Precision Calculations for Associated WH and ZH Production at Hadron Colliders

0.17 MB·English

by  O. Brein

PRECISION CALCULATIONS FOR ASSOCIATED WH AND ZH
PRODUCTION AT HADRON COLLIDERS
O.Brein1,M.Ciccolini2,S.Dittmaier3,A.Djouadi4,R.Harlander5 andM.Kra¨mer2
1 Institut fu¨rTheoretische Physik,RWTHAachen, Germany
2 SchoolofPhysics,TheUniversityofEdinburgh, Scotland
3 Max-Planck-Institut fu¨rPhysik(Werner-Heisenberg-Institut), Mu¨nchen,Germany
4 Laboratoire dePhysiqueMathe´matique etThe´orique, Universite´ deMontpellier II,France
5 Institut fu¨rTheoretische Teilchenphysik, Universita¨tKarlsruhe, Germany
4
0
Abstract
0
2 Recently the next-to-next-to-leading order QCD corrections and the electro-
weak (α)correctionstotheHiggs-strahlungprocessespp¯/pp WH/ZH+
n
O →
a X have been calculated. Bothtypes ofcorrections are oftheorder of5–10%.
J
In this article the various corrections are briefly discussed and combined into

1
state-of-the-art predictions for the cross sections. The theoretical uncertain-
3
tiesfromrenormalization/factorization scalesandfromtheparton distribution

1 functions arediscussed.
v
3
0 1. INTRODUCTION
0
2 AttheTevatron,Higgs-bosonproductioninassociationwithW orZ bosons,pp¯ WH/ZH+X,isthe
0 →
mostpromisingdiscoverychannelforaSMHiggsparticlewithamassbelowabout135GeV,wherede-
4
0 caysintob¯bfinalstatesaredominant[1,2]. AttheppcolliderLHCotherHiggs-production mechanisms
/ playtheleadingrole[3],butnevertheless theseHiggs-strahlung processes shouldbeobservable.
h
p Atleading order (LO),theproduction ofaHiggsboson inassociation withavector boson, pp¯
- ′ ∗ →
p VH +X,(V = W,Z) proceeds through qq¯annihilation [4], qq¯ V VH. The next-to-leading
→ →
e order(NLO)QCDcorrectionscoincidewiththosetotheDrell-Yanprocessandincreasethecrosssection
h
byabout 30%[5]. BeyondNLO,theQCDcorrections toVH production differfromthose totheDrell-
:
v
Yan process by contributions where the Higgs boson couples to a heavy fermion loop. The impact of
i
X theseadditionaltermsis,however,expectedtobesmallingeneral[6]. Moreover,forZH productionthe
r one-loop-induced process gg ZH contributes at next-to-next-to-leading order (NNLO). The NNLO
a
→
corrections corresponding totheDrell-Yanmechanism aswellasthegg ZH contribution havebeen
→
calculated in Ref.[7]. These NNLO corrections further increase the cross section by the order of 5–
10%. Mostimportant, asuccessive reduction oftherenormalization and factorization scale dependence
is observed when going from LO to NLO to NNLO. The respective scale uncertainties are about 20%
(10%),7%(5%),and3%(2%)attheTevatron(LHC).Atthislevelofaccuracy, electroweak corrections
become significant andneed tobeincluded tofurther improvethe theoretical prediction. InRef.[8]the
electroweak (α)correctionshavebeencalculated;theyturnouttobenegativeandabout–5%or–10%
dependingonOwhethertheweakcouplingsarederivedfromG orα(M2),respectively. Inthispaperwe
µ Z
summarize and combine the results of the NNLO corrections of Ref.[7] and of the electroweak (α)
O
corrections ofRef.[8].
Thearticleisorganized asfollows. InSects.2.and3.wedescribethesalientfeaturesoftheQCD
andelectroweakcorrections, respectively. Section4.containsexplicitnumericalresultsonthecorrected
WH and ZH production cross sections, including a brief discussion of the theoretical uncertainties
originating fromthepartondistribution functions (PDFs). Ourconclusions aregiveninSect.5.
