Table Of Content¯ ¯
NLO QCD corrections to Wbb and Zbb production
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J FernandoFebresCordero
5 UniversityofCalifornia,LosAngeles
1
E-mail: [email protected]
]
h LauraReina∗
p FloridaStateUniversity
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p E-mail: [email protected]
e
h DoreenWackeroth
[ UniversityatBuffalo-TheStateUniversityofNewYork
1 E-mail: [email protected]
v
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7 WepresentNLOQCDresultsforWbb¯ andZbb¯ productionattheTevatronincludingfullbottom-
3
quark mass effects. We study the impact of QCD corrections on both total cross-section and
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. invariant mass distribution of the bottom-quark pair. Including NLO QCD corrections greatly
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0 reduces the dependence of the tree-level cross-section on the renormalizationand factorization
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scales. We also compareourcalculationto a calculationthatconsidersmassless bottomquarks
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: andfindthatthebottom-quarkmasseffectsamounttoabout8-10%ofthetotalNLOQCDcross-
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i sectionandcanimpacttheshapeofthebottom-quarkpairinvariantmassdistribution,inparticular
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inthelowinvariantmassregion.
r
a
8thInternationalSymposiumonRadiativeCorrections
October1-5,2007
Florence,Italy
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina
1. Introduction
Precisetheoreticalpredictions fortheproduction ofaweakgaugeboson(W/Z)inassociation
with a bb¯ pair are very important both in the search for a light Standard Model (SM)-like Higgs
boson andinthesearch forsingle-top production athadron colliders, inparticular attheTevatron.
Indeed, W/Z+bb¯ represent the major irreducible backgrounds to the associated production of a
lightHiggsbosonwithaweakgaugeboson(WH andZH withH→bb¯),whicharethemainsearch
modes for a SM-like Higgs boson at the Tevatron [1, 2]. At the same time,Wbb¯ is an irreducible
background forsingle-top production, whichisbeingmeasuredforthefirsttimeattheTevatronas
pp¯→tb¯,t¯bwitht(t¯)→b(b¯)W [3,4].
The signal cross sections for both WH/ZH and single-top production are known including
higher order QCD and electroweak corrections. It is therefore important to also control the back-
ground to a good level of theoretical precision. In the present experimental analyses the effects
ofNext-to-Leading (NLO)QCDcorrections onthetotalcross-section andthedijetinvariant mass
distribution of the Wbb¯ and Zbb¯ background processes have been taken into account by using
the MCFM package [5], which implements the zero bottom-quark mass (m = 0) approxima-
b
tion[6,7,8].
In this proceedings we present results for the NLO QCDcross-sections and bb¯-pair invariant
mass distributions at the Tevatron, including full bottom-quark mass effects [9]. We separately
studytheimpactofNLOQCDcorrections andofnon-zerobottom-quark masseffects. NLOQCD
corrections stabilize the theoretical prediction of total cross-sections and distributions, reducing
the dependence on the renormalization and factorization scales. On the other hand, the presence
of a non-zero bottom-quark mass mainly affect the cross-section in the region where the bb¯-pair
invariant massissmall,bothatLeadingOrder(LO)andatNLOinQCD,andamounttoanoverall
8-10% differencewithrespecttothezerobottom-quark massapproximation.
2. NLO calculation
The hadronic production of a W boson with a bb¯ pair occurs at tree level in QCD via the
qq¯′ →Wbb¯ partonic process. On the other hand, the hadronic production of a Z boson with a bb¯
pairconsists, atthetreelevelinQCD,oftwopartonicchannels, namelyqq¯→Zbb¯ andgg→Zbb¯.
TheNLOQCDcorrections tothetreelevelpartoniccross-section consistsofbothone-loop O(a )
s
virtual corrections and O(a ) real corrections corresponding to the emission of one extra parton
s
from the tree level parton processes. The NLO hadronic cross-section is obtained by convoluting
theparton-level NLOcross-sections withNLOPartonDistribution Functions(PDF).
The O(a ) virtual corrections to the partonic cross-section contain ultraviolet (UV) and in-
s
frared (IR)singularities. TheUVsingularities arecalculated usingdimensional regularization and
cancelledbyintroducingasuitablesetofcounterterms(seeRef.[9]fordetails). IRsingularitiesare
isolated using dimensional regularization andcancelled against theanalogous singularities arising
intheO(a )realcorrections tothepartonic cross-section.
s
The O(a ) virtual corrections to the partonic cross-section consist of one-loop self-energy,
s
vertex, box andpentagon diagrams. Weapply techniques developed intheNLOQCDcalculation
of Htt¯[10, 11] for the calculation of scalar and tensor loop-integrals. In particular, we calculate
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NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina
the tensor loop-integrals via Passarino-Veltman reduction (PV) [12]. We encounter instabilities
duetothequasi-vanishing oftheGramdeterminant(s) oftheprocess onlyinoneboxdiagram and
in several pentagon diagrams. We are able to obtain stable numerical results by combining, when
necessary, setsofgaugeinvariant diagrams.
