Associated production of a Higgs boson decaying into bottom quarks at the LHC in full NNLO QCD

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Associated

production

of

a

Higgs

boson

decaying

into

bottom

quarks

at

the

LHC

in

full

NNLO

QCD

Giancarlo Ferrera

a

,

∗

,

Gábor Somogyi

b

,

Francesco Tramontano

c

a_{DipartimentodiFisica,UniversitàdiMilanoandINFN,SezionediMilano,I-20133Milan,Italy} b_{MTA-DEParticlePhysicsResearchGroup,H-4010Debrecen,POBox105,Hungary}

c_{DipartimentodiFisica,UniversitàdiNapoliandINFN,SezionediNapoli,I-80126Naples,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received15September2017

Receivedinrevisedform24January2018 Accepted6March2018

Availableonline12March2018 Editor:G.F.Giudice

WeconsidertheproductionofaStandardModelHiggsbosondecayingtobottomquarksinassociation withavectorbosonW±/Z inhadroncollisions.WepresentafullyexclusivecalculationofQCDradiative corrections bothfor the productioncross sectionand forthe Higgs bosondecay rate upto next-to-next-to-leading order (NNLO)accuracy. Ourcalculation alsoincludes the leptonicdecayofthe vector boson withfinite-widtheffects and spincorrelations.We consider typicalkinematicalcuts appliedin the experimental analyses at the Large Hadron Collider (LHC) and we find that the full NNLO QCD correctionssignificantlydecreasetheacceptedcrosssectionandhaveasubstantialimpactontheshape of distributions. We point outthat theseadditional effects are essential to obtainprecise theoretical predictionstobecomparedwiththeLHCdata.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

The discovery of the long sought Higgs boson (H ) [1,2] by the ATLAS and CMS Collaborations at the Large Hadron Collider (LHC) [3,4] paved the way for the experimental investigation of theelectroweak(EW)symmetrybreakingmechanismand,in par-ticular, for the measurements of the Higgs boson couplings to the StandardModel (SM) particles. In thisrespect the increasing amountofpreciseexperimentaldatacollectedattheLHCdemands acorrespondingimprovementoftheoreticalpredictions.

Oneofthe mainproduction mechanismsofthe SMHiggs bo-sonathadroncollidersistheassociatedproductionwithavector boson(V

=

W±

,

Z ).Thisprocess offerstheuniqueopportunityto studyboththeHiggsbosoncouplingtomassivegaugebosonsand tobottom(b)quarksviathedecayH

→

bb.

¯

A direct search for the SM Higgs boson through associated

VH production and H

→

bb decay

¯

has been carried out at the LHC at a centre–of–mass energy of

√

s

=

7

/

8 TeV [5,6] and at

√

s

=

13 TeV [7,8]. The ATLAS and CMS Collaborations observed an excess ofeventsabove the expectedbackground witha mea-sured signal strength relative to the SM expectation of 0

.

90

±

0

.

18

(

stat

.)

+₋0_0..21₁₉

(

syst

.)

[7] and

μ

=

1

.

06+₋₀0_..31₂₉[8] respectively.

High precision theoretical predictions require detailed com-putations of radiative corrections for cross sections and

corre-*

Correspondingauthor.

E-mailaddress:giancarlo .ferrera @mi .infn .it(G. Ferrera).

sponding distributions. The total cross section for associated VH

production is known at next-to-next-to-leading order (NNLO) in QCD [9–12] and next-to-leading order (NLO) in the electroweak theory [13,14].Fullydifferentialcalculationshavebeenperformed in NNLO QCD forthe VH production crosssection, together with theNLO QCDcorrectionsfortheHiggsbosondecayrateinto bot-tom quarks [15–18]. Thefullydifferential H

→

bb decay

¯

ratehas beencomputeduptoNNLOinQCD [19,20] whiletheinclusiverate isknownupto

O(

α

4

S

)

[21] anduptoNLOintheelectroweak

the-ory [22,23]. Resummation andhigher order (beyond NNLO)QCD effects have been investigated inRefs. [24–30] while the combi-nation offixed-orderQCDcalculationswithpartonshowerMonte CarloalgorithmshasbeenconsideredinRefs. [31–33].NLOEW ef-fectsatfullydifferentiallevelforVH productionwiththeleptonic decayofthevectorbosonhavebeenconsideredinRefs. [14,34].

Inthisletterwepresentthefullydifferentialcalculationofthe NNLOQCDcorrectionsfortheproductioncrosssectionand forthe Higgsbosondecayratetobottomquarks,exploitingtheverygood accuracy ofthenarrow widthapproximationforthe Higgsboson (



H



mH). InRefs. [16,35] it was shown that, when the set of

kinematical cutsapplied in the LHC analyses are considered, the effect of QCD corrections to the Higgs boson decay process can be large.Motivatedby thesefindings, weextendexisting calcula-tionsonhigherorderQCDpredictionsbyconsideringthecomplete

second order terms. Together withthe NNLO correctionsfor the productioncross section,we includetheNNLO correctionstothe https://doi.org/10.1016/j.physletb.2018.03.021

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

H

→

bb decay

¯

rateand thecombinationoftheNLO termsforthe productionanddecaystages.

We implemented our computation in the parton level Monte Carlonumerical program

HVNNLO

which allows the user to ap-ply arbitrary kinematical cuts on final-state leptons, b jets and associatedQCD radiation,andto compute thecorresponding dis-tributionsintheformofhistograms.

Themain resultof ourstudyis that fora typical setof kine-maticalcutsappliedintheLHCanalysesweobserveasubstantial decrease of the complete NNLO QCD prediction with respect to lowerordercalculations.ThereforetheinclusionoftheQCDeffects wehavecalculatedcouldberelevanttoimprovetheagreement be-tweentheSM predictionsandthecurrentLHCdata.Inthisletter wediscussthemainingredientsofourcomputation,amore com-prehensivephenomenologicalanalysiswillappearelsewhere.

We considerthe inclusive hard scatteringreaction h1

+

h2

→

V H

+

X

→

l1l2bb

¯

+

X , wherethecollision ofthehadronsh1 and

h2 producestheVH system(V

=

W±

,

Z ) whichsubsequently de-caysintothelepton pairl1l2 (l1l2

≡

l

ν

l inthecaseofW± decay)

andthebottomquark–antiquarkpairbb,

¯

while X denotesthe ac-companyingQCD radiation. We consider a highvalue of the VH

invariantmass (MVH), whichsets thehard-scatteringscale ofthe

process,andwetreatthecollidinghadrons,theleptonsandtheb

quarksinthemasslessapproximation.

ByusingthenarrowwidthapproximationfortheHiggsboson, theperturbativeQCDexpansion ofthefullydifferentialcross sec-tioncanbewritteninthefollowingfactorizedform1:

d

σ

_{h1h2→VH→V bb¯}

=

d

σ

h1h2→VH

×

d



_H→bb¯



H

=



_∞



k=0 d

σ

_h1h2(k) _→VH



×

⎡

⎣



_∞ k=0d



(k) H→bb¯



_∞ k=0



(k) H→bb¯

⎤

⎦ ×

Br

(

H

→

bb

¯

) ,

(1)

where



_H→bb¯ and



H are the Higgs boson partial decay width

tobottomquarks andthetotaldecay widthrespectively,andthe expansionin powersof

α

S is givenby theexponentk. Eq. (1) is

arrangedinaformsuchthatwecanexploitthepreciseprediction ofthe Higgs boson branching ratio into b quarks Br

(H

→

bb)

¯

=



_H→bb¯

/

H (seeforinstanceRef. [36]),bywhichwenormalizethe

contributionstothedifferentialdecayrateoftheHiggsboson.2 ByexpandingEq. (1) uptothesecondorderin

α

S wehave:

d

σ

NNLO h1h2→VH→V bb¯

=

⎡

⎣

d

σ

_h(0) 1h2→VH (2)

×

d



(0) H→bb¯

+

d



(1) H→bb¯

+

d



(2) H→bb¯



(0) H→bb¯

+ 

(1) H→bb¯

+ 

(2) H→bb¯

+

d

σ

_h1h2(1) _→VH

×

d



(0) H→bb¯

+

d



(1) H→bb¯



(0) H→bb¯

+ 

(1) H→bb¯

1 _{InordertosimplifythenotationtheleptonicdecayV→l1l2ofthe V boson}

(includingspincorrelations)hasbeenunderstood,sinceithasnoeffectfromthe pointofviewofQCDcorrections.

2 _{Indeed,byconsideringobservablesthatareinclusiveovertheHiggsboson}

de-cayproducts,weobtain the production crosssection timesthe branching ratio

dσh1h2→VH×Br(H→bb¯).

+

d

σ

_h1h(2) 2→VH

×

d



(0) H→bb¯



(0) H→bb¯

⎤

⎦ ×

Br

(

H

→

bb

¯

) .

Eq. (2) contains the complete NNLO contributions, which include the

α

2

S corrections:(i) tothedecayrate(includedinthefirstterm

ontherighthandside),(ii) fromthecombinationoftheNLO con-tributions forproductionanddecay(included inthesecond term on the righthand side), and(iii) to the productioncross section (the third termon the right handside). The novel resultof this letter compared with previous approximations concerns the full computationoftheterms(i) and(ii).

In order to compare with the partial NNLO calculations con-sideredsofar [16–18],we alsoconsiderthe truncationofEq. (1) definedas:

d

σ

NNLO(prod)+NLO(dec)

h1h2→VH→V bb¯

=

⎡

⎣

d

σ

_h1h2(0) _→VH

×

d



(0) H→bb¯

+

d



(1) H→bb¯



(0) H→bb¯

+ 

(1) H→bb¯ (3)

+

d

σ

_h1h2(1) _→VH

+

d

σ

_h1h2(2)_→VH



×

d



(0) H→bb¯



(0) H→bb¯

⎤

⎦ ×

Br

(

H

→

bb

¯

) ,

whichcontainstheNNLOcorrectionsfortheproductioncross sec-tiontogetherwiththeNLOcorrectionsforthe H

→

bb decay

¯

rate. Each term in the Eqs. (2) and (3) includes all the relevant contributions from (double-)real, real-virtual and (double-)virtual

corrections.InourimplementationwehaveemployedtheqT

sub-traction method forthe VH production cross section [15,17] and theCoLoRFulNNLO methodforthe H

→

bb decay

¯

rate [20].Details oftheseformalismscanbefound inRefs. [37,38] and[39–41] re-spectively.

While the full NLO and partof the NNLO QCD correctionsto

VH productionare thesame asthose of theDrell–Yan (DY) pro-cess [45],withtheHiggsbosonradiatedbytheV boson,additional contributionsappearatNNLO,withtheHiggsbosoncoupledtoa heavy-quark loop. In the case of ZH production at the LHC, the impactofthegluon–gluon initiatedsubprocessinvolvinga heavy-quark loop is substantial, due to the large gluon luminosity.We havetakenintoaccountthesecorrectionswiththefulldependence onthetopandbottomheavy-quarkmasses [17].AtNNLOthereis yet another set of non DY like contributions involving quark in-ducedheavy-quarkloops both forZH and WH production. These correctionshavebeencomputedinRef. [11],relyinginsomecases onthe large-mt approximation,andhavebeen showntohavean

impact on the VH cross section at the LHC at the 1% level (for

mH

∼

125 GeV). However since the validity of the large-mt

ap-proximationischallengedinthehighinvariantmassregionprobed bytheVH kinematics,weconsideredinourcomputationonlythe termswhichcan bepresently calculatedretainingthefullmt

de-pendence.In particularweincludedthe NNLOtermsobtainedby radiatingtheHiggsbosonoffatop-quarkbubble-insertionintoan externalgluon line.Theseterms,calledRI inRef. [11],contribute

bothtoZH andWH production.Ontheotherhandthe RI I terms

ofRef. [11],whicharepresentonlyforZH production,havebeen showntocontributeatthesub-per-millelevelandhavebeenthus neglectedinthispaper.

Weareinterestedintheidentificationoftheb-quarkjetswhich originatefromtheHiggsbosondecay.Besidestheb-quarkpair di-rectlyproducedintheHiggsbosondecay,weconsistentlyinclude the effect of b-quark emissions from initial and final state

par-Table 1

Crosssectionsandtheirscaleuncertaintiesforpp→V H+X→l1l2bb¯+X atLHC with√s=13 TeV.Theappliedkinematicalcutsaredescribedinthetext.

σ(fb) NNLO(prod)+NLO(dec) full NNLO

pp→W+H+X→lνlbb¯+X 3.94+₋1%1.5% 3.70+ 1.5% −1.5% pp→Z H+X→ννbb¯+X 8.65+₋43..5%5% 8.24+ 4.5% −3.5%

tons.3 _{However thestandardjetclustering algorithms [42] donot} providean infraredandcollinear safedefinition offlavoured jets withmassless quarks.In the presentcase, atNNLO, thesplitting ofagluoninasoftorcollinear(massless)bb pair

¯

mayaffectthe

flavour of a jet. While the collinearunsafety can be removed by definingasa“b-jet”ajetcontaining anumberofb quarks differ-ent fromthe number ofb quarks,

¯

the definitionof infrared safe

b-jetsusingstandardjetclusteringalgorithms islesstrivial. In or-dertodealwithaninfraredandcollinearsafeb-jetdefinition,we considerthesocalledflavourkT algorithm [43].Accordingtothis

algorithm,the definitionof thekT-distancemeasure inthe

pres-enceofflavouredpartons(particles)ismodifiedinsuchawaythat theflavourofajetisinsensitivetosoftpartonemissions.

We now presentnumerical results for pp collisions ata cen-tre–of–massenergyof

√

s

=

13 TeV.Fortheelectroweakcouplings, weusetheGμ schemeandthefollowinginputparameters:GF

=

1

.

1663787

×

10−5 GeV−2,mZ

=

91

.

1876 GeV,mW

=

80

.

385 GeV,



Z

=

2

.

4952 GeV,



W

=

2

.

085 GeV, mt

=

172 GeV and mb

=

4

.

18 GeV.4 _{The mass andthe width of the SM Higgs boson are} settomH

=

125 GeVand



H

=

4

.

070 MeVrespectively,whilethe H

→

bb branching

¯

ratioissettoBr

(H

→

bb)

¯

=

0

.

578 [36].

As for the parton distribution functions (PDFs), we use the NNLO PDF4LHC set [44] with

α

S

(m

Z

)

=

0

.

118.We set the

renor-malizationand factorization scales to the dynamical value

μ

R

=

μ

F

=

MVH(i.e.theinvariantmassoftheVH system)andthe

renor-malizationscalefortheH

→

bb coupling

¯

tothevalue

μ

r

=

mH.To

assessthe impact ofscale variation,we fix

μ

r

=

mH varying

μ

R

and

μ

F independently in the range MV H

/

2

≤ {

μ

R

,

μ

F

}

≤

2MV H,

with the constraint 1

/

2

≤

μ

R

/

μ

F

≤

2. We then fix

μ

R

=

μ

F

=

MV H andvarythedecayrenormalisationscale

μ

r betweenmH

/

2

and2mH.Thefinaluncertaintyisobtainedbytakingtheenvelope

ofthetwo(productionanddecay)scaleuncertainties. Jetsare re-constructed withtheflavour-kT algorithm with R

=

0

.

5 [43]. We

defineab-jetasajetwhichcontainsanumberofb quarks differ-entfromthenumberofanti-b quarks(N

(b)

=

N(¯b)).

We start the presentation of our results by considering W+H

productionand decayat the LHC at

√

s

=

13 TeV. Our choice of kinematical selection cuts on the final states closely follows the fiducialsetupconsideredintheCERNYellowReport oftheLHCHiggs CrossSectionWorkingGroup [46]. We require the charged lepton to have transverse momentum pl_T

>

15 GeV and pseudorapidity

|

η

l

|

<

2

.

5 whilethe missingtransverseenergyoftheeventis

re-quiredto be Emiss

T

>

30 GeV. The W boson isrequiredto havea

transversemomentum pW_T

>

150 GeV.Finally we requireatleast twob-jetseachwithpb_T

>

25 GeVand

|

η

b

|

<

2

.

5.The

correspond-ing cross sections in the fiducialregion are reportedin the first rowof Table 1, where we present the full NNLO prediction (see Eq. (2))comparedwiththepartial NNLOprediction(seeEq. (3)).5 Weobservethattheinclusionofthefull NNLOcorrectionsreduces

3 _{Therefore,withinourNNLOcalculation,wehaveuptofourb quarksinthefinal}

state.

4 _{Weconsiderthepolemassforthetopquark(m}

t)andtheM S schemeforthe

bottomquarkmassmb=mb(mb).

5 _{TheresultsforthecaseofW}−_{H productionanddecayarequalitativesimilar,} withanumericalreductionoffiducialcrosssectionaround40%.

the crosssection byaround 6% withrespect tothe partial NNLO

result.6

We next consider differential distributions. In Fig. 1 (left) we presentthetransverse-momentum distribution pbb

T oftheleading b-jetpair(i.e. thetwob-jets withlargest pT). Inthelowerpanel

weshowtheratioofthetwotheoreticalpredictionsdefinedabove. We observethat theadditional

α

2

S correctionsincludedinthe full NNLO prediction havean important effectalso on the shape

ofthe pbb_T distribution.Inparticularthecrosssectionisincreased by around2–5%for pbb_T



140 GeVanditisdecreasedby around 6–8% for pbb_T



140 GeV. The corresponding K -factor, definedas theratiobetweenthefull NNLOpredictioninEq. (2) andthe par-tial NNLO prediction in Eq. (3), is thus remarkably not constant (see thelower panel ofFig.1(left)). The qualitativebehaviour of these effects is not unexpected. The additional QCD radiation in theHiggsbosondecay,whichisincludedinthefullNNLO calcula-tion,hastheeffectofdecreasingthetransverse-momentumofthe leadingb-jetpair,makingthepbb

T distributionsofter.

In Fig. 1(right)we presentthe invariant massdistribution of theleadingb-jetpair,Mbb.Weconsideragainthecomparison

be-tween the full NNLO QCD prediction in Eq. (2) and the partial

NNLO prediction in Eq. (3) and we show the ratio of the two predictions in the lower panel. For this observable the effect of the NNLO corrections to the decay rateare even more substan-tial. While the position of the peak is rather stable around the value of theHiggs bosonmass Mbb



mH,the spectrum receives

large positive corrections (up to

+

60%) for Mbb

<

mH and

size-able negative corrections (from

−

30% to

−

10%) for Mbb



mH.

The large impact ofthese correctionscan be understood by not-ing that the leading order (LO) computation would produce an invariant mass distribution which exactly fulfils the constraint

Mbb

=

mH.Higher-order correctionstothedecaydecreasethe

in-variant mass of the leading b-jet pair. In the Mbb

<

mH region

the partial NNLOprediction(which contains justtheNLO correc-tion to the decay rate)is effectively a first-ordercalculation and thenext-ordertermiscontainedonlyinthefull NNLOcorrection. Conversely, higher-order correctionsto the production cross sec-tiontypicallyincrease theinvariantmassoftheleading b-jetpair andtheregion Mbb

>

mH receivescontributionsonlyfrompartons

emitted fromthe initialstate. Inthiscasethe effectofthe addi-tional

α

2

S corrections contained in the full NNLO calculation has

a sizeable butmoderateimpactwithrespect tothe partial NNLO

calculation.

As fortheperturbativescale variationwe havefoundthat the scale dependence isdominated by the effectof the renormaliza-tionscaleofthedecayprocess

μ

randisparticularlysmall:atthe

1% level forthe fiducial cross section. The scale variation of the “full” NNLO resultis around

±

5% in the caseof pbb

T distribution

andaround

±

10% inthecaseofMbb distribution.Thescale

depen-dence ofthe “partial” NNLOresult isquantitatively similar being significantly larger (around

±

17%)only in the region Mbb

<

mH,

wherethe“partial”NNLOresultisafirst-ordercalculation. We observe that the uncertainty bands for the “partial” and “full”NNLOresultsfailtooverlapforthefiducialcrosssectionand invariousregionsofdifferentialdistributions.

We next turn to the caseof ZH production and decayat the LHCat

√

s

=

13 TeV. Weconsidertheinvisible Z decayinto neu-trinos ( Z

→

ν

ν)

¯

andwe requireto haveatleast twob-jets each with pb

T

>

25 GeV and

|

η

b

|

<

2

.

5 andamissingtransverseenergy Emiss_T

>

150 GeV.The corresponding crosssectionsin thefiducial

6 _{Inparticularwenotethatroughly40% ofthereductionisduetothe}

combina-tionoftheNLOcontributionsforproductionanddecayand60% isduetotheNNLO contributionstothedecayrate(seeEq. (2) andsubsequentcomments).

Fig. 1. pp→W+H+X→lνlbb¯+X atLHCwith√s=13 TeV.Transverse-momentumdistribution(leftpanel)andinvariantmassdistribution(rightpanel)oftheleading

b-jetpaircomputedatfullNNLO(red)andpartialNNLO(blue).Thelowerpanelsshowtheratiosoftheresults.Theappliedcutsaredescribedinthetext.(Forinterpretation ofthecoloursinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

Fig. 2. pp→Z H+X→ννbb¯+X atLHCwith√s=13 TeV.Transverse-momentumdistribution(leftpanel)andinvariantmassdistribution(rightpanel)oftheleadingb-jet

paircomputedatfullNNLO(red)andpartialNNLO(blue).Thelowerpanelsshowtheratiosoftheresults.Theappliedcutsaredescribedinthetext. regionarereportedinthesecondrowofTable1.Weobservethat

theinclusionofthefull NNLOcorrectionsreducesthecrosssection byaround5% withrespecttothepartial NNLOresult.

InFig. 2(left) we presentthetransverse-momentum distribu-tion of the leading b-jet pair, pbb_T . As in the previous case we compare the full NNLO QCD prediction (Eq. (2)) withthe partial

NNLOprediction(Eq. (3))andinthelowerpanelweshowthe ra-tioofthetwopredictions.

In thiscase the inclusion ofthe NNLO correctionsto the de-cay rate decreases the cross section up to about 10% below the peakandaround5%abovethepeak.ThecorrespondingK -factoris showninthelowerpanelofFig.2(left).

Finally in Fig. 2 (right) we consider, for the ZH case, the in-variant mass distribution of the leading b-jet pair, Mbb. The

ef-fect of the full NNLO corrections is similar to the W+H case.

Thespectrumreceives largepositive corrections(upto

+

70%)for

Mbb

<

mH andsizeablenegativecorrections(from

−

30% to

−

10%)

for125



Mbb



150 GeV.

Wefinallyobservethatwhenakinematicalboundaryispresent ata givenorder in perturbation theory,higher order corrections areaffected byinstabilitiesof Sudakovtype [47], whichspoilthe reliabilityofthefixed-orderexpansion aroundtheboundary. This isthecaseforboth the pbb

T distribution (pVT

>

150 GeVLO

kine-maticalboundary) andthe Mbb distribution(LO condition Mbb

=

mH). While a proper treatment of this misbehaviour requires an

allorderresummationofperturbativelyenhancedterms,theeffect oftheseinstabilitiescanbemitigatedbyincreasingthebinsizeof thedistributionaroundthecriticalpoint.

Inthe caseof Z H production, dueto thesubstantial effectof thegluon–gluoninitiatedsubprocessinvolvingaheavy-quarkloop, scale uncertainty is dominated by the effect of the renormaliza-tionscale

μ

R andtheensuingscalevariationbandturnsouttobe

larger(atthe4% level).

Thescale variationof the“full”NNLO resultisaround

±

3–5% inthecaseofpbb

T distribution andaround

±

10% inthecaseofMbb

distribution.AsintheW+H case,thescaledependenceofthe “par-tial”NNLOresultissignificantlylarger(around

±

17%)onlyinthe

Mbb

<

mH region andtheuncertaintybandsforthe“partial” and

“full”NNLOresultsfailtooverlapinvariousregionsofdifferential distributions.

AsalreadypointedoutinRef. [17],weareinterestedina spe-cific “boosted” kinematical regime where the size of the NNLO corrections tends to be underestimated by the customary NLO scale uncertainty band. Therefore theNLO scale variation cannot beregardedasareliableapproximationofthe“true”perturbative uncertaintyanditcastssomedoubtsalsoonthereliabilityofthe NNLOscalevariationband.

Ahinton thereliability ofthecustomary scaleuncertainty at NNLO can be obtained considering missing higher-order contri-butions that canbe calculated through asuitable combinationof individualpartsofourcomputation.Wehavethereforecalculated the

O(

α

3

S

)

contributions proportional to (i) d

σ

(1)

h1h2→VH

×

d (2) H→bb¯

and(ii) d

σ

_h1h2(2) _→VH

×

d(1)

H→bb¯ inEq.1.Wehavefoundthatthe

nu-mericalimpact tothefiducialcross-section oftheN3_LOterms(i) and(ii) aboveisrespectivelyaround

−

0

.

4% and

+

0

.

4% (

−

0

.

3% and

+

1

.

5%) for W H ( Z H )production. The fact that theseeffects are coveredby the scalevariation inTable 1suggeststhat theNNLO scaledependence(contraryto theNLO case)couldbe considered as a trustable estimate of the “true” perturbative uncertainty of thecalculation.However amoreconservativeestimate ofthe un-certaintycan beobtainedby comparingtheNNLO resulttowhat isobtainedatthepreviousorder.

We briefly comment on expected PDF uncertainties and on electroweak effects. PDF uncertainty has been calculated, within a similar setup, in Ref. [46] and has been shown to be at the

±

1

.

5% level forfiducial cross sections.The NLO EW effectshave beencalculatedfor pp

→

V H

+

X

→

l1l2H

+

X only(i.e.without theinclusion ofEWeffects for H

→

bb decay) [14

¯

,34] andit has beenshowntobesignificant(

∼ −

10%)[46].

Inconclusion,wehavepresentedafullydifferentialQCD com-putation for the associated production of a vector boson and a Standard Model Higgs boson in hadron collisions including the QCDradiative correctionsup tonext-to-next-toleading order (NNLO)bothfortheVH productioncrosssectionandforthe differ-entialHiggsbosondecaywidthintobottomquarks.Ourcalculation also includes the leptonic decay of thevector boson with finite-width effects andspin correlationsand it isimplemented in the partonlevelMonteCarlonumericalcode

HVNNLO

.

We havestudiedthe impactofthefull NNLO QCDcorrections to the VH production anddecay at the LHC by focusing on the mostrelevantdistributions,namelythetransversemomentumand invariantmassoftheHiggsbosoncandidate.Wehavestudiedthe renormalizationandfactorizationscale dependenceofthe results in order to estimate the perturbative uncertainty of our predic-tions.Wehavefoundthattheadditionalsecond-ordercorrections included inthe presentcalculationhave asubstantial effectboth forWH andZH production.Thereforetheinclusionoftheseeffects turns out to be essential in order to obtain a precise theoretical predictionforassociatedVH productionanddecayattheLHC.

Acknowledgements

We gratefully acknowledge Massimiliano Grazzini for discus-sions and comments on themanuscript, Raoul Röntsch for com-parisonwiththeresultsofRef. [48],andthecomputingresources provided bytheSCOPE FacilityattheUniversityofNapoliandby the TheoryFarmattheUniversity ofMilanoandINFN Sezionedi Milano.ThisresearchwassupportedinpartbyFondazioneCariplo underthegrantnumber2015-0761,bytheItalianMinistryof Ed-ucation and Research(MIUR) underproject number2015P5SBHT andbygrantK125105oftheNationalResearch,Developmentand InnovationFundinHungary.

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