Dominant $\mathcal{O}(\alpha_s\alpha)$ corrections to Drell-Yan processes in the resonance region

0.64 MB·

by  Stefan Dittmaier

O
Dominant (α α) corrections to Drell–Yan
s
processes in the resonance region
Stefan Dittmaier
Albert-Ludwigs-UniversitätFreiburg,PhysikalischesInstitut,D-79104Freiburg,Germany
E-mail: [email protected]
6
1 Alexander Huss
0 InstituteforTheoreticalPhysics,ETH,CH-8093Zürich,Switzerland
2
DepartmentofPhysics,UniversityofZürich,CH-8057Zürich,Switzerland

n E-mail: [email protected]
a
J
Christian Schwinn
8
InstituteforTheoreticalParticlePhysicsandCosmology,RWTHAachenUniversity,D-52056

] Aachen,Germany
h
E-mail: [email protected]
p
-
p
Apart from the well-known NNLO QCD and NLO electroweak corrections to W- and Z-boson
e
h productionathadroncolliders,themostimportantfixed-ordercorrectionsaregivenbythemixed
[
QCD–electroweak corrections of O(αsα). The knowledge of these corrections is of particular
1
importancetocontrolthetheoreticaluncertaintiesintheupcominghigh-precisionmeasurements
v
7 oftheW-bosonmassandtheeffectiveweakmixingangleattheLHC.Sincetheseobservablesare
2 dominated by the phase-space regions of resonant W/Z bosons, we address the O(α α)correc-
0 s
2 tionsintheframeworkofanexpansionabouttheW/Zpoles. Retainingonlytheleading,resonant
0
contributionintheso-calledpoleapproximation,thecorrectionscanbeclassifiedintofactorizable
.
1
and non-factorizable contributions. In this article we review our calculation of the numerically
0
6 dominant corrections which arise from factorizable corrections of “initial–final” type, i.e. they
1
combinetheQCDcorrectionstotheproductionwiththelargeelectroweakcorrectionstothede-
:
v cayoftheW/Zboson. Moreover, wecompareourresultstosimplerapproximatecombinations
i
X of electroweak and QCD corrections based on naive products of NLO QCD and electroweak
r correction factors and using leading-logarithmic approximations for QED final-state radiation.
a
Finally,weestimatetheshiftintheW-bosonmassthatresultsfromtheO(α α)correctionstothe
s
transverse-massdistribution.
12thInternationalSymposiumonRadiativeCorrections(Radcor2015)andLoopFestXIV(Radiative
CorrectionsfortheLHCandFutureColliders)
15-19June,2015
UCLADepartmentofPhysics&AstronomyLosAngeles,USA
c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
(cid:13)Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
O(α α)correctionstoDrell–Yanprocesses
s
1. Introduction
TheDrell–Yan-likeproductionofWandZbosons,pp/pp¯ →V →(cid:96) (cid:96)¯ +X,isoneofthemost
1 2
prominentclassesofparticlereactionsathadroncolliders. Thelargeproductionrateandtheclean
experimental signature of the leptonic vector-boson decay allow these processes to be measured
withgreatprecisionandrenderthemoneofthemostimportant“standard-candle”processesatthe
LHC. Not only do these processes represent powerful tools for detector calibration, but they can
also be used as luminosity monitor and further deliver important constraints in the fit of parton
distributionfunctions(PDFs). OfparticularrelevanceforprecisiontestsoftheStandardModelis
thepotentialoftheDrell–YanprocessattheLHCforhigh-precisionmeasurementsintheresonance
regions. Inparticular,theeffectiveweakmixinganglemightbemeasuredwithLEPprecisionand
theW-bosonmassisexpectedtobeextractedfromkinematicfitswithasensitivitybelow10 MeV
(seeRef.[1]andreferencestherein).