2. QCDCORRECTIONS
The NNLO corrections, i.e. the contributions at (α2), to the Drell-Yan process pp¯/pp V∗ + X
s
O →
consistofthefollowingsetofradiativecorrections:
1.4 1.6
) )
C n
H1.35 NNLO ro 1.5 NNLO
(LWH1.12.53 Tevat 1.4
K NLO (
H1.3
1.2 W NLO
K
1.15 1.2
LO
1.1
1.1
1.05
LO 1
1
0.9
0.95
0.9 0.8
100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300
M [GeV] M [GeV]
H H
Fig.1: QCDK-factorsforWH production(i.e.fromthesumofW+H andW−H crosssections)attheLHC(l.h.s.) and
theTevatron(r.h.s.). Thebandsrepresentthespreadofthecrosssectionwhentherenormalizationandfactorizationscalesare
variedintherange 1M µ (µ ) 3M ,theotherscalebeingfixedatµ (µ )=M .(TakenfromRef.[7].)
3 VH ≤ R F ≤ VH F R VH
∗
two-loopcorrections toqq¯ V ,whichhavetobemultipliedbytheBornterm,
• →
∗ ∗
one-loop corrections to the processes qg qV and qq¯ gV , which have to be multiplied by
• → →
thetree-levelgq andqq¯terms,
∗
tree-level contributions from qq¯,qq,qg,gg V + 2 partons in all possible ways; the sums of
• →
thesediagramsforagiveninitialandfinalstatehavetobesquaredandadded.
These corrections have been calculated a decade ago in Ref.[9] and have recently been updated [10].
Theyrepresent abasic building block inthe NNLOcorrections toVH production. There are, however,
twoothersourcesof (α2)corrections:
O s
′
irreducible two-loop boxes forqq¯ VH where the Higgsboson couples via heavy-quark loops
• →
totwogluonsthatareattached totheq line,
the gluon–gluon-initiated mechanism gg ZH [11] at one loop; it is mediated by closed quark
• →
loops which induce ggZ and ggZH couplings and contributes only to ZH but not to WH pro-
duction.
InRef.[7]theNNLOcorrectionstoVH productionhavebeencalculatedfromtheresults[10]onDrell-
Yan production and completed by the (recalculated) contribution of gg ZH. The two-loop contri-
→
butions with quark-loop-induced ggZ or ggH couplings are expected to be very small and have been
neglected.
The impact of higher-order (HO) QCD corrections is usually quantified by calculating the K-
factor, which is defined as the ratio between the cross sections for the process at HO (NLO or NNLO),
withthevalueofαsandthePDFsevaluatedalsoatHO,andthecrosssectionatLO,withαsandthePDFs
consistently evaluated alsoatLO:KHO = σHO(pp¯/pp VH +X)/σLO(pp¯/pp VH +X). AK-
→ →
factor forthe LOcross section, KLO, mayalso be defined byevaluating the latter atgiven factorization
andrenormalizationscalesandnormalizingtotheLOcrosssectionsevaluatedatthecentralscale,which,
inourcase,isgivenbyµ = µ = M ,whereM istheinvariant massoftheVH system.
F R VH VH
The K-factors at NLO and NNLO are shown in Fig.1 (solid black lines) for the LHC and the
Tevatron as a function of the Higgs mass M for the process pp¯/pp WH +X; they are practically
H
→
the same for the process pp¯/pp ZH +X when the contribution of the gg ZH component isnot
→ →
included. Inclusion ofthiscontribution addssubstantially totheuncertainty oftheNNLOprediction for
ZH production. Thisisbecausegg ZH appearsat (α2)inLO.
s
→ O
Thescaleshavebeenfixedtoµ = µ = M ,andtheMRSTsetsofPDFsforeachperturbative
F R VH
order(including theNNLOPDFsofRef.[12])areusedinaconsistent manner.
The NLO K-factor is practically constant at the LHC, increasing only from KNLO = 1.27 for
MH = 110GeVtoKNLO = 1.29forMH = 300GeV.TheNNLOcontributions increase theK-factor
byamere1%forthelowM valueandby3.5%forthehighvalue. AttheTevatron,theNLOK-factor
H
issomewhathigherthanattheLHC,enhancing thecrosssectionbetweenKNLO = 1.35forMH = 110
GeVandKNLO = 1.3forMH = 300GeVwithamonotonicdecrease. TheNNLOcorrections increase
theK-factoruniformlybyabout10%. Thus,theseNNLOcorrectionsaremoreimportantattheTevatron
thanattheLHC.
The bands around the K-factors represent the cross section uncertainty due to the variation of
either the renormalization or factorization scale from 1M µ (µ ) 3M , with the other
3 VH ≤ F R ≤ VH
scalefixedatµ (µ ) = M ;thenormalization isprovided bytheproduction crosssection evaluated
R F VH
at scales µ = µ = M . As can be seen, except from the accidental cancellation of the scale
F R VH
dependence oftheLOcrosssection attheLHC,thedecrease ofthescalevariation isstrongwhengoing
from LO to NLOand then to NNLO.ForM = 120 GeV, the uncertainty from the scale choice at the
H
LHCdropsfrom10%atLO,to5%atNLO,andto2%atNNLO.AttheTevatronandforthesameHiggs
boson mass, the scale uncertainty drops from 20% at LO, to 7% at NLO, and to 3% at NNLO. If this
variation ofthecross section withthetwoscales istaken asanindication oftheuncertainties due tothe
not yet calculated higher-order corrections, one concludes that once the NNLO QCD contributions are
included in theprediction, the QCDcorrections to thecross section for thepp¯/pp VH +X process
→
areknownattheratheraccuratelevelof2to3%relativetotheLO.
3. ELECTROWEAKCORRECTIONS
The calculation of the electroweak (α) corrections, which employs established standard techniques,
O
is described in detail in Ref.[8]. The virtual one-loop corrections involve a few hundred diagrams,
including self-energy, vertex, andboxcorrections. Inordertoobtain IR-finitecorrections, real-photonic
bremsstrahlung has to be taken into account. In spite of being IR finite, the (α) corrections involve
O
logarithms of the initial-state quark masses which are due to collinear photon emission. These mass
singularitiesareabsorbedintothePDFsinexactlythesamewayasinQCD,viz.byMSfactorization. As
amatteroffact, thisrequires also theinclusion ofthecorresponding (α)corrections intothe DGLAP
O
evolution ofthese distributions andintotheir fittoexperimental data. Atpresent, thisfull incorporation
of (α) effects in the determination of the quark distributions has not been performed yet. However,
O
an approximate inclusion of the (α) corrections to the DGLAP evolution shows [13] that the impact
O
ofthesecorrections onthequarkdistributions intheMSfactorization schemeiswellbelow1%,atleast
in the x range that is relevant for associated VH production at the Tevatron and the LHC. This is also
supported by a recent analysis of the MRSTcollaboration [14] who took into account the (α) effects
O
totheDGLAPequations.
The size of the (α) corrections depends on the employed input-parameter scheme for the cou-
O
pling α. This coupling can, for instance, be derived from the fine-structure constant α(0), from the
effective running QEDcoupling α(M2) at the Z resonance, or from the Fermi constant G via α =
Z µ Gµ
√2G M2 s2 /π. The corresponding schemes are known as α(0)-, α(M2)-, and G -scheme, respec-
µ W W Z µ
tively. In contrast tothe α(0)-scheme, where the (α) corrections are sensitive to the non-perturbative
regimeof thehadronic vacuum polarization, intheOα(M2)-and G -schemes these effects areabsorbed
Z µ
intothecouplingconstantα. IntheG -schemelargerenormalizationeffectsinducedbytheρ-parameter
µ
areabsorbedinadditionviaα . Thus,theG -schemeispreferableoverthetwootherschemes(atleast
Gµ µ
overtheα(0)-scheme).
Figure 2 shows the relative size of the (α) corrections as a function of the Higgs-boson mass
for pp¯ W+H +X and pp¯ ZH +X atOthe Tevatron. The numerical results have been obtained
→ →
+
pp(cid:22) ! W H + X pp(cid:22) ! ZH + X
p p
s = 1:96TeV s = 1:96TeV
0 0
-10 -10
% %
δ/ δ/
-20 -20
-30 -30
α( 0 ) α( 0 )
Gµ Gµ
α( M2 ) α( M2 )
-40 Z -40 Z
80 100 120 140 160 180 200 80 100 120 140 160 180 200
M / GeV M / GeV
H H
Fig.2: Relativeelectroweak correctionδ asafunctionof M for thetotalcrosssectionof pp¯ W+H +X (l.h.s.) and
H
→
pp¯ ZH+X(r.h.s.)attheTevatroninvariousinput-parameterschemes.(TakenfromRef.[8].)