The O(a ) real corrections to the partonic cross-section consist of the qq¯′ →Wbb¯+g and
s
qg(q¯g)→Wbb¯+q(q¯) subprocesses for Wbb¯ production and of the qq¯ →Zbb¯+g, gg→Zbb¯+
g, and qg(q¯g)→Zbb¯+q(q¯) subprocesses for Zbb¯ production. We have extracted both soft and
collinear IR singularities by implementing a Phase Space Slicing method with two cutoffs [13]
in order to isolate the soft (d ) and collinear (d ) singularities respectively. Both the soft, hard-
s c
collinear, and hard-non-collinear parts of the cross-section depend on the cutoffs, but their sum is
cutoffindependent overalargerangeofvaluesofthecutoffs(seeRef.[9]).
Both analytical and numerical results for the NLOhadronic cross-section have been checked
withtwoindependent calculations basedondifferent programming languages andpublic/in-house
packages. The analytical reduction of the calculation has been obtained using FORM [14] and
Maple codes, while the numerical results have been obtained using Fortran and C codes. The FF
package [15]has been used to check some ofthe IR-finite scalar and tensor integrals. Finally, the
hardnon-collinear realcorrections havebeendouble-checked usingMadgraph [16,17,18].
3. Numerical results
Intheseproceedings wepresentresultsforWbb¯ andZbb¯ production attheTevatronincluding
NLO QCD corrections and a non-zero bottom-quark mass fixed at m =4.62 GeV. Results for
b
Wbb¯ have been published inRef. [9], while results forZbb¯ are currently being cross-checked and
shouldbeconsideredaspreliminary. BothW andZ bosonareconsideredon-shellandtheirmasses
are taken to be M = 80.41 GeV for the Wbb¯ runs and M = 91.1876 GeV for the Zbb¯ runs,
W Z
while,ineachcase,theotherweakgaugebosonmassiscalculatedviatherelationM =cosq M
W w Z
with sin2q = 0.223. The LO results use the 1-loop evolution of a and the CTEQ6L set of
w s
PDF[19],whiletheNLOresultsusethe2-loopevolutionofa andtheCTEQ6MsetofPDF,with
s
a NLO(M )=0.118. TheW boson coupling to quarks is proportional to the Cabibbo-Kobayashi-
s Z
Maskawa (CKM) matrix elements. We takeV =V =0.975 andV =V =0.222, while we
ud cs us cd
neglectthecontributionofthethirdgeneration,sinceitissuppressedeitherbytheinitialstatequark
densities orbythecorresponding CKMmatrixelements.
We implement the k jet algorithm [20, 21, 22, 23] with a pseudo-cone size R = 0.7 and
T
we recombine the parton momenta within a jet using the so called covariant E-scheme [21]. We
checked that our implementation of the k jet algorithm coincides with the one in MCFM. We
T
require all events to have a bb¯ jet pair in the final state, with a transverse momentum larger than
15 GeV (pb,b¯ >15 GeV) and a pseudorapidity that satisfies |h b,b¯|<2. We impose the same p
T T
and|h |cutsalsoontheextrajetthatmayariseduetohardnon-collinear realemissionofaparton,
i.e. in the processesW/Zbb¯+gorW/Zbb¯+q(q¯). This hard non-collinear extra parton is treated
eitherinclusivelyorexclusively, followingthedefinitionofinclusiveandexclusiveasimplemented
in the MCFM code [5]. In the inclusive case we include both two- and three-jet events, while in
the exclusive case we require exactly two jets in the event. Two-jet events consist of a bottom-
quark jet pair that may also include a final-state light parton (gluon or quark) due to the applied
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NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina
recombinationprocedure. Resultsinthemasslessbottom-quarkapproximationhavebeenobtained
usingtheMCFMcode[5].