On the theory side, the Drell–Yan-like production of W or Z bosons is one of the best un-
derstood and most precisely predicted processes. The current state of the art includes QCD cor-
rectionsatnext-to-next-to-leading-order(NNLO)accuracy,supplementedbyleadinghigher-order
soft-gluoneffectsormatchedtoQCDpartonshowersuptoNNLO.Theelectroweak(EW)correc-
tionsareknownatnext-to-leadingorder(NLO)andleadinguniversalcorrectionsbeyond(see,e.g.,
references in Ref. [2]). Thus, in addition to the N3LO QCD corrections, the next frontier in theo-
reticalfixed-ordercomputationsisgivenbythecalculationofthemixedQCD–EWcorrectionsof
O(α α),whichcanaffectobservablesrelevantfortheW-massdeterminationatthepercentlevel.
s
While leading universal QCD and EW corrections are known to factorize from each other, a
full NNLO calculation at O(α α) is necessary for a proper combination of NLO QCD and NLO
s
EW corrections without ambiguities. Here some partial results for two-loop amplitudes [3–5] as
well as the full O(α α)corrections to the W/Z decay widths [6,7] are known. A complete calcu-
s
lationoftheO(α α)correctionsrequirestocombinethedouble-virtualcorrectionswiththeO(α)
s
EWcorrectionstoW/Z+jetproduction,theO(α )QCDcorrectionstoW/Z+γ production,and
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thedouble-realcorrections.
Inaseriesoftworecentpapers[2,8],wehaveinitiatedthecalculationoftheO(α α)correc-
s
tionstoDrell–Yanprocessesintheresonanceregionviatheso-calledpoleapproximation(PA).It
is based on a systematic expansion of the cross section about the resonance pole and is suitable
for theoretical predictions in the vicinity of the gauge-boson resonance. In detail, the PA splits
the corrections into factorizable and non-factorizable contributions. The former can be separately
attributedtotheproductionandthesubsequentdecayofthegaugeboson, whilethelatterlinkthe
productionanddecaysubprocessesbytheexchangeofsoftphotons.
In this article we motivate the general idea of the pole expansion and outline the salient fea-
tures of the PA at O(α α). We discuss our numerical results for the factorizable corrections of
s
“initial–final” type, which are the dominant contribution at this order, as they combine sizable
QCD corrections to the production with the large EW corrections to the W/Z decays. Finally, we
presenttheimpactoftheO(α α)correctionsontheW-bosonmassextractionfromakinematicfit
s
tothetransverse-massspectrum.
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O(α α)correctionstoDrell–Yanprocesses
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2. Structureofthepoleapproximation
ThePAforDrell–YanprocessesprovidesasystematicclassificationofcontributionstoFeyn-
man diagrams that are enhanced by the resonant propagator of a vector bosonV =W,Z. To this
end,weschematicallywritethetransitionamplitudeoftheprocessinthefollowingform,
W(p2)
M = V +N(p2), (2.1)
p2 −M2+Σ(p2) V
V V V
where Σ denotes the self-energy ofV and the functionsW and N represent the resonant and non-
resonantparts,respectively. Inordertoisolatetheresonantcontributionsinagauge-invariantway,
wefurtherrewritetheaboveexpressionas
W(µ2) 1 (cid:20) W(p2) W(µ2) 1 (cid:21)
M = V + V − V +N(p2) (2.2)
p2 −µ2 1+Σ(cid:48)(µ2) p2 −M2+Σ(p2) p2 −µ2 1+Σ(cid:48)(µ2) V
V V V V V V V V V
withµ2 =M2−iM Γ denotingthegauge-invariantlocationofthepropagatorpoleinthecomplex
V V V V
p2 plane.
V
TheamplitudeinthePAisobtainedfromEq.(2.2)byomittingthelast,non-resonanttermand
systematicallyexpandingtheterminsquarebracketsaboutthepoint p2 =M2 andonlykeepingthe
V V
leading, resonant contribution. The first term in Eq. (2.2) corresponds to the factorizable correc-
tionsinwhichon-shellproductionanddecaysubamplitudesarelinkedbytheoff-shellpropagator.
