Table Of ContentHiggs pair production at the LHC with NLO and parton-shower effects
R. Frederixa, S. Frixionea, V. Hirschib, F. Maltonic, O. Mattelaerc, P. Torriellid, E. Vryonidouc, M. Zaroe,f
aPH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland
bSLAC, National Accelerator Laboratory
2575 Sand Hill Road, Menlo Park, CA 94025-7090, USA
cCentre for Cosmology, Particle Physicsand Phenomenology (CP3),
UniversitéCatholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
dInstitut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, Switzerland
eSorbonne Universités, UPMCUniv. Paris 06, UMR 7589, LPTHE, F-75005, Paris, France
fCNRS, UMR 7589, LPTHE, F-75005, Paris, France
4
1
0
Abstract
2
r We present predictions for the SM-Higgs-pair production channels of relevance at the LHC: gluon-gluon fusion, VBF,
aand top-pair, W, Z and single-top associated production. All these results are at the NLO accuracy in QCD, and
M
matched to partonshowersby means of the MC@NLO method; hence, they are fully differential. With the exceptionof
the gluon-gluon fusion process, for which a special treatment is needed in order to improve upon the infinite-top-mass
0
limit, our predictions are obtained in a fully automatic way within the publicly available MadGraph5_aMC@NLO
2
framework. We show that for all channels in general, and for gluon-gluon fusion and top-pair associated production in
]particular,NLO correctionsreducethe theoreticaluncertainties,andareneededinordertoarriveatreliablepredictions
h
for total rates as well as for distributions.
p
-
p
e1. Introduction dimensionalinteractionsaswellastheexistenceofheavier
h
states coupled to the Higgs [5, 6, 7, 8, 9, 10, 11]. Unfortu-
[ PresentLHCdataprovideevidencethatthescalarpar-
nately,intheSMtheratesforHiggspairproductionatthe
ticle observed at the LHC is the one predicted by the
2 LHCarequitesmall[12,13,14,15]. Sounlessnewphysics
vBrout-Englert-Higgssymmetry breaking mechanism [1, 2]
produces sizable enhancements (something quite possible
0ofSU(2) U(1) asimplementedintheStandardModel
4(SM) [2].L×In thisYcase, the strengths of the Higgs boson in several scenarios), a measurement of the HH produc-
3 tion cross sections will necessitate considerable integrated
couplings are uniquely determined by the masses of the
7 luminosity even at 14 TeV centre-of-mass energy. In any
elementary particles. The measuredcouplings to fermions
. case,precisepredictionsforratesanddistributionswillbe
1
and vector bosons agree within 10-20% with the SM pre-
0 neededin orderto be ableto extractvaluable information
dictions [3, 4]. No information, however, has been col-
4 on λ or on new physics effects in general.
1lected so far on the Higgs self-coupling λ. In the SM the Analogously to single-Higgs production, several chan-
:Higgs boson mass itself fixes the value of this self cou-
v nels can lead to a final state involving two Higgs bosons.
pling in the scalar potential whose form, in turn, is de-
i They entail the Higgs coupling to either the top quark
Xtermined by the global symmetries and the requirement
(as in the case of gluon-gluon fusion and of tt¯associated
rof renormalisability. These conditions, however, have no
araison d’être once experimental indications (such as the production), or vector bosons (in VBF, and in W and
Z associated production), or both (for single-top associ-
existence of dark matter) as well as theoreticalarguments
ated production). The dominant production mechanism
(such as naturalness) are put forward. In this respect, it
is gluon-gluonfusion via a top loop, exactly as in the case
is appropriate(and,infact,advantageous)toconsiderthe
of single-Higgs production. Cross sections corresponding
SM as the subset of operators of dimension less than or
to the other channels are at least one order of magnitude
equal to four ofan effective field theory (EFT) lagrangian
smaller, even though possibly interesting because of dif-
withanSU(2) U(1) symmetry. Directinformationon
L× Y ferent sensitivity to λ or to new physics, and because of
the Higgs three- and four-point interactions could there-
the possibility of exploiting a wider range of Higgs decay
fore provide a key indication of the structure of the scalar
signatures.
potential, and of where the scale Λ characterisingsuch an
In this Letter we present results accurate to NLO in
EFT might lie.
QCD for the six production channels mentioned before,
In this context, Higgs pair production could play a which are the largest in the SM. For all of them our pre-
key role. Not only it is the simplest production process
dictionsimproveuponexistingonesinatleastoneaspect.
that is sensitive to the self-coupling λ, but it also pro-
We shall discuss this point in more details in what fol-
vides one with a wealthofpossibilities forprobing higher-
Preprint submittedto Elsevier March 21, 2014
SM and the MSSM, but is only capable of computing to-
H H tal cross sections. In HPAIR the NLO calculation is es-
sentially performed with EFT techniques; the exact one-
loop Born amplitudes are however employed as leading-
order contribution to the NLO cross section, and used to
H H
reweight (after the integration over the polar scattering
angle) the HEFT virtual- and real-emission matrix ele-
Figure 1: Classes of diagrams for Higgs pair production in hadron ments. In this work we improve on the HPAIR approach
hadron collisions: double Higgs production without HHH vertices
on several counts. Firstly, we include the exact one-loop
on the left-hand side, and, on the right-hand side, the contribution
results not only for the 2 2 Born amplitudes, but also
duetotheHiggsselfinteraction. Finalstateparticlesotherthanthe
→
Higgsbosonsareunderstood. for the 2 3 real-emission processes, which we compute
→
withMadLoop[19]. Inotherwords,the onlyapproxima-
tion made at the level of matrix elements is that for the
finite partofthe two-loopvirtualcorrectionswhich,being
lows. Here, we limit ourselves to pointing out that HH
presentlyunknown,is approximatedby the corresponding
productionviagluon-gluonfusioniscomputedattheNLO
one-loopHEFT resultreweighted(withoutanyintermedi-
in a “loop-improved”EFT approach,using the exact one-
ate integration) with the exact one-loop Born amplitude.
loop real-emission and improved one-loop virtual matrix
elements; that in the case of tt¯HH and tjHH produc- Secondly, in the loops we make use of the complex mass
(and Yukawa) scheme for the top quark [20, 21]. Thirdly,
tion exact NLO QCD results are presented in this paper
our results are fully differential, and can be used to ob-
for the first time; and that by matching NLO computa-
tain any distribution after matching with parton shower.
tions to parton showers we generate samples of events,
In summary, our predictions improve both on the total
also for the first time, for each of the production chan-
cross sections that can be obtained with HPAIR, and on
nels, which can be used for fully realistic simulations, in-
the differential, hadron-level (i.e., showered) observables
cluding those at detector level. With the exception of
recently presented in ref. [22, 23] (which do not include
the gluon-gluonfusion process which, being loop-induced,
virtual effects, and are therefore akin to tree-level merged
needs an ad-hoc treatment, our results are obtained auto-
results). Wealsostresswedonotmakeuseoftherecently-
matically with the publicly-available version of the Mad-
derived 1/m effects at the NLO accuracy [24], of the
Graph5_aMC@NLO framework [16, 17]. t
NNLO HEFT results for total rates [25] and of threshold
In the next section we introduce and review the main
resummation[26]. Moredetailsontheprocedureemployed
features of the Higgs-pair production channels. In sec-
in this work will be presented elsewhere [27].
tion 3 we present the calculation and simulation frame-
The second-largest production channel is vector bo-
work, and in section 4 we collect results for some selected
son fusion (VBF). In this case the NLO QCD corrections
observables together with their uncertainties. We sum-
are trivial, as they involve the same contributions as for
marise our findings and prospects in the conclusions.
single-Higgs production. In VBF we compute only vertex
loop corrections, i.e., the finite part of the pentagon and
2. Higgs pair production channels hexagonloopdiagramsarediscardedforsimplicity. These
contributions only affect interferences between diagrams
IntheSM,thediagramscontributingtoHiggspairpro-
that feature identical quarks, which are negligibly small
duction can be organised in two classes (see fig. 1): those
already at the LO. NLO results have been presented in
where both Higgs bosons couple only to vector bosons or
the literature (see e.g. [15]) only for total rates. In this
toheavyquarks,andthosethatfeaturetheHiggsselfcou-
paper we study, for the first time, differential observables
pling.
for VBF in the SM at fixed NLO and matched to parton
The dominant channel for Higgs pair production is
showers, showing distributions for the latter. Distribu-
gluon-gluon fusion via virtual top quarks, i.e., box and
tions at fixed NLO in the two Higgs doublet model have
triangle diagrams. This process therefore starts at the
appeared in ref. [28]. We point out that, although NNLO
leading order with a loop, exactly as single-Higgs produc-
correctionstothetotalVBFcrosssectionsarenotknown,
tion. In contrast with the latter, however, the effective
they could be easily computed following the approach of
field theory approach(where Higgs-gluons vertices are in-
ref. [29].
cluded in the lagrangian =α /(3πv2)(φ†φ)GG, G
LHEFT S At variance with single-Higgs production, the produc-
being the QCD field tensor) provides only a rough ap- tionofaHiggspairinassociationwithatt¯pairisthethird
proximation for total rates, and a very poor one for dis-
most importantprocessand, in fact, it is evenlargerthan
tributions [18, 10]. Better predictions, which take loop
VBF at high Higgs-pair transverse momenta, or for col-
effectsintoaccountexactly,needthereforetobeemployed
lider centre-of-massenergieshigherthan thatofthe LHC.
inactualphenomenologicalandexperimentalstudies. Re-
TheinclusionofNLOQCDcorrectionsinthis processhas
sults have been available for some time, and implemented
never been achieved prior to this work, even at a fully
in the code HPAIR [12, 13], which deals with both the
inclusive level, as it involves thousands of Feynman dia-
2
grams of high complexity, such as pentagon and hexagon ules: MadFKS[34]takescareoftheBornandofthereal-
loops. Our framework, however, has no problems in han- emissionamplitudes,anditalsocarriesoutthesubtraction
dling it in a fully automatic way. For instance, the total oftheinfraredsingularitiesaccordingtotheFKSprescrip-
(sequential) CPU time required to generate one million tion [35, 36] as well as the generation of the Monte Carlo
unweighted events and to obtain a cross section accurate subtractiontermsrequiredbytheMC@NLOmethod[37].
at the per-mil level is about one hundred and sixty CPU MadLoop [19] computes one-loop amplitudes, using the
hours on a 2.3GHz machine. This renders the computa- OPP integrand-reductionmethod [38] (as implemented in
tion feasible on a medium-size (30 core) cluster in a few CutTools [39]) and the OpenLoops method [40]. In the
hours. caseofVBFandtjHH production,someminimalinternal
Thechannelsofvectorbosonassociatedproductionare manipulations make use of FJcore [41].
technically the easiest ones, as all QCD corrections fac- InoursimulationswesettheHiggsmassequaltom =
H
torise and are relevant only to the initial state. As in 125 GeV. Parton distributions functions (PDFs) are eval-
the case of VBF, we improve upon existing NLO results uated by using the MSTW2008 (LO and NLO) set in
by giving one the possibility of studying fully-differential the five-flavour scheme [42]. b-quark masses as well as
observables; in this work, we do not include the finite their coupling to the Higgs are neglected. For the sake
one-loop, gg-initiated contributions to ZHH production, of brevity, we only show observables related to the Higgs
which however can also be handled by MadLoop. Our bosons and therefore we have left the latter stable in the
results correspond to on-shell final state vector bosons. simulations. We stress, however, that the top quarks and
NNLO QCD corrections to total cross sections are known the vector bosons that appear in the final states can be
to be small [15]. decayed with the built-in MadSpin package [43], which
Finally,inordertoprovidethecompletesetofpossibly allows one to include all spin-correlation effects. On the
interestingfinalstates,wealsocomputeforthefirsttimeat other hand, Higgs decays can be handled correctly also
the NLO the single-top associated production, by includ- by the Monte Carlos, thanks to the Higgs being a spin-0
ingboths-andt-channelcontributionsandbyconsidering particle.
both top and anti-top in the final state. The correspond- Thecodeallowsfullflexibilityasfarasthechoiceofthe
ing crosssectionsaretiny atthe LHC,andofverylimited renormalisationandfactorisationscalesµ isconcerned.
R,F
phenomenologicalrelevance inthe SM. However,this pro- The centralvalues of these scales have been chosen as fol-
cess is at least of academic interest because it is sensitive lows. Forgluon-gluonfusion, VBF,andVHH production
to couplingsto bothvectorbosonsandtopquarks,andto we set µ = m /2, m and m , respectively. For
0 HH W VHH
theirrelativephases. Inadditiontothat,giventhatithas tt¯HH wechooseµ0 =(cid:0)mT(H1)mT(H2)mT(t)mT(t¯)(cid:1)1/4,
the largest sensitivity to the self-coupling λ, it might be- m being the transverseenergy of the corresponding par-
T
come relevant at a future proton-proton 100 TeV collider. ticle,aswefindthatinthiswaythecrosssectiondisplaysa
ratherstablebehaviour. Forsingle-topassociatedproduc-
3. Setup tiontjHH wesimplyusethefixedvalueµ0 =mH+mt/2.
Scale and PDF uncertainties can be evaluated at no
As was mentioned above, apart from the gluon-gluon extra computational cost thanks to the reweighting tech-
fusionchannel,allresultspresentedinthisworkhavebeen niqueintroducedinref.[44],theuserdecidingtherangeof
obtainedinafullyautomaticwaywithMadGraph5_aMC- variation. In addition, such information is available on an
@NLO [16, 17]. This program is designed to perform the event-by-event basis and therefore uncertainty bands can
computation of tree-level and NLO cross sections, includ- be plotted for any differential observable of interest. In
ing their matching to parton showers and the merging of ouranalysiswevaryindependently the scalesinthe range
samples with different parton multiplicities. A user can 1/2µ < µ ,µ < 2µ . PDF uncertainties at the 68%
0 R F 0
generate a given process through a simple shell interface C.L. are obtained by following the prescription given by
(inamannerfullyanalogoustothatofMadGraph5[30]), the MSTW collaboration [42].
with the corresponding self-contained code being gener- For the studies shown in this paper we employ HER-
atedonthefly. Whileitispossibletoobtainpredictionsat WIG6[45]andPythia8[46]forpartonshowerandhadro-
the ME+PSlevel(i.e., with the MLM-kT tree-levelmerg- nisation. The matching to HERWIG++ [47] and Py-
ing techniqueofrefs.[31,32,33]anditsanalogues)inthis thia6 [48] (virtuality ordered, plus p ordering for pro-
T
work we limit ourselves to NLO+PS results. This is be- cesses with no final-state radiation) is also available in
cause the smallness of the Higgs-pair crosssections rather MadGraph5_ aMC@NLO.
emphasisesobservableswhichareinclusivewithrespectto
extraradiation,andforwhichNLO-levelresultshavetobe
4. Results
preferredtotree-levelmergedones,sincetheyprovideone
withbetterpredictionsforabsolutenormalisationsandfor
We start by presenting in fig. 2 the predictions for the
theoretical uncertainties.
total rates at proton-proton colliders with up to 100 TeV
Within MadGraph5_aMC@NLO, any NLO compu-
c.m.energy. Thethicknessofthecurvescorrespondstothe
tation is performed by means of two independent mod-
3
4
10
HH production at pp colliders at NLO in QCD
M =125 GeV, MSTW2008 NLO pdf (68%cl)
3 H
10
102 →HH (EFT loop-improved)
pp
1
10
σ[fb]NLO 100 pppp→→HttHHHjj (VBF) pp→WHH
O
pp→ZHH NL
@
C
10-1 pp→tjHH _aM
5
h
p
-2 a
10 r
G
d
a
M
-3
10
8 13 14 25 33 50 75 100
√s[TeV]
Figure 2: Total cross sections at the NLO in QCD for the six largest HH production channels at pp colliders. The thickness of the lines
corresponds tothescaleandPDFuncertainties addedlinearly.
scale and PDF uncertainties added linearly. More details processes, the quark-gluon initiated channel which opens
are available in table 1 for selected LHC energies, i.e., 8, up at the NLO can be important.
13 and 14 TeV. The first uncertainties (in percent) corre- In fig. 3 we display total LO and NLO cross sections
sponds to scalevariation,while the second(only shownat forthesixdominantHH productionchannelsattheLHC
theNLO)toPDFssystematics. Severalobservationsarein with√s=14TeV,asafunctionoftheself-interactioncou-
order. Firstly, contrary to what happens in single-Higgs pling λ. The dashed (solid) lines and light- (dark-)colour
production, the top-pair associated channel is the third- bandscorrespondtotheLO(NLO)resultsandtothescale
largest starting at about √s =10 TeV, and becomes the and PDF uncertainties added linearly. The SM value of
second-largestwhenc.m.energiesapproach√s=100TeV. the cross section corresponds to λ/λ = 1. The sensi-
SM
Secondly, the theoretical uncertainties due to scale varia- tivity of the total cross sections to the actual value of λ
tions in the three most important processes (gluon-gluon depends in a non-trivial way on the relative couplings of
fusion, VBF, and tt¯associatedproduction) are sizably re- the Higgs to vector bosons and top quarks, and on the
duced by the inclusion of the NLO corrections. Thirdly, kinematics in a way that is a difficult to predict a priori,
the K-factor is always slightly larger than one, except for i.e., without an explicit calculation. The reduction of the
gluon-gluonfusionwhereitisofordertwo,andforthetop- scale uncertainties that affect the gg HH, VBF, and
→
pair associated channel where it is smaller than one. Fi- tt¯HH rates, due to the inclusion of NLO corrections,and
nally,PDFuncertaintiesarecomparabletoNLOscaleun- pointed out in table 1 for the SM, is seen here also for
certainties,exceptinthecaseofgluon-gluonfusion,where values of λ=λ .
SM
6
the latter are dominant. In the case of VHH and tjHH We then plot typical distributions for all channels and
production it is manifest that the standard procedure of at the 14 TeV LHC, which we obtain by generating sam-
determining uncertainties due to missing higher orders by ples of events at parton level, which are then showered
varying the scales does not give a reliable estimate, as with Pythia8 (solid) and HERWIG6 (dashes). Being
NLO corrections for these processes are much larger than tiny at the 14 TeV LHC, we do not show the results for
the LO scale dependence band. This is due to two facts: single-top associated production. We present observables
these processes are purely electro-weak processes at the at the NLO+PSaccuracyin the main frames ofthe plots:
LO, and therefore the scale uncertainties are artificially thetransversemomentumofthehardest(softest)Higgsin
small;furthermoreinthekinematicregionprobedbythese fig.4(fig.5),andthetransversemomentum(fig.6)andthe
4
√s=8TeV √s=13TeV √s=14TeV
(LO) NLO (LO) NLO (LO) NLO
HH (EFT loop-improv.) (5.44+38%) 8.73+17+2.9% (19.1+33%) 29.3+15+2.1% (22.8+32%) 34.8+15+2.0%
−26% −16−3.7% −23% −14−2.5% −23% −14−2.5%
HHjj (VBF) (0.436+12%) 0.479+1.8+2.8% (1.543+9.4%) 1.684+1.4+2.6% (1.839+8.9%) 2.017+1.3+2.5%
−10% −1.8−2.0% −8.0% −0.9−1.9% −7.7% −1.0−1.9%
tt¯HH (0.265+41%) 0.177+4.7+3.2% (1.027+37%) 0.792+2.8+2.4% (1.245+36%) 0.981+2.3+2.3%
−27% −19−3.3% −25% −10−2.9% −25% −9.0−2.8%
W+HH (0.111+4.0%) 0.145+2.1+2.5% (0.252+1.4%) 0.326+1.7+2.1% (0.283+1.1%) 0.364+1.7+2.1%
−3.9% −1.9−1.9% −1.7% −1.2−1.6% −1.3% −1.1−1.6%
W−HH (0.051+4.2%) 0.069+2.1+2.6% (0.133+1.5%) 0.176+1.6+2.2% (0.152+1.1%) 0.201+1.7+2.2%
−4.0% −1.9−2.2% −1.7% −1.2−2.0% −1.4% −1.1−1.8%
ZHH (0.098+4.2%) 0.130+2.1+2.2% (0.240+1.4%) 0.315+1.7+2.0% (0.273+1.1%) 0.356+1.7+1.9%
−4.0% −1.9−1.9% −1.7% −1.1−1.6% −1.3% −1.2−1.5%
tjHH(10−3) (5.057+2.0%) 5.606+4.4+3.9% (23.20+0.0%) 29.77+4.8+2.8% (28.79+0.0%) 37.27+4.7+2.6%
· −3.2% −2.3−4.2% −0.8% −2.8−3.2% −1.2% −2.7−3.0%
Table 1: LO and NLO total cross sections (in fb) for the six largest production channels at the LHC, with √s = 8,13,14TeV. The first
uncertaintyquotedreferstoscalevariations,whilethesecond(onlyattheNLO)toPDFs. Uncertaintiesareinpercent. Nocutsareapplied
tofinalstateparticlesandnobranchingratiosareincluded.
invariant mass (fig. 7) of the Higgs pair. The insets show, 5. Conclusions
channel by channel, the ratios of NLO+Pythia8 (solid),
NLO+HERWIG6(dashes),LO+HERWIG6(dashedwith Assessingthenatureofthenewlydiscoveredbosonwill
open boxes) results over the LO+Pythia8 ones. The need a campaign of measurements to be performed at the
dark-colour(light-colour)bandsdisplaythescale(red)and LHC at an unprecedented accuracy. One of the key pro-
PDF(blue)uncertaintiesaddedlinearlyfortheNLO(LO) cessesinthisendeavourisHiggs-pairproduction. Notonly
simulations. it gives one the possibility of measuring the value of the
NLO effects appear as overall rescaling factors only Higgs self-coupling λ, but also of putting constraints on
in some distributions and on a channel-dependent basis. several, still viable, new-physics scenarios. All such mea-
Moreover, differences between results obtained with the surementswillneedaccurateSMpredictionsfortotalcross
two different shower programs are very mild for all ob- sections(inordertoextractinformationonthe couplings)
servables and anyway decreasing when going from LO to and differential distributions (in order to establish accep-
NLO. In addition, we have checkedthat differences in the tances and identify optimal selection cuts), including reli-
distributions betweenNLO+Pythia8/NLO+HERWIG6 able estimates of the theoretical uncertainties.
andNLO fixed-orderresultsarequite small(typically less InthisLetterwehavepresentedthefirstpredictionsat
than a few percent), and that in general the NLO+PS the NLO accuracymatchedwithpartonshowerforallthe
results are slightly closer to each other than to the corre- relevant Higgs-pair production channels in the Standard
sponding NLO fixed-order results. The only exception to Model. We find that, as expected, including NLO correc-
this general behaviour is seen in some distributions rele- tions leads to a reduction of the theoretical uncertainties,
vant to the tt¯HH channel, where the NLO curves lie be- especially significant in the gluon-initiated channels, and
tweentheNLO+PSones,whichcandifferupto10%(still provides one with reliable predictions for the kinematic
withinthescaleuncertainties). However,inallthesecases, distributions of final state particles. With the exception
the corresponding differences at the LO+PSlevel are sys- of the gluon-gluon fusion process, which needs an ad-hoc
tematically largerthan forNLO+PS,whichthus confirms treatment and for which a dedicated procedure and code
thestabilisationtrendusuallyseenwhenhigher-ordercor- have been developed [27], all the results presented here
canbe easily andautomatically reproducedwith the pub-
rections are included.
licly available version of the MadGraph5_aMC@NLO
NLO corrections in the gluon-gluonfusion channel are
code [17].
important rate-wise, yet the shapes are not strongly af-
Theextensionofourstudytomodelsthatfeaturephysics
fected, asisapparentfromtheratherflatK-factors. NLO
beyond the SM is in progress.
corrections in VBF production are of order 10-20%, and
modify the shape of the distributions towards low mass-
scale values. NLO effects in tt¯HH production lead to a Acknowledgements
drastic reduction of the scale uncertainties, and to mi-
nor changes in the shapes, except for m . The asso- This work has been supported in part by the ERC grant
HH
ciatedvectorbosonchannelsdisplayverysimilarfeatures: 291377"LHCTheory",bytheSwissNationalScienceFoun-
rather flat K-factors for all the distributions studied, ex- dation (SNF) under contracts 200020-138206and 200020-
ceptforp (HH)wheretheNLOcorrectionsbecomemore 149517,and grantPBELP2_146525,by the ResearchEx-
T
and more important at high p . ecutive Agency (REA) of the European Union under the
T
5
HH production at 14 TeV LHC at (N)LO in QCD
M =125 GeV, MSTW2008 (N)LO pdf (68%cl)
H
2
10
pp→
HH (EFT loop-improved)
1
[fb]O 10 pp→HHjj (VBF)
L
N)
σ( pp→ttHH O
0 L
10 N
@
C
→WHH M
pp a
_
5
h
p
10-1 pp→ZHH p p→tjH H Gra
d
a
M
-4 -3 -2 -1 0 1 2 3 4
λ/λ
SM
Figure 3: Total cross sections at the LO and NLO in QCD for HH production channels, at the √s =14 TeV LHC as a function of the
self-interactioncouplingλ. Thedashed(solid)linesandlight-(dark-)colourbandscorrespondtotheLO(NLO)resultsandtothescaleand
PDFuncertainties addedlinearly. TheSMvaluesofthecrosssectionsareobtainedatλ/λSM=1.
GrantAgreementnumbersPITN-GA-2010-264564(LHCPhe- [11] M.Gouzevitch,A.Oliveira,J.Rojo,R.Rosenfeld,G.P.Salam,
noNet)andPITN-GA-2012-315877(MCNet). Theworkof et al.,JHEP1307,148(2013), arXiv:1303.6636[hep-ph].
[12] T.Plehn,M.Spira, andP.Zerwas,Nucl.Phys.B479,46(1996),
FM and OM is supported by the IISN “MadGraph” con-
arXiv:hep-ph/9603205 [hep-ph].
vention4.4511.10,bytheIISN“Fundamentalinteractions” [13] S.Dawson,S.Dittmaier, andM.Spira,Phys.Rev.D58,115012
convention 4.4517.08, and in part by the Belgian Federal (1998), arXiv:hep-ph/9805244 [hep-ph].
Science Policy Office through the Interuniversity Attrac- [14] T. Binoth, S. Karg, N.Kauer, and R.Ruckl, Phys.Rev. D74,
113008 (2006), arXiv:hep-ph/0608057[hep-ph].
tion Pole P7/37. OM is "Chercheur scientifique logistique
[15] J.Baglio,A.Djouadi,R.Gröber,M.Mühlleitner,J.Quevillon,
postdoctoral F.R.S.-FNRS". et al.,JHEP1304,151(2013), arXiv:1212.5581[hep-ph].
[16] R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer,
H.-S.Shao,T.Stelzer,P.Torrielli, andM.Zaro,toappear .
References [17] The code can be downloaded at:
https://launchpad.net/madgraph5,http://amcatnlo.cern.ch.
[1] F.EnglertandR.Brout,Phys.Rev.Lett.13,321(1964). [18] U. Baur, T. Plehn, and D. L. Rainwater, Phys.Rev.Lett. 89,
[2] P.W.Higgs,Phys.Rev.Lett.13,508(1964). 151801 (2002), arXiv:hep-ph/0206024[hep-ph].
[3] CMS-HIG-13-003. CMS-HIG-13-004. CMS-HIG-13-006. CMS- [19] V.Hirschietal.,JHEP05,044(2011),arXiv:1103.0621[hep-ph]
HIG-13-009. (2013). .
[4] ATLAS-CONF-2013-009. ATLAS-CONF-2013-010. ATLAS- [20] A. Denner, S. Dittmaier, M. Roth, and D. Wackeroth,
CONF-2013-012.ATLAS-CONF-2013-013. (2013). Nucl.Phys.B560,33(1999), arXiv:hep-ph/9904472[hep-ph].
[5] E. Asakawa, D. Harada, S. Kanemura, Y. Okada, and [21] A.Denner,S.Dittmaier,M.Roth, andL.Wieders,Nucl.Phys.
K. Tsumura, Phys.Rev. D82, 115002 (2010), arXiv:1009.4670 B724,247(2005), arXiv:hep-ph/0505042[hep-ph].
[hep-ph]. [22] Q.Li,Q.-S.Yan, andX.Zhao, (2013), arXiv:1312.3830[hep-
[6] S. Dawson, E. Furlan, and I. Lewis, Phys.Rev. D87, 014007 -ph].
(2013), arXiv:1210.6663[hep-ph]. [23] P.MaierhöferandA.Papaefstathiou, (2013), arXiv:1401.0007
[7] R. Contino, M. Ghezzi, M. Moretti, G. Panico, F. Piccinini, [hep-ph].
et al.,JHEP1208,154(2012), arXiv:1205.5444[hep-ph]. [24] J.Grigo,J.Hoff,K.Melnikov, andM.Steinhauser,Nucl.Phys.
[8] G. D. Kribs and A. Martin, Phys.Rev. D86, 095023 (2012), B875,1(2013), arXiv:1305.7340[hep-ph].
arXiv:1207.4496[hep-ph]. [25] D. de Florianand J. Mazzitelli, Phys. Rev. Lett. 111, 201801
[9] M.J.Dolan,C.Englert, andM.Spannowsky,Phys.Rev.D87, (2013), 10.1103/PhysRevLett.111.201801, arXiv:1309.6594
055002(2013), arXiv:1210.8166[hep-ph]. [hep-ph].
[10] M.J.Dolan,C.Englert, andM.Spannowsky,JHEP1210,112 [26] D. Y. Shao, C. S. Li, H. T. Li, and J. Wang, JHEP07, 169
(2012), arXiv:1206.5001[hep-ph]. (2013), arXiv:1301.1245[hep-ph].
6
-2 HH production at the LHC14, NLO+PS pp→HH (EFT loop-improved) PY8
10
pp→HH (EFT loop-improved) HW6
pp→HHjj (VBF) PY8
pp→HHjj (VBF) HW6
pp→ttHH PY8
pp→ttHH HW6
-3
10 pp→WHH PY8
pp→WHH HW6
pp→ZHH PY8
pp→ZHH HW6
b] -4
p 10
n[
bi
σ/
d O
L
N
10-5 @
C
M
a
_
5
h
p
10-6 ra
G
d
a
M
pp→HH (EFT loop-improved) LO+PY8 LO+HW6
2
1
1.4 pp→HHjj (VBF) NLO+PY8 NLO+HW6
1.2
1
0.8
0.6
1.4 pp→ttHH PDF unc sc. unc
1.2
1
0.8
0.6
pp→WHH
1.4
1.2
1
pp→ZHH
1.4
1.2
1
0 100 200 300 400 500 600
p (H ) [GeV]
T 1
Figure 4: Transverse momentum distribution of the hardest Higgs boson in HH production in the gluon-gluon fusion, VBF, tt¯HH, WHH
and ZHH channels, at the 14 TeV LHC. The main frame displays the NLO+PS results obtained after showering with Pythia8 (solid)
and HERWIG6 (dashes). The insets show, channel by channel, the ratios of the NLO+Pythia8 (solid), NLO+HERWIG6 (dashes), and
LO+HERWIG6 (open boxes) results over the LO+Pythia8 results (crosses). The dark-colour (light-colour) bands represent the scale (red)
andPDF(blue)uncertainties addedlinearlyfortheNLO(LO) simulations.
7
-2 HH production at the LHC14, NLO+PS pp→HH (EFT loop-improved) PY8
10
pp→HH (EFT loop-improved) HW6
pp→HHjj (VBF) PY8
pp→HHjj (VBF) HW6
pp→ttHH PY8
pp→ttHH HW6
-3
10 pp→WHH PY8
pp→WHH HW6
pp→ZHH PY8
pp→ZHH HW6
b] -4
p 10
n[
bi
σ/
d O
L
N
10-5 @
C
M
a
_
5
h
p
10-6 ra
G
d
a
M
pp→HH (EFT loop-improved) LO+PY8 LO+HW6
2
1
1.4 pp→HHjj (VBF) NLO+PY8 NLO+HW6
1.2
1
0.8
0.6
1.4 pp→ttHH PDF unc sc. unc
1.2
1
0.8
0.6
pp→WHH
1.4
1.2
1
pp→ZHH
1.4
1.2
1
0 100 200 300 400 500 600
p (H ) [GeV]
T 2
Figure5: Asinfig.4,forthesoftestHiggsbosons.
8
-2 HH production at the LHC14, NLO+PS pp→HH (EFT loop-improved) PY8
10
pp→HH (EFT loop-improved) HW6
pp→HHjj (VBF) PY8
pp→HHjj (VBF) HW6
pp→ttHH PY8
pp→ttHH HW6
-3
10 pp→WHH PY8
pp→WHH HW6
pp→ZHH PY8
pp→ZHH HW6
b] -4
p 10
n[
bi
σ/
d O
L
N
10-5 @
C
M
a
_
5
h
p
10-6 ra
G
d
a
M
pp→HH (EFT loop-improved) LO+PY8 LO+HW6
2
1
1.4 pp→HHjj (VBF) NLO+PY8 NLO+HW6
1.2
1
0.8
0.6
1.4 pp→ttHH PDF unc sc. unc
1.2
1
0.8
0.6
pp→WHH
1.4
1.2
1
pp→ZHH
1.4
1.2
1
0 100 200 300 400 500 600
p (HH) [GeV]
T
Figure6: Asinfig.4,forthetransversemomentum oftheHiggspair.
9
-2 HH production at the LHC14, NLO+PS pp→HH (EFT loop-improved) PY8
10
pp→HH (EFT loop-improved) HW6
pp→HHjj (VBF) PY8
pp→HHjj (VBF) HW6
pp→ttHH PY8
pp→ttHH HW6
-3
10 pp→WHH PY8
pp→WHH HW6
pp→ZHH PY8
pp→ZHH HW6
b] -4
p 10
n[
bi
σ/
d O
L
N
10-5 @
C
M
a
_
5
h
p
10-6 ra
G
d
a
M
pp→HH (EFT loop-improved) LO+PY8 LO+HW6
2
1
1.4 pp→HHjj (VBF) NLO+PY8 NLO+HW6
1.2
1
0.8
0.6
1.4 pp→ttHH PDF unc sc. unc
1.2
1
0.8
0.6
pp→WHH
1.4
1.2
1
pp→ZHH
1.4
1.2
1
0 200 400 600 800 1000 1200 1400 1600 1800
m(HH) [GeV]
Figure7: Asinfig.4,fortheinvariantmassoftheHiggspair.
10
