NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC

0.11 MB·English

by  Axel Bredenstein

NLO QCD corrections to t¯tbb¯ production at the LHC
0
1
0
2
AxelBredenstein
n
a HighEnergyAcceleratorResearchOrganization(KEK),
J Tsukuba,Ibaraki305-0801,Japan

6 E-mail: [email protected]
2

AnsgarDenner
]
h PaulScherrerInstitut,WürenlingenundVilligen,
p
CH-5232VilligenPSI,Switzerland
-
p E-mail: [email protected]
e
h
Stefan Dittmaier
[
Albert-Ludwigs-UniversitätFreiburg,PhysikalischesInstitut,

1 D-79104Freiburg,Germany
v
E-mail: [email protected]
7
2
StefanoPozzorini
7 ∗
4 CERN,TheoryDivision
1. CH-1211GENEVA23
0 E-mail: [email protected]
0
1
: Thenext-to-leadingorder(NLO)QCDcorrectionstopp t¯tbb¯ attheLHCrevealthatthescale
v →
i choice adopted in previous lowest-order simulations underestimates the t¯tbb¯ cross section by a
X
factor two. We discuss a new dynamical scale that stabilizes the perturbative predictions and
r
a describe the impact of the corrections on the shape of distributions. We also account for the
techniquesemployedtocomputethesix-particleone-loopamplitudeswithhighCPUefficiency.
RADCOR2009-9thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumField
TheorytoPhenomenology)
October25-302009
Ascona,Switzerland
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
1. Introduction
ThediscoveryoftheHiggsbosonandthemeasurementofitsinteractionswithmassivequarks
and vectorbosons represent acentral goaloftheLargeHadronCollider (LHC).Inthemassrange
114GeV<M <130GeV,associatedt¯tHproductionprovidestheopportunitytoobservetheHiggs
H
boson intheH ∼ bb¯ decaychannel andtomeasurethetop-quark Yukawacoupling. However,the
→
extraction of the t¯tH(H bb¯) signal from its large QCD backgrounds, pp t¯tbb¯ and t¯tjj, repre-
→ →
sents a serious challenge. The selection strategies elaborated by ATLAS and CMS [1, 2], which
assume 30fb 1 and 60fb 1, respectively, anticipate a statistical significance around 2s (ignoring
− −
systematicuncertainties) andasignal-to-background ratioaslowas1/10. Thiscallsforbetterthan
10% precision inthebackground description, avery demanding requirement bothfrom theexper-
imental and theoretical point of view. Very recently, a novel selection strategy based on highly
boosted Higgs bosons has opened new and very promising perspectives [3]. This approach might
increase thesignal-to-background ratiobeyond1/3.
The calculation of the NLO QCD corrections to the irreducible t¯tbb¯ background, first pre-
sentedin [4,5]andsubsequently confirmedin [6],constitutes anotherimportant steptowardsthe
observability oft¯tH(H bb¯)attheLHC.TheseNLOpredictionsaremandatoryinordertoreduce
→
thehugescaleuncertaintyofthelowest-order(LO)t¯tbb¯ crosssection,whichcanvaryuptoafactor
four if the QCD scales are identified with different kinematic parameters [7]. Previous results for
five-particle processes that feature asignature similar tot¯tbb¯ indicate that setting the renormaliza-
tionandfactorization scalesequaltohalfthethreshold energy, m =E /2,isareasonable scale
R,F thr
choice. Atthis scale the NLOQCDcorrections to pp t¯tH(K 1.2) [8], pp t¯tj(K 1.1)[9],
→ ≃ → ≃
andpp t¯tZ(K 1.35)[10],arefairlymoderate. Thismotivatedexperimentalgroupstoadoptthe
→ ≃
scalem =E /2=m +m /2fortheLOsimulationofthet¯tbb¯ background[1]. However,atthis
R,F thr t bb¯
scale theNLOcorrections topp t¯tbb¯ turn out tobeunexpectedly large (K 1.8) [5,6]. Aswe
→ ≃
argue, areliable perturbative description oft¯tbb¯ production requires adifferentscalechoice[11].
The calculation of the NLO corrections to pp t¯tbb¯ constitutes also an important techni-
→
cal benchmark. The description of many-particle processes at NLO plays an central role for the
LHCphysics programme, and thetechnical challenges raised by these calculations have triggered
an impressive amount of conceptual and technical developments. Within the last few months,
this progress has lead to the first NLO results for six-particle processes at the LHC, namely for
pp t¯tbb¯ [5,6], theleading- [12]andthefull-colour contributions [13]topp Wjjj, andforthe
→ →
qq¯contribution topp bb¯bb¯ [14].
→
To compute the virtual corrections to t¯tbb¯ production we employ explicit diagrammatic rep-
resentations of the one-loop amplitudes and numerical reduction of tensor integrals [15, 16]. The
factorization of colour matrices, the algebraic reduction of helicity structures, and the systematic
recyclingofamultitudeofcommonsubexpressions—both insideindividualdiagramsandintensor
integrals of different diagrams that share common sub-topologies—strongly mitigate the factorial
complexity that is inherent in Feynman diagrams and lead to a remarkably high CPU efficiency.
The real corrections are handled with the dipole subtraction method [17]. Our results have been
confirmed with the HELAC-1LOOP implementation of the OPP method [18, 19, 20] within the
statistical MonteCarloerrorof0.2%[6].
2
NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
2. Description ofthe calculation
In NLO QCD, hadronic t¯tbb¯ production involves the 2 4 partonic channels qq¯ t¯tbb¯
→ →
(7trees and 188 loop diagrams) and gg t¯tbb¯ (36 trees and 1003 loop diagrams). The 2 5
→ →
bremsstrahlung contributions comprise the crossing-symmetric channels qq¯ t¯tbb¯g, qg t¯tbb¯q,
→ →
and gq¯ t¯tbb¯q¯ (64 diagrams each), and the partonic process gg t¯tbb¯g (341 diagrams). Each
→ →
contribution has been worked out twice and independently, resulting in two completely indepen-
dent computer codes. The treatment of the qq¯- and gluon-induced reactions are described in [4]
and [11],respectively. Herewemainlyfocusonthevirtualcorrections intheggchannel.
Feynman diagrams are generated with two independent version of FEYNARTS [21, 22] and
handled with two in-house MATHEMATICA programs that perform algebraic manipulations and
generateFORTRAN77codefullyautomatically. OneofthetwoprogramsreliesonFORMCALC[23]
forpreliminaryalgebraicmanipulations. Theinterferenceoftheone-loopandLOmatrixelements,
summedovercoloursandhelicities, iscomputed onadiagram-by-diagram basis,
(cid:229) (cid:229) M(1-loop) M(LO) ∗ = (cid:229) (cid:229) (cid:229) M(G ) M(LO) ∗ . (2.1)
colhel G "colhel #
(cid:16) (cid:17) (cid:16) (cid:17)
Individual loop diagrams(G )areevaluated byseparate numerical routines andsummedexplicitly.
The cost related to the large number of diagrams is compensated by the possibility to perform
colour sums very efficiently thanks to colour factorization [see (2.2)]. Individual (sub)diagrams
consist ofasinglecolour-stripped amplitude A(G ) multiplied byasimplecolour structure,1 which
is easily reduced to a compact colour basis C . The LO amplitude is handled as a vector in
{ k}
colourspace, andcoloursumsareencoded onceandforallinacolour-interference matrixI ,
kl
M(G ) = A(G ) (cid:229) c(G )C , M(LO) =(cid:229) M(LO)C , I =(cid:229) C C . (2.2)
k k l l kl k l∗
k ! l col
These ingredients yield colour-summed results by means of a single evaluation of the colour-
strippedamplitudeA(G ) ofeach(sub)diagram. TensorintegralswithN propagators andPLorentz
indicesareexpressedintermsoftotallysymmetriccovariantstructures g...gp...p m 1...m P involv-
{ }j1...jP
ingexternal momenta p ,...,p andgmn inD=4 2e dimensions [16],
1 N 1
− −
(2pm )4−D dDq qm 1...qm P = N(cid:229) −1 TN g...gp...p m 1...m P. (2.3)
ip 2 (cid:213) N 1 (q+p)2 m2+i0 j1...jP { }j1...jP
Z i=−0 i − i j1,...,jP=0
(cid:2) (cid:3)
The gg channel involves tensor integrals up to rank P=4. The coefficients TN are related to
j1,...,jP
scalarintegralsbymeansofnumericalalgorithmsthatavoidinstabilities frominverseGramdeter-
minantsandotherspurioussingularities[15,16]. Thetensorrankandthenumberofpropagatorsof
integrals with N >4 are simultaneously reduced without introducing inverse Gram determinants.
Tensor integrals with N = 4,3 are handled with the Passarino–Veltman algorithm as long as no
1More precisely, each quartic gluon coupling generates three independent colour structures that are handled as
separatesubdiagrams. However, mostdiagramsdonotinvolvequarticcouplings, andtheircolourstructurefactorizes
completely.
3
NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
smallGramdeterminantappearsinthereduction. Otherwise,expansionsaboutthelimitofvanish-
ingGramdeterminantsandpossiblyotherkinematicaldeterminantsareapplied[16]. Thereduction
isstrongly boostedbyacachesystemthatrecyclestensorintegralsamongdiagramswithcommon
subtopologies. Theggchannel involves about 350scalar integrals, whichrequire 10msCPUtime
per phase-space point.2 The calculation of all scalar and tensor integrals with and without cache
system takes40msand200ms,respectively. Rationaltermsarisingfromultraviolet (UV)polesof
tensorintegralswithD-dependent coefficientsareautomatically extractedbymeansofacatalogue
ofresidues RN ,
j1...jP
RN
f(D)TN = f(D) TˆN + j1...jP = f(4)TN 2f (4)RN . (2.4)
j1...jP j1...jP e j1...jP− ′ j1...jP
UV !
Rational terms resulting from infrared poles must be taken into account only in wave-function
renormalization factors, sincetheycancelintruncated one-loopamplitudes [4].
Thehelicity-dependent partsofalldiagramsarereducedtoacommonbasisofso-calledStan-
dardMatrixElements(SMEs)oftheform
Mˆm =Qmm 1m 2r1...rle m 1(p1)e m 2(p2) v¯(p3)gr1...grku(p4) v¯(p5)grk+1...grlu(p6) , (2.5)
where Qm 1m 2r1...rl are combinations of m(cid:2)etric tensors and ex(cid:3)te(cid:2)rnal momenta. The co(cid:3)lour-stripped
m
partofeachloopdiagram[see(2.2)]yieldsalinearcombination ofSMEsandtensorintegrals,
N 1
A(G ) = (cid:229) F(G )Mˆ , F(G ) =(cid:229) (cid:229) − K (G ) TN + rationalparts. (2.6)
m m m m;j1...jP j1...jP
m P j1,...,jP=0
TheuseofSMEsenablesveryefficienthelicitysummations. Helicityandcoloursumsareencoded
intheinterference ofSMEsMˆ andcolourstructures C withtheLOamplitude,
m k
M = (cid:229) (cid:229) Mˆ C M(LO) ∗=(cid:229) I (cid:229) Mˆ M(LO) ∗. (2.7)
km m k kl m l
colhel l hel
(cid:16) (cid:17) (cid:16) (cid:17)
This matrix, which must be computed only once per phase-space point, links the colour/helicity-
independent formfactorsF(G ) ofeachdiagram toitscolour/helicity-summed contribution
m
(cid:229) (cid:229) M(G ) M(LO) ∗ = (cid:229) F(G ) (cid:229) c(G )M . (2.8)
m k km
colhel m k !
(cid:16) (cid:17)
The SME-reduction starts with process-independent D-dimensional relations such as momentum
conservation, Dirac algebra, transversality, and gauge-fixing conditions for the gluon-polarization
vectors. Once rational terms are extracted, we further reduce SMEs with two alternative algo-
rithms in four dimensions. The first algorithm splits each fermion chain into two contributions,
u(pi)= (cid:229) l = w l u(pi), via insertion of chiral projectors w = (1 g 5)/2. This permits to em-
ployvariousr±elationsoftypeg m g a g b w gm =g m w gm g±b g a w ±+g a g b gm w ,whichconnect
±⊗ ±⊗ ± ∓
Diracmatricesofdifferentfermionchains[4,24]. Inthiswayarichvarietyofnon-trivialidentities
(cid:0) (cid:1)
2AllCPUtimesrefertoa3GHzIntelXeonprocessor.
4
NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
are obtained that reduce the full amplitude to 502 SMEs [11]. Besides this procedure, which de-
pends onprocess-specific aspects, weimplemented asimpleprocess-independent reduction based
onafour-dimensional identity oftypeg m 1g m 2g m 3g m 4g m 5 =gm 1m 2g m 3g m 4g m 5 gm 1m 2gm 3m 4g m 5+perm.,
−
which eliminates spinor chains with more than three Dirac matrices without introducing g [11].
5
Thisleadsto970SMEs. Inspiteofthefactor-twodifferenceinthenumberofSMEs,wefoundthat
thenumericalcodes basedonthetwodifferent reductions havethesame—andremarkably high—
CPU speed: about 180 ms per phase-space point. This unexpected result means that the obtained
CPUperformance, atleastforthisprocess, doesnotdependonprocess-dependent optimisations.
Tohandletherealcorrectionsweemployedthedipolesubtractionmethod[17,25]. The2 5
→
matrix elements were generated with MADGRAPH [26] and checked against analytic calculations
andin-house codebasedonoff-shellrecursions. Moredetailsaregivenin [11].
3. Predictions forthe LHC
Wepresentresultsforpp t¯tbb¯+X at√s=14TeVform =172.6GeVandm =0. Collinear
t b
→
final-state partons are recombined into jets with D f 2+D y2 > 0.4 using a k -algorithm. We
T
imposethecuts p >20GeVand y <2.5onbjetsandusetheCTEQ6setofPDFswithN =5
T,b | b| p F
activeflavoursandL MS=226MeV. Furtherdetailsaregivenin [11].
5
In all recent ATLAS studies of t¯tH(H bb¯) [1] the signal and its t¯tbb¯ background are sim-
→
ulated by setting the renormalization and factorization scales equal to half the threshold energy,
Ethr =2mt+mbb¯. Being proportional to a s4, these LO predictions are extremely sensitive to the
scalechoice, andin [5]wefoundthatatm =E /2theNLOcorrections topp t¯tbb¯ areclose
R,F thr
→
to a factor of two. This enhancement is due to the fact that pp t¯tbb¯ is a multi-scale process
→
involving various scaleswellbelowE /2. Inparticular, thecross sectionissaturated bybquarks
thr
with p m [11]. Inordertoavoidlargelogarithmsweadvocatetheuseofthedynamicalscale
T,b t
≪
m 02=mt pT,bpT,b¯, (3.1)
which improvestheperturbative convergence anpdminimises NLOeffects intheshape ofdistribu-
tions [11]. In Fig.1 we show results for the kinematic region mbb¯ >100GeV, which is relevant
for ATLAS/CMS studies of t¯tH(H bb¯). The left plot displays the dependence of the LO and
→
NLOcrosssections withrespecttoscalevariations m =m =xm . Atthecentralscaleweobtain
R F 0
s = 786.3(2)fb and s = 978(3)fb. This NLO result is 2.18 times larger as compared to
LO NLO
the LOcross section based on theATLASscale choice [11]. Thescale choice (3.1)reduces the K
factorto1.24,andtheNLO(LO)uncertaintycorresponding tofactor-twoscalevariationsamounts
to21% (78%). Theimprovement withrespect to [5]isevident also from thestability oftheNLO
curve in Fig.1a. The right plot in Fig.1 displays LO (blue) and NLO (red) scale-dependent pre-
dictionsforthem distribution. TheresultsarenormalizedtotheLOdistributionatm =m and
bb¯ R,F 0
the bands correspond to factor-two scale variations. The NLO predictions perfectly fit within the
LO band and exhibit little kinematic dependence. Inspecting various other distributions wefound
similarly smallNLOcorrections totheirshapes. InFig.2weshow similarplots forthe kinematic
region p >200GeV, which permits to increase the separation between the Higgs signal and
T,bb¯
its t¯tbb¯ background [3]. At the central scale we obtain s =451.8(2)fb and s =592(4)fb.
LO NLO
The K factor (1.31), and the LO(79%) and NLO(22%) scale dependence behave similarly as for
5
NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
σ[fb] pp→t¯tbb¯+X ddσσNLLOO pp→t¯tbb¯+X
6000 2
LO
NLO 1.8
5000
1.6
µ2R =mt√pT,bpT,b¯ξ2
4000 µ2F =mt√pT,bpT,b¯ξ2 1.4
1.2
3000 mbb¯ >100GeV
1
2000 0.8
0.6
1000
0.4 mbb¯ >100GeV
0 0.2
0.125 0.25 0.5 1 2 4 8 0 50 100 150 200 250 300 350 400
ξ mbb¯[GeV]
Figure1: pp→t¯tbb¯+X crosssectionattheLHCforstandardcutsandmbb¯ >100GeV: scaledependence
oftheLOandNLOcrosssection(leftplot)andrelativeNLOcorrectionstothem distribution(rightplot).
bb¯
σ[fb] pp→t¯tbb¯+X ddσσNLLOO pp→t¯tbb¯+X
3500 2
LO
NLO 1.8
3000
1.6
2500 µ2R =mt√pT,bpT,b¯ξ2
µ2F =mt√pT,bpT,b¯ξ2 1.4
2000 1.2
pT,bb¯ >200GeV
1500 1
0.8
1000
0.6
500
0.4 pT,bb¯ >200GeV
0 0.2
0.125 0.25 0.5 1 2 4 8 0 50 100 150 200 250 300 350 400
ξ mbb¯[GeV]
Figure2: pp→t¯tbb¯+X crosssectionattheLHCforstandardcutsand pT,bb¯ >200GeV:scaledependence
oftheLOandNLOcrosssection(leftplot)andrelativeNLOcorrectionstothem distribution(rightplot).
bb¯
mbb¯ > 100GeV. But in this case the NLO corrections are rather sensitive to mbb¯. In the physi-
cally interesting region of m 100GeV, the shape of the m distribution is distorted by about
bb¯ ∼ bb¯
20%. This effect tends to mimic a Higgs signal and should be carefully taken into account in the
t¯tH(H bb¯)analysis.
→
4. Conclusions
The observation of the t¯tH(H bb¯) signal and the direct measurement of the top-quark
→
Yukawacoupling attheLHCrequire averyprecise description ofthepp t¯tbb¯ irreducible back-
→
ground. The NLO QCD corrections to t¯tbb¯ production reveal that the scale choice adopted in
previous LOstudies underestimates thiscrosssection byafactor oftwo. Weadvocate theuseofa
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NLOQCDcorrectionstot¯tbb¯ productionattheLHC StefanoPozzorini
dynamical scale that stabilizes the perturbative predictions reducing the K factor to about 1.2. In
presence of standard cuts NLO effects feature small kinematic dependence. But in the regime of
highly boostedHiggsbosonsweobservesignificant distortions intheshapeofdistributions.
The calculation is based on process-independent algebraic techniques, which reduce loop di-
agrams to standard colour/helicity structures, and numerically stable tensor-reduction algorithms.
TheveryhighnumericalstabilityandCPUefficiencyofthisapproachareveryencouraginginview
offutureNLOcalculations formulti-particle processes.
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