The evaluation of the subamplitudes using on-shell kinematics is essential in order to guarantee
gauge invariance. The specification of the on-shell projection is not unique and different vari-
(cid:16) (cid:17)
ants lead to differences within the intrinsic accuracy of the PA which is of O α × ΓV in cross-
π MV
sectioncontributionsthatcorrecttheLOpredictionbytermsofO(α). Despitethefreedominthe
choice of the on-shell projection, they have to match between the virtual and real corrections in
theinfrared-singularlimitstoensurethepropercancellationofthesingularitiesinthefinalresult.
The non-factorizable corrections arise from the term on the r.h.s. of Eq. (2.2) in square brackets
andaredeeplylinkedtothesoft-singularstructureofW(p2)andΣ(cid:48)(p2)inthelimit p2 →µ2. As
V V V V
explained in more detail in Ref. [8], these corrections solely arise from soft-photon exchange that
linkproductionanddecay.
As a result, the corrections to the production and decay stages of the intermediate unstable
particle are separated in a consistent and gauge-invariant way by the PA. This is particularly rele-
vantforthecharged-currentDrell-Yanprocess,wherephotonradiationofftheintermediateWbo-
son contributes simultaneously to the corrections to production and decay of a W boson, and to
the non-factorizable contributions. Applications of different variants of the PA to NLO EW cor-
rections [8–10] have been validated by a comparison to the complete NLO EW calculations and
show excellent agreement at the order of some 0.1% in kinematic distributions dominated by the
resonance region. In particular, the bulk of the NLO EW corrections near the resonance can be
attributed to the factorizable corrections to the W/Z decay subprocesses, while the factorizable
corrections to the production process are mostly suppressed below the percent level, and the non-
factorizablecontributionsareevensmaller.
The quality of the PA at NLO justifies the application of this approach in the calculation of
the O(α α) corrections to observables that are dominated by the resonances. The structure of
s
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O(α α)correctionstoDrell–Yanprocesses
s
the PA for the O(α α) correction has been worked out in Ref. [8], where details of the method
s
and our setup can be found. The corrections can be classified into the four types of contributions
shown in Fig. 1 for the case of the double-virtual corrections. For each class of contributions
with the exception of the final–final corrections (c), also the associated real–virtual and double-
real corrections have to be computed, obtained by replacing one or both of the labels α and α
s
in the blobs in Fig. 1 by a real photon or gluon, respectively. The corresponding crossed partonic
channels,e.g.withquark–gluoninitialstateshavetobeincludedinaddition.
q ‘ q ‘
a 1 a 1
αααααααααααααααααsssssssssssssssssααααααααααααααααα αααααααααααααααααsssssssssssssssss ααααααααααααααααα
V V
qb ‘2 qb ‘2
(a)Factorizableinitial–initialcorrections (b)Factorizableinitial–finalcorrections
γ
q ‘
a 1 q ‘
a 1
αααααααααααααααααsssssssssssssssssααααααααααααααααα α
s
V
V
qb ‘2 qb ‘2
(c)Factorizablefinal–finalcorrections
(d)Non-factorizablecorrections
Figure 1: The four types of corrections that contribute to the mixed QCD–EW corrections in the
PA illustrated in terms of generic two-loop amplitudes. Simple circles symbolize tree structures,
doublecirclesone-loopcorrections,andtriplecirclestwo-loopcontributions.
Indetail,thefourtypesofcorrectionsarecharacterizedasfollows:
(a) The initial–initial factorizable corrections are given by two-loop O(α α) corrections to on-
s
shellW/Zproductionandthecorrespondingone-loopreal–virtualandtree-leveldouble-real
contributions, i.e. W/Z+jet production at O(α), W/Z+γ production at O(α ), and the
s
processesW/Z+γ+jetattreelevel. Resultsforindividualingredientsoftheinitial–initial
part are known, however, a consistent combination of these building blocks requires also a
subtractionschemeforinfrared(IR)singularitiesatO(α α)andhasnotbeenperformedyet.
s
NotethatcurrentlynoPDFsetincludingO(α α)correctionsisavailable,whichisrequired
s
to absorb IR singularities of the initial–initial corrections from QCD and photon radiation
collineartothebeams.
Results of the PA at O(α) show that observables such as the transverse-mass distribution
in the case of W production or the lepton-invariant-mass distributions for Z production are
extremely insensitive to photonic initial-state radiation (ISR) [8]. Since these distributions
alsoreceiverelativelymoderateQCDcorrections,wedonotexpectsignificantinitial–initial
NNLO O(α α) corrections to such distributions. For observables sensitive to initial-state
s
recoileffects,suchasthetransverse-lepton-momentumdistribution,theO(α α)corrections
s
shouldbelarger,butstillverysmallcomparedtothehugeQCDcorrections.
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O(α α)correctionstoDrell–Yanprocesses
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(b) Thefactorizableinitial–finalcorrectionsconsistoftheO(α )correctionstoW/Zproduction
s
combinedwiththeO(α)correctionstotheleptonicW/Zdecay. Thelattercomprisetheby
fardominantcontributiontothePAatO(α)withasubstantialimpactontheshapeofdiffer-
ential distributions. Further given that the NLO QCD correction are sizable with no appar-
ent suppression, we expect the factorizable corrections of “initial–final” type to capture the
dominantO(α α)effects. TheseNNLOcorrectionsreceivedouble-virtual,real–virtual,and
s
double-real contributions, as illustrated in Fig. 2 in terms of generic interference diagrams.
ThecancellationofIRsingularitiesbetweentheseindividualcontributionsisachievedusing
a local subtraction scheme resulting in a fully differential calculation. In order to construct
ourNNLOsubtractiontermsweemployatwo-foldapplicationofthedipolesubtractionfor-
malism for NLO QCD and EW corrections. An important ingredient for this construction
was the recent generalization of the formalism to cover decay kinematics, which has been
workedoutinRef.[11]. FurtherdetailsonthecomputationcanbefoundinRef.[2]. Inthe
followingsectionswereviewthemainresultspresentedthere.
(c) Factorizablefinal–finalcorrectionsarisesolelyfromtheO(α α)countertermsofthelepton–
s
W/Z-bosonvertices,whichinvolveonlycorrectionstothevector-bosonself-energiesatthis
order. NotonlyarethesecorrectionsfreeofIRdivergences, butowingtothefactthatthere
arenocorrespondingrealcontributions,thefinal–finalcorrectionshavepracticallynoimpact
on the shape of distributions. The explicit calculation carried out in Ref. [2] further reveals
thatthosecorrectionsareinfactphenomenologicallynegligible.
(d) The non-factorizable O(α α) corrections are given by soft-photon corrections connecting
s
the initial state, the intermediate vector boson, and the final-state leptons, combined with
QCD corrections toV-boson production. As shown in detail in Ref. [8], these corrections
canbeexpressedintermsofsoft-photoncorrectionfactorstosquaredtree-levelorone-loop
QCD matrix elements by using gauge-invariance arguments. Furthermore, exploiting this
factorization property, the IR cancellation can be accomplished by using a composition of
twoNLOmethods: Weemploythedipole-subtractionformalismforthetreatmentoftheIR
singularitiesassociatedwiththeQCDcorrectionstogetherwiththesoft-slicingapproachfor
the photonic corrections. The numerical impact of these corrections was found to be below
the0.1%levelandisthereforenegligibleforallphenomenologicalpurposes.
3. Numericalresults
3.1 Setupandconventions
Thedetailedsetupandtheinputparametersusedtoobtaintheresultsdiscussedinthefollow-
ingcanbefoundinRef.[2]. Herewejustrepeatthebasicdefinitionoftheindividualcomponents
oftheQCDandEWcorrections.
OurdefaultpredictionfortheDrell–YancrosssectionatmixedQCD–EWNNLOisobtained
byaddingtheO(αsα)corrections∆σpNroNdL×Odse⊗cew tothesum∆σNLOs+∆σNLOew ofthefullNLOQCD
andEWcorrections,
σNNLOs⊗ew =σ0+∆σNLOs+∆σNLOew+∆σNNLOs⊗ew, (3.1)
prod×dec
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O(α α)correctionstoDrell–Yanprocesses
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q ‘ q ‘
a 1 a 1
ααααααααααααααααα ααααααααααααααααα
V V
qb ‘2 qb ‘2
(a)Factorizableinitial–finaldouble-virtualcorrections
q ‘ q ‘
a 1 a 1
V V
ααααααααααααααααα ααααααααααααααααα
qb g ‘2 g q ‘2
b
(b)Factorizableinitial–final(realQCD)×(virtualEW)corrections
γ
q ‘
a 1
V
qb ‘2
(c)Factorizableinitial–final(virtualQCD)×(realphotonic)corrections
γ γ
q ‘ q ‘
a 1 a 1
V V
qb g ‘2 g q ‘2
b
(d)Factorizableinitial–finaldouble-realcorrections
Figure 2: Interference diagrams for the various contributions to the factorizable initial–final cor-
rectionsofO(α α),withblobsrepresentingallrelevanttreestructures. Theblobswith“α”inside
s
representone-loopcorrectionsofO(α),andthedoubleslashonapropagatorlineindicatesthatthe
correspondingmomentumissetonitsmassshellintherestofthediagram(butnotontheslashed
lineitself).
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O(α α)correctionstoDrell–Yanprocesses
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wherealltermsareconsistentlyevaluatedwithNLOPDFsincludingtheleading-order(LO)contri-
butionσ0. Thenon-factorizablecorrectionsaswellasthefactorizablecorrectionsof“final–final”
typearenottakenintoaccountduetotheirnegligiblesize,asdiscussedabove.
In order to validate estimates of the NNLO QCD–EW corrections based on a naive product
ansatz,wedefinethenaiveproductoftheNLOQCDcrosssectionandtherelativeEWcorrections,
σNNLOs⊗ew =σNLOs(1+δ )=σ0+∆σNLOs+∆σNLOew+∆σNLOs δ . (3.2)
naivefact α α
TherelativeNLOEWcorrections
∆σNLOew
δ ≡ (3.3)
α σ0
are defined in two different versions: First, based on the full O(α) correction (δ ), and second,
α
basedonthedominantEWfinal-statecorrectionofthePA(δdec).
α
Definingthecorrectionfactors,1
δprod×dec≡ ∆σpNroNdL×Odse⊗cew, δ(cid:48) ≡ ∆σNLOs, (3.4)
αsα σLO αs σLO
wecancasttherelativedifferenceofourbestprediction(3.1)andtheproductansatz(3.2)intothe
followingform,
σNNLOs⊗ew−σNNLOs⊗ew
naivefact =δprod×dec−δ(cid:48) δ , (3.5)
σLO αsα αs α
wherethe LOpredictionσLO inthe denominatorsisevaluatedwith theLOPDFs. Thedifference
oftherelativeNNLOcorrectionδprod×decandthenaiveproductδ(cid:48) δ(dec)thereforeallowstoassess
αsα αs α
thevalidityofanaiveproductansatz.
Most contributions to the factorizable initial–final corrections take the reducible form of a
productoftwoNLOcorrections,withtheexceptionofthedouble-realemissioncorrectionswhich
are defined with the full kinematics of the 2→4 phase space. It is only in the double-real con-
tributions where the final-state leptons receive recoils from both QCD and photonic radiation, an
effect that cannot be captured by naively multiplying NLO QCD and EW corrections. Any large
deviations between δprod×dec and δ(cid:48) δ(dec) can therefore be attributed to this type of contribution.
αsα αs α
Thedifferenceofthenaiveproductsdefinedintermsofδdec andδ allowsustoassesstheimpact
α α
of the missing O(α α) corrections beyond the initial–final corrections considered in our calcula-
s
tionandthereforealsoprovidesanerrorestimateofthePA,andinparticularoftheomissionofthe
correctionsof“initial–initial”type.
3.2 ResultsonthedominatingO(α α)corrections
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Figure3showsthenumericalresultsfortherelativeO(α α)initial–finalfactorizablecorrec-
s
tionsδprod×dec tothetransverse-mass(M )andthetransverse-lepton-momentum(p )distribu-
αsα T,ν(cid:96) T,(cid:96)
tionsforW+ productionattheLHC.Theresultsfortheneutral-currentprocessaregiveninFig.4,
whichdisplaystheresultsforthelepton-invariant-mass(M )distributionandatransverse-lepton-
(cid:96)(cid:96)
momentum(p )distribution.
T,(cid:96)+
1Notethatthecorrectionfactorδα(cid:48)s differsfromthatinthestandardQCDKfactorKNLOs =σNLOs/σLO≡1+δαs
duetotheuseofdifferentPDFsetsintheBorncontributionsappearinginthenormalization.
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O(α α)correctionstoDrell–Yanprocesses
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0.5 pp→W+→µ+νµ √s=14TeV 20 pp→W+→µ+νµ √s=14TeV
δαprsoαd×dec 10 baremuons
0 δδαα00ss××δδααdec 0
−0.5 −10
δ[]% δ[]% −20
1
− 30
−
40 δαprsoαd×dec
−1.5 − δα0s×δαdec
baremuons −50 δα0s×δα
2 60
− 65 70 75 80 85 90 95 − 32 34 36 38 40 42 44 46 48
MT,µ+νµ[GeV] pT,µ+[GeV]
Figure3: RelativefactorizablecorrectionsofO(α α)inducedbyinitial-stateQCDandfinal-state
s
EW contributions to the transverse-mass (left) and transverse-lepton-momentum (right) distribu-
tionsforW+ productionattheLHC.ThenaiveproductsoftheNLOcorrectionfactorsδ(cid:48) andδ
αs α
areshownforcomparison. (TakenfromRef.[2].)
pp→Z→µ+µ− √s=14TeV pp→Z→µ+µ− √s=14TeV
10 20
8 δαprsoαd×dec 10 baremuons
6 δα0s×δαdec 0
4 δα0s×δα
10
2 −
]% 0 ]% 20
δ[ δ[ −
2
− 30
−
−4 40 δαprsoαd×dec
−6 − δα0s×δαdec
−8 baremuons −50 δα0s×δα
10 60
− 75 80 85 90 95 100 105 − 36 38 40 42 44 46 48 50 52 54
Mµ+µ−[GeV] pT,µ+[GeV]
Figure 4: Relative factorizable corrections of O(α α) induced by initial-state QCD and final-
s
state EW contributions to the lepton-invariant-mass distribution (left) and a transverse-lepton-
momentum distribution (right) for Z production at the LHC. The naive products of the NLO cor-
rectionfactorsδ(cid:48) andδ areshownforcomparison. (TakenfromRef.[2].)
αs α
In Figs. 3 and 4 we compare to the two different implementations of a naive product of cor-
rection factors discussed after Eq. (3.5). In both figures, we assume that final-state leptons and
collinearphotonscanberesolvedcompletely(defining“bareleptons”), asituationthatisrealistic
formuons,butnotforelectrons,whichappearinshowerstogetherwiththephotonsintheelectro-
magnetic calorimeter in the detector. For the latter case, results based on some recombination of
leptonswithcollinearphotonsaremorerealistic(defining“dressedleptons”). Suchresultscanbe
found in Ref. [2]. They show the same features as the ones for bare leptons, with corrections that
aretypicallysmallerbyafactoroftwo.
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O(α α)correctionstoDrell–Yanprocesses
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FortheM distributionforW+ production(leftplotinFig.3),themixedNNLOQCD–EW
T,ν(cid:96)
corrections are moderate and amount to approximately −1.7 % around the resonance, which is
about an order of magnitude smaller than the NLO EW corrections.2 Both variants of the naive
productprovideagoodapproximationtothefullresultintheregionaroundandbelowtheJacobian
peak, which is dominated by resonant W production. For larger M , the product δ(cid:48) δ based
T,ν(cid:96) αs α
on the full NLO EW correction factor deviates from the other curves. Due to the well-known
insensitivityoftheobservableM toISReffectsseenfortheNLOcorrections[8],thisdifference
T,ν(cid:96)
signalsthegrowingimportanceofeffectsbeyondthePA.However,thedeviationsamounttoonly
fewper-milleforM (cid:46)90 GeV. Theoverallgoodagreementbetweentheδprod×dec corrections
T,ν(cid:96) αsα
and both naive products can be attributed to well-known insensitivity of the observable M to
T,ν(cid:96)
ISReffectsalreadyseeninthecaseofNLOcorrectionsinRef.[8].
For the p distributions (right plots in Figs. 3 and 4 for W+/Z production, respectively) we
T,(cid:96)
observecorrectionsthataresmallfarbelowtheJacobianpeak,butwhichrisetoabout15%(20%)
on the Jacobian peak at p ≈M /2 for the case of the W+ boson (Z boson) and then display
T,(cid:96) V
a steep drop reaching almost −50% at p =50 GeV. This enhancement stems from the large
T,(cid:96)
QCD corrections above the Jacobian peak familiar from the NLO QCD results (see e.g. Fig. 8
in Ref. [8]) where the recoil due to real QCD radiation shifts events with resonant W/Z bosons
above the Jacobian peak. The naive product ansatz fails to provide a good description of the full
prod×dec
result δ and deviates by 5–10% at the Jacobian peak, where the PA is expected to be the
αsα
most accurate. This can be attributed to the strong influence of the recoil induced by ISR on
the transverse momentum, which implies a larger effect of the double-real emission corrections
on this distribution that are not captured correctly by the naive products. The two versions of
the naive products display larger deviations than in the M distribution discussed above, which
T,ν(cid:96)
signalsalargerimpactofthemissingO(α α)initial–initialcorrections. However,thesedeviations
s
should be interpreted with care, since a fixed-order prediction is not sufficient to describe this
distribution around the peak region p ≈ M /2, which corresponds to the kinematic onset for
T,(cid:96) V
V+jetproductionandisknowntorequireQCDresummationforaproperdescription.
IncaseoftheM distributionforZproduction(leftplotinFig.4),correctionsupto10%are
(cid:96)(cid:96)
observed below the resonance for the case of bare muons. This is consistent with the large EW
correctionsatNLOinthisregion,whicharisefromphotonicfinal-stateradiation(FSR)thatshifts
thereconstructedvalueoftheinvariantlepton-pairmassawayfromtheresonancetolowervalues.
prod×dec
The naive product approximates the full initial–final corrections δ reasonably well at the
αsα
resonance itself (M =M ) and above, but completely fails already a little below the resonance
(cid:96)(cid:96) Z
prod×dec
where the naive products do not even reproduce the sign of the full δ correction. This
αsα
deviationoccursalthoughtheinvariant-massdistributioniswidelyunaffectedbyISReffects. The
fact that we obtain almost identical corrections from the two versions of the product δ(cid:48) δdec and
αs α
δ(cid:48) δ demonstratestheinsensitivityofthisobservabletophotonicISR.Theoriginofthefailureof
αs α
thenaiveproductansatz,whichisdiscussedinRef.[2]indetail,canbeunderstoodasfollows. The
largeEWcorrectionsbelowtheresonanceariseduetotheredistributionofeventsneartheZpole
2Thestructureobservedinthecorrectionδαpsrαod×dec aroundMT,ν(cid:96)≈62 GeVcanbeattributedtotheinterplayof
thekinematicsofthedouble-realemissioncorrectionsandtheeventselection.Itarisesclosetothekinematicboundary
MT,ν(cid:96)>50 GeVimpliedbythecutpT,(cid:96)±,ETmiss>25 GeVfortheback-to-backkinematicsofthenon-radiativeprocess.
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O(α α)correctionstoDrell–Yanprocesses
s
tolowerleptoninvariantmassesbyphotonicFSR,sothatitwouldbemoreappropriatetoreplace
theQCDcorrectionfactorδ(cid:48) inthenaiveproductbyitsvalueattheresonanceδ(cid:48) (M =M )≈
αs αs (cid:96)(cid:96) Z
6.5%,whichcorrespondstothelocationoftheeventsthatareresponsibleforthebulkofthelarge
EW corrections below the resonance. In contrast, the naive product ansatz simply multiplies the
correctionslocallyonabin-by-binbasis. Theobservedmismatchisdramaticallyenhancedbythe
factthattheQCDcorrectionδ(cid:48) exhibitsasignchangeatM ≈83 GeV.
αs (cid:96)(cid:96)
Contrary to the lepton-invariant-mass distribution, the transverse-mass distribution is domi-
natedbyeventswithresonantWbosonsevenintherangebelowtheJacobianpeak, M (cid:46)M ,
T,ν(cid:96) W
sothatitislesssensitivetotheredistributionofeventstolowerM . Thisexplainswhythenaive
T,ν(cid:96)
productcanprovideagoodapproximationofthefullinitial–finalNNLOcorrections. Itshouldbe
emphasized,however,thateveninthecaseoftheM distributionanyeventselectioncriteriathat
T,ν(cid:96)
depleteeventswithresonantWbosonsbelowtheJacobianpeakwillresultinincreasedsensitivity
totheeffectsofFSRandcanpotentiallyleadtoafailureofanaiveproductansatz.
In conclusion, simple approximations in terms of products of correction factors have to be
usedwithcareandrequireacarefulcase-by-caseinvestigationoftheirvalidity.
3.3 ApproximatingO(α α)correctionsbyleadinglogarithmicfinal-stateradiation
s
As is evident from Figs. 3 and 4, a naive product of QCD and EW correction factors (3.2)
isnotadequatetoapproximatetheNNLOQCD–EWcorrectionsforallobservables. Apromising
approachtoafactorizedapproximationforthedominantinitial–finalcorrectionscanbeobtainedby
combiningthefullNLOQCDcorrectionstovector-bosonproductionwiththeleading-logarithmic
(LL)approximationforFSR.Thebenefitinthisapproximationliesinthefactthattheinterplayof
the recoil effects from jet and photon emission is properly taken into account. On the other hand,
the LL approximation neglects certain (non-universal) finite contributions, which are, however,
suppressedwithrespecttothedominatingradiationeffects.
InthefollowingwecomparetwostandardapproachestoincludeFSRoffleptonsinLLapprox-
imation: thestructure-functionandtheparton-showerapproaches. Intheformer,generatedevents
are dressed with FSR effects by convoluting the differential cross section by a structure function
describingtheenergylossoftheleptonsbycollinearphotonemission. Sincethestructure-function
approach works with strictly collinear photon emission, by construction the impact of LL FSR is
zero for dressed leptons, where the mass-singluar logarithm of the lepton cancels by virtue of the
KLNtheorembecauseoftheinclusivetreatmentofthecollinearlepton–photonsystem. Incontrast,
photons generated through parton-shower approaches to photon radiation (see e.g. Refs. [12–14])
also receive momentum components transverse to the original lepton momentum, following the
differentialfactorizationformula,sothatthemethodisalsoapplicabletothecaseofcollinear-safe
observables,i.e.tothedressed-leptoncase. Forthispurpose,wehaveimplementedthecombination
of the exact NLO QCD prediction for vector-boson production with the simulation of FSR using
PHOTOS [15]. Since we are interested in comparing to the O(α α) corrections in our setup, we
s
onlygenerateasinglephotonemissionusