators before the matching to a parton shower is performed, while Table 4166

reports the predictions matched to a parton shower simulation. The last row of the tables shows the167

result of HJ-MiNLO including mass effects, as implemented in ref. [17]. The results include only the168

– 6 –

Table 3: Results from the indicated event generators before the matching to parton showers isperformed (labelled as Fixed order level in the table). Predictions are expressed in [pb] units. Thetotal cross section for gg → H obtained with the indicated event generator is also reported wheneveravailable.

Table 4: Results matched to parton shower for the indicated event generators. Predictions areexpressed in [pb] units. The total cross section for gg → H obtained with the indicated eventgenerator is also reported whenever available.

top contribution, implemented through a rescaling of the EFT result by the exact LO spectrum, and169

hence very similar in spirit to the prescription introduced in Section 2.1, in Eq. (2.1). In the large170

Higgs transverse momentum region, the generator HJ-MiNLO reproduces exactly the NNLOPS [11]171

which is currently the baseline for many Higgs analyses in gluon fusion at the LHC. Uncertainties172

are obtained through a 7-point scale variation around the central renormalisation and factorisation173

sales by a factor of two.174

By inspecting the last two rows of Tables 3 and 4, we observe that the inclusion of the parton175

shower impacts the result at the 2− 5% level, as one expects for the considered kinematics regime.176

The results obtained with MG5−MC@NLO are obtained with top mass corrections included exactly177

in the Born and real corrections, and approximately in the virtual corrections by rescaling the EFT178

virtual corrections by the LO result in the full theory. Exact bottom quark mass effects are not179

– 7 –

included as they are negligible in the considered region. The events are showered with the Pythia180

8 Monte Carlo [22]. The results for some relevant pT cuts are summarized in Table 5, together181

with a comparison to the results of the HJ-MiNLO generator, and to our best prediction described182

in Section 2.1. The quoted uncertainties have been obtained by a 9-point scale variation around183

the central renormalisation and factorisation scales by a factor of two.184

pcutT NNLOapproximatequad.unc. [fb] HJ-MINLO [fb] MG5_MC@NLO [fb]

400 GeV 32.0+9.1%−11.6% 29+24%

−21% 31.5+31%−25%

430 GeV 22.1+9%−11.4% - 21.8+31%

−25%

450 GeV 17.4+8.9%−11.5% 16.1+22%

−21% 17.1+31%−25%

Table 5: Comparison of predictions at fixed order, with HJ-MINLO and with MG5−MC@NLO. See textfor details.

We observe that the predictions obtained with the more accurate generators used in the study185

( HJ-MiNLO and MG5−MC@NLO) are in very good agreement with one another. Moreover, they both186

reproduce, within uncertainties, the best prediction obtained in the previous section. We conclude187

that the above two generators can be safely used to perform accurate studies in the boosted regime.188

However, state of the art QCD predictions reach a higher level of precision and novel methods are189

necessary to exploit such calculations in the context of Monte Carlo simulations.190

3 Predictions from Vector Boson Fusion191

We conclude this note by reporting the results for the vector boson fusion (VBF) contribution to192

the boosted-Higgs cross section. This is obtained using the predictions of ref. [12, 13], where the193

VBF cross section is computed to approximate N3LO accuracy in perturbative QCD. Estimated194

uncertainties due to imprecise knowledge of coupling constants, parton distribution functions, miss-195

ing higher perturbative orders and corrections due to the chosen approximation amount in total to196

roughly 3%. Note, that also electro-weak corrections to the inclusive VBF cross section start to197

play an important role and are at the level of ∼ −20%, as discussed in Appendix A.198

We observe that the VBF contribution at the relevant p⊥ values is a significant fraction of199

the total cross section. We consider its value at p⊥ = 450 GeV as a reference value. Using a200

BR(H → bb̄) = 0.5824, we obtain the results reported in Table 6, where we observe that the VBF201

contribution accounts for 30% of the total cross section in this region. The errors indicate only the202

scale uncertainty. However, as mentioned above, further sources of uncertainties are present. The203

cumulative cross section as a function of p⊥ is reported in Fig. 4.

pcut⊥ ΣggF(pcut⊥ ) × BR [fb] ΣVBF(pcut⊥ ) × BR [fb] ΣggF+VBF(pcut⊥ ) × BR [fb]

450 GeV 10.13+8.9%−11.5% 4.71+0.14%

−0.14% 14.84+6.08%−7.85%

Table 6: Predictions including the VBF contribution. The uncertainties in the gluon fusion crosssection are determined by combining the various sources in quadrature as described in the previoussection. The errors in the VBF contribution indicates the QCD scale uncertainty.

204

It should also be noted that other production modes gain in importance relative to the gluon205

fusion production mechanism as the transverse momentum of the Higgs boson is increased. The final206

– 8 –

1

10-1

101

102

103

0 100 200 300 400 500 600 700 800 900 1000

dσ[f

b]

pT,H[GeV]

AccumulatedptHiggs(H->bb)

Figure 4: Cumulative cross section from vector-boson fusion in the boosted regime. The Higgsbranching ratio into bottom quarks is included.

state signature arising from the production of a Higgs boson in association with a vector boson [23–207

27] or a top quark pair [23, 24] is however significantly different such that we do not include explicit208

values for the respective cross sections here. For reference we display the contributions of the209

different production modes in Appendix A.210

4 Summary and conclusions211

In this note we studied the inclusive production of a boosted Higgs boson at the LHC. We performed212

a rough combination of state of the art QCD predictions for the gluon-fusion and vector-boson-213

fusion channels, and provide a recommendation for the cumulative distribution at large transverse214

momenta. The resulting predictions are reported in Table 7 for different values of the lower cut215

on the Higgs transverse momentum. Moreover, we compare the resulting predictions to those of216

Monte-Carlo event generators in Table 5 and find good agreement within the quoted uncertainties.217

This implies that one can safely use the predictions from the considered event generators with the218

associated theoretical errors in the simulation of the boosted Higgs cross section.219

We stress that the results presented in this note constitute only an approximate analysis,220

and additional sources of theoretical uncertainties (such as the top mass scheme) must be studied221

carefully. Therefore, further in-depth studies are required for future precise determinations of the222

boosted Higgs cross section.223

– 9 –

pcut⊥ [GeV] ΣggF(pcut

⊥ ) [fb] ΣVBF(pcut⊥ ) [fb] ΣggF+VBF(pcut

⊥ ) [fb]400 32.03+9.09%

−11.55% 14.26+0.11%−0.11% 46.29+6.29%

−7.99%

410 28.15+9.00%−11.49% 12.70+0.12%

−0.12% 40.84+6.20%−7.92%

420 24.89+9.00%−11.46% 11.32+0.12%

−0.12% 36.20+6.18%−7.87%

430 22.05+8.99%−11.45% 10.10+0.13%

−0.13% 32.16+6.17%−7.85%

440 19.57+8.97%−11.52% 9.04+0.13%

−0.13% 28.60+6.14%−7.88%

450 17.37+8.90%−11.50% 8.09+0.14%

−0.14% 25.45+6.07%−7.85%

460 15.34+8.85%−11.50% 7.26+0.14%

−0.14% 22.60+6.01%−7.81%

470 13.63+8.93%−11.51% 6.52+0.15%

−0.15% 20.15+6.04%−7.79%

480 12.14+8.90%−11.48% 5.87+0.16%

−0.16% 18.01+6.00%−7.74%

490 10.83+8.96%−11.52% 5.29+0.16%

−0.16% 16.12+6.02%−7.74%

500 9.66+8.86%−11.49% 4.77+0.17%

−0.17% 14.43+5.93%−7.69%

510 8.62+8.80%−11.51% 4.31+0.18%

−0.18% 12.93+5.87%−7.67%

520 7.71+8.74%−11.55% 3.90+0.18%

−0.18% 11.61+5.80%−7.67%

530 6.88+8.69%−11.52% 3.54+0.19%

−0.19% 10.42+5.74%−7.61%

540 6.19+8.68%−11.51% 3.21+0.20%

−0.20% 9.39+5.72%−7.58%

550 5.54+8.76%−11.45% 2.91+0.21%

−0.21% 8.45+5.74%−7.51%

560 4.96+8.78%−11.38% 2.65+0.22%

−0.22% 7.61+5.72%−7.42%

570 4.46+8.71%−11.35% 2.41+0.22%

−0.22% 6.87+5.66%−7.37%

580 4.03+8.69%−11.33% 2.20+0.23%

−0.23% 6.22+5.62%−7.33%

590 3.61+8.74%−11.36% 2.01+0.24%

−0.24% 5.62+5.61%−7.30%

Table 7: Predictions for the cumulative Higgs boson cross section as a function of the lowestallowed p⊥. We show predictions due to the ggF and VBF production mechanism and their com-bination. The uncertainties of the combined prediction are computed by combining the individualuncertainties in quadrature.

A Contribution from other production modes224

In this appendix we display the breakdown of the boosted Higgs cross section into different produc-225

tion modes. Table 8 reports the QCD predictions for the contribution to the inclusive cross section226

from gluon fusion at approximate NNLO (as estimated in this note), from VBF and associated227

production VH at NNLO, and from tt̄H at NLO. Table 9 displays the percentage decrease of the228

corresponding cross sections of Table 8 due to the inclusion of electro-weak corrections. Finally, the229

absolute and relative contributions of the different production modes are summarised in Fig. 5.230

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pcut⊥ [GeV] Σ

NNLOapprox

ggF (pcut⊥ ) [fb] ΣNNLO

VBF (pcut⊥ ) [fb] ΣNNLO

VH (pcut⊥ ) [fb] ΣNLO

tt̄H (pcut⊥ ) [fb]

400 32.03+9.09%−11.55% 14.23+0.15%

−0.19% 11.16+4.12%−3.68% 6.89+12.62%

−12.97%

450 17.37+8.90%−11.50% 8.06+0.24%

−0.23% 6.87+4.6%−3.49% 4.24+12.84%

−13.15%

500 9.66+8.86%−11.49% 4.75+0.33%

−0.29% 4.39+4.43%−4.04% 2.66+12.85%

−13.22%

550 5.54+8.76%−11.45% 2.90+0.34%

−0.36% 2.87+4.44%−3.74% 1.76+14.23%

−13.93%

600 3.24+8.73%−11.28% 1.82+0.41%

−0.39% 1.91+5.22%−4.71% 1.11+12.99%

−13.4%

650 1.94+8.66%−11.28% 1.17+0.49%

−0.39% 1.30+4.67%−4.28% 0.72+12.6%

−13.26%

700 1.15+8.56%−11.24% 0.77+0.57%

−0.45% 0.90+4.15%−5.4% 0.47+11.42%

−12.74%

750 0.69+8.53%−11.27% 0.51+0.69%

−0.56% 0.62+5.15%−4.66% 0.32+11.53%

−12.84%

800 0.41+8.47%−11.18% 0.35+0.71%

−0.6% 0.44+5.64%−4.13% 0.22+11.42%

−13.3%

Table 8: Predictions for the cumulative Higgs boson cross section as a function of the lowest p⊥cut. We show QCD predictions for the various channels contributing to Higgs production. Thetable does not contain the EW corrections whose contribution can be sizeable in the consideredregion of p⊥.

pcut⊥ [GeV] VBF VH tt̄H

400 −17.80% −19.05% −6.95%

450 −19.43% −20.83% −7.75%

500 −21.05% −22.50% −8.49%

550 −22.34% −24.07% −9.11%

600 −23.73% −25.56% −9.91%

650 −25.03% −26.98% −10.67%

700 −26.29% −28.30% −11.37%

750 −27.35% −29.60% −11.94%

800 −28.42% −30.83% −12.51%

Table 9: Percentage decrease of the cross sections of Table 8 due to the inclusion of electro-weakcorrections as a function of the cut in p⊥.

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10-4

10-3

10-2

10-1

400 450 500 550 600 650 700 750 800

√s = 13 TeVmH = 125 GeVPDF4LHC15_nnlo_mc

Σ(p

tH) [

pb]

ptH [GeV]

ggF (EFT-improved(1), NNLO)VBF (NNLO-QCD x NLO-EW)

VH (NLO-QCD x NLO-EW)ttH (NLO-QCD x NLO-EW)

Total

0

0.2

0.4

0.6

0.8

1

400 450 500 550 600 650 700 750 800

√s = 13 TeVmH = 125 GeVPDF4LHC15_nnlo_mc

Frac

tiona

l con

tribu

tion

ptH [GeV]

ggF (EFT-improved(1), NNLO)VBF (NNLO-QCD x NLO-EW)

VH (NLO-QCD x NLO-EW)ttH (NLO-QCD x NLO-EW)

Figure 5: Cumulative cross section for the production of a Higgs boson as a function of the lowestHiggs boson transverse momentum. The cross section due to the gluon-fusion (green), VBF (red),vector boson associated (blue) and top-quark pair associated (magenta) production mode are shownin absolute values (left) and relative size (right).

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