5 5
LO LO
NLO inclusive NLO inclusive
NLO exclusive NLO exclusive
4 4
(pb)total3 (pb)total3
s s
2 2
cuts: p > 15 GeV
|h|t < 2 c u t s : p|ht |> < 1 25 GeV
1 R = 0.7 m 0 = Mw/2 + mb 1 R = 0.7 m0 = MZ/2 + mb
0.5 1 2 4 0.5 1 2 4
m /m m /m
f 0 f 0
Figure1: DependenceoftheLO(blacksolidband),NLOinclusive(bluedashedband),andNLOexclusive
(reddottedband)totalcross-sectionsontherenormalization/factorizationscales,includingfullbottom-quark
masseffects. Thel.h.s. plotisfor pp¯→Wbb¯ andther.h.s. plotfor pp¯→Zbb¯ . Thebandsareobtainedby
varyingboth m and m independentlybetween m /2and 4m (with m =m +M /2forV =W,Z in the
R F 0 0 0 b V
pp¯→Wbb¯ and pp¯→Zbb¯ casesrespectively).
In Fig. 1 we illustrate the renormalization and factorization scale dependence of the LO and
NLO total cross-sections, both in the inclusive and exclusive case. The bands are obtained by
varyingbothm andm independently betweenm /2and4m (withm =m +M /2forV =W,Z
R F 0 0 0 b V
in the pp¯ →Wbb¯ and pp¯ → Zbb¯ cases respectively), including full bottom-quark mass effects.
Wenotice thattheNLOcross-sections haveareduced scale dependence overmostoftherangeof
scalesshown,andtheexclusive NLOcross-section ismorestablethantheinclusive oneespecially
atlowscales. WhiletheLOcross-section stillhasa40%uncertainty duetoscaledependence, this
uncertainty is reduced at NLO to about 20% for the inclusive and to about 10% for the exclusive
cross-section respectively. This is consistent with the fact that the inclusive NLO cross-section
integrates over the entire phase space of the qg(q¯g)→bb¯W/Z+q(q¯) channels that are evaluated
withNLOa andNLOPDF,butareactuallytree-levelprocessesandretainthereforeastrongscale
s
dependence. Intheexclusivecaseonlythe2→3collinearkinematicoftheseprocessesisretained,
since3-jetseventsarediscarded,andthismakestheoverallrenormalizationandfactorizationscale
dependence milder. This is better illustrated in Fig. 2 for the case of pp¯→Zbb¯, where the r.h.s.
plots show the scale dependence of the cross-sections due to the single partonic channels (qq¯, gg
and qg+q¯g). The strong residual scale dependence of the inclusive NLO cross-section is clearly
drivenbytheqg+q¯gchannel. Similarresultsareobtainedfor pp¯→Wbb¯,asreportedinRef.[9].
A first illustration of the impact of keeping a non-zero bottom-quark mass in the calculation
of the NLO QCD cross-section is given in the l.h.s. plots of Fig. 2, where LO and NLO total
cross-sections for pp¯ →Zbb¯ are given, both for m =0 and m =4.62 GeV, as functions of the
b b
renormalizationandfactorizationscales(identifiedforthepurposeofthisplot). Neglectingbottom-
quark mass effects overestimate the NLOcross-section by about 8-10%, depending on the choice
of the scale. In Fig. 3 we analyze the impact of a non-zero bottom-quark mass on the bb¯-pair
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NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina
NLO massless NLO massive
5 5 _
NLO massive qq initiated
4.5 LO massless gg initiated
LO massive 4
qg initiated
4
b) b)
(ptotal3.53 (ptotal23
s cuts: p > 15 GeV s Inclusive case
2.5 t
|h | < 2 1
2 m = M/2 + m
0 Z b
1.5 R = 0.7 0
0.5 1 2 4 0.5 1 2 4
m /m m /m
0 0
4.5 NLO massless NLO massive
NLO massive 4 qq initiated
4 LO massless gg initiated
LO massive
3.5 3 qg initiated
b) b)
p p
(total 3 (total2
s 2.5 s Exclusive case
cuts: p > 15 GeV
t 1
2 |h | < 2
m = M/2 + m
0 Z b
R = 0.7 0
1.5
0.5 1 2 4 0.5 1 2 4
m /m m /m
0 0
Figure2: Dependenceofthe LO andNLO inclusive(upperplots) andexclusive (lowerplots)totalcross-
section for pp¯→Zbb¯ onthe renormalization/factorizationscale, when m =m . The l.h.s. plotscompare
R F
bothLOandNLOtotalcross-sectionsforthecaseinwhichthebottomquarkistreatedasmassless(MCFM)
ormassive(ourcalculation).Ther.h.s.plotsshowseparately,forthemassivecaseonly,thescaledependence
oftheqq¯,ggandqg+q¯gcontributions,aswellastheirsum.
invariant-mass (m ) distribution. We give results for both inclusive and exclusive distribution in
bb¯
the pp¯→Wbb¯ case. Inbothcasesdistributions mostoftheimpactisinthelowm region,which
bb¯
canbeimportantinavarietyofdifferent analyses.
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NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina
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