→
using the CTEQ6L1 [15] parton distribution function, but the dependence of the relative electroweak
correction δ displayed inFig. 2onthe PDFisinsignificant. Results arepresented for thethree different
input-parameter schemes. The corrections in the G - and α(M2)-schemes are significant and reduce
µ Z
thecrosssectionby5–9%andby10–15%, respectively. Thecorrections intheα(0)-scheme differfrom
those in the G -scheme by 2∆r 6% and from those in the α(M2)-scheme by 2∆α(M2) 12%.
The quantitiesµ∆r and ∆α(M2)≈denote, respectively, the radiative cZorrections to muon deZcay≈and the
Z
correctiondescribingtherunningofα(Q2)fromQ = 0toM (seeRef.[8]fordetails). Thefactthatthe
Z
relativecorrectionsintheα(0)-schemearerathersmallresultsfromaccidentalcancellationsbetweenthe
running of the electromagnetic coupling, which leads to a contribution of about 2∆α(M2) +12%,
Z ≈
and other (negative) corrections of non-universal origin. Thus, corrections beyond (α) in the α(0)-
O
schemecannotbeexpected tobesuppressed aswell. Inallschemes, thesizeofthecorrections doesnot
dependstrongly ontheHiggs-boson mass.
FortheLHCthecorrectionsaresimilarinsizetothoseattheTevatronandreducethecrosssection
by5–10%intheG -schemeandby12–17%intheα(M2)-scheme(seeFigs.13and14inRef.[8]).
µ Z
In Ref.[8] the origin of the electroweak corrections was further explored by separating gauge-
invariant building blocks. Itturns outthat fermionic contributions (comprising alldiagrams withclosed
fermion loops) and remaining bosonic corrections partly compensate each other, but the bosonic cor-
rections are dominant. The major part of the corrections is of non-universal origin, i.e. the bulk of the
corrections isnotduetocoupling modifications, photonradiation, orotheruniversaleffects.
Figure 3 shows the K-factor after inclusion of both the NNLO QCD and the (α) electroweak
O
corrections for pp¯/pp WH +X and pp¯/pp ZH +X at the Tevatron and the LHC. The larger
→ →
uncertainty bandfortheZH production processattheLHCisduetothecontribution ofgg HZ.
→
4. CROSS-SECTIONPREDICTIONS
Figure 4 shows the predictions for the cross sections of WH and ZH production at the LHC and the
Tevatron, including the NNLO QCD and electroweak (α) corrections as discussed in the previous
O
sections. AttheLHCtheprocessgg ZH addsabout10%totheZH production crosssection,which
→
isduetothelargegluonflux;attheTevatronthiscontribution isnegligible.
Finally,webrieflysummarizethediscussion[8]oftheuncertaintyinthecross-section predictions
1.35 1.6
) )
C n
LH 1.3 QCD atro 1.5 QCD
(WH1.25 Tev
K ( 1.4
H
W
1.2 K QCD+EW
1.3
QCD+EW
1.15
1.2
1.1
1.1
1.05
1 1
80 100 120 140 160 180 200 80 100 120 140 160 180 200
M [GeV] M [GeV]
H H
1.6 1.6
) )
C n
H o
L 1.5 atr 1.5 QCD
K(ZH QCD Tev
1.4 ( 1.4
H
Z
K QCD+EW
1.3 1.3
QCD+EW
1.2 1.2
1.1 1.1
1 1
80 100 120 140 160 180 200 80 100 120 140 160 180 200
M [GeV] M [GeV]
H H
Fig.3: K-factorsforWH productionandZH productionattheLHC(l.h.s.) andtheTevatron(r.h.s.) afterinclusionofthe
NNLOQCDandelectroweak (α)corrections.TheoreticalerrorsasdescribedinFigure1.
O
10 1
σ(pp→VH)[pb] σ(pp→VH)[pb]
WH √s = 14 TeV √s = 1.96 TeV
WH
1 ZH
ZH
QCD 0.1
QCD+EW
0.1
QCD
gg→ZH
QCD+EW
0.01 0.01
80 100 120 140 160 180 200 80 100 120 140 160 180 200
M [GeV] M [GeV]
H H
Fig.4: Cross-sectionpredictions(intheG -scheme)forWHandZHproductionattheLHC(l.h.s.)andtheTevatron(r.h.s.),
µ
includingNNLOQCDandelectroweak (α)corrections.
O
Table1: Totalcrosssections(infb)attheTevatron(√s = 1.96 TeV)includingNLOQCDandelectroweakcorrectionsin
theG -schemefordifferentsetsofPDFs. Theresultsincludeanestimateoftheuncertaintyduetotheparametrizationofthe
µ
PDFsasobtained withtheCTEQ6[15]andMRST2001[17]eigenvector sets. Therenormalizationandfactorizationscales
havebeensettotheinvariantmassoftheHiggs–vector-bosonpair,µ=µ0 =MVH.(TakenfromRef.[8].)
pp¯ WH +X pp¯ ZH +X
→ →
M /GeV CTEQ6M[15] MRST2001[17] CTEQ6M[15] MRST2001[17]
H
100.00 268.5(1) 11 269.8(1) 5.2 158.9(1) 6.4 159.6(1) 2.0
± ± ± ±
120.00 143.6(1) 6.0 143.7(1) 3.0 88.20(1) 3.6 88.40(1) 1.1
± ± ± ±
140.00 80.92(1) 3.5 80.65(1) 1.8 51.48(1) 2.1 51.51(1) 0.66
± ± ± ±
170.00 36.79(1) 1.7 36.44(1) 0.91 24.72(1) 1.0 24.69(1) 0.33
± ± ± ±
190.00 22.94(1) 1.1 22.62(1) 0.60 15.73(1) 0.68 15.68(1) 0.21
± ± ± ±
Table2: SameasinTable1,butfortheLHC(√s=14TeV)(TakenfromRef.[8].)
pp WH +X pp ZH +X
→ →
M /GeV CTEQ6M[15] MRST2001[17] CTEQ6M[15] MRST2001[17]
H
100.00 2859(1) 96 2910(1) 35 1539(1) 51 1583(1) 19
± ± ± ±
120.00 1633(1) 55 1664(1) 21 895(3) 30 9217(3) 11
± ± ± ±
140.00 989(3) 34 1010(1) 12 551(2) 19 568.1(2) 6.7
± ± ± ±
170.00 508(1) 18 519.3(1) 6.3 290(1) 10 299.4(1) 3.6
± ± ± ±
190.00 347(1) 12 354.7(2) 4.3 197.8(1) 6.9 204.5(1) 2.5
± ± ± ±
duetotheerrorintheparametrization oftheparton densities (seealso [16]). TothisendtheNLOcross
section evaluated using the default CTEQ6 [15] parametrization with the cross section evaluated using
theMRST2001[17]parametrization arecompared. TheresultsarecollectedinTables1and2. Boththe
CTEQandMRSTparametrizations includeparton-distribution-error packageswhichprovideaquantita-
tiveestimateofthecorrespondinguncertaintiesinthecrosssections.1 Usingtheparton-distribution-error
packagesandcomparingtheCTEQandMRST2001parametrizations,wefindthattheuncertaintyinpre-
dicting the WH and ZH production processes at the Tevatron and the LHCdue to the parametrization
ofthepartondensities islessthanapproximately 5%.
5. CONCLUSIONS
AftertheinclusionofQCDcorrections uptoNNLOandoftheelectroweak (α)corrections, thecross-
O
section predictions for WH and ZH production are by now the most precise for Higgs production at
hadroncolliders. Theremaininguncertainties shouldbedominatedbyrenormalization andfactorization
scale dependences and uncertainties in the parton distribution functions, which are of the order of 3%
and5%,respectively. Theseuncertainties maybereducedbyformingtheratiosoftheassociated Higgs-
productioncrosssectionwiththecorrespondingDrell-Yan-likeW-andZ-bosonproductionchannels,i.e.
byinspecting σpp¯/pp→VH+X/σpp¯/pp→V+X,rendering theirmeasurements particularly interesting atthe
Tevatronand/ortheLHC.
1Inaddition,theMRST[18]parametrizationallowstostudytheuncertaintyoftheNLOcrosssectionduetothevariationof
αs.ForassociatedWHandZHhadroproduction,thesensitivityofthetheoreticalpredictiontothevariationofαs(αs(MZ2)=
0.119 0.02)turnsouttobebelow2%.
±
ACKNOWLEDGEMENTS
Wewould like tothank the organizers ofthe LesHouches workshop for their invitation and hospitality.
M. Kra¨mer would like to thank the DESY Theory Group for their hospitality and financial support.
This work has been supported in part by the European Union under contract HPRN-CT-2000-00149.
M. L. Ciccolini is partially supported by ORS Award ORS/2001014035. R. Harlander is supported by
DFG,contract HA2990/2-1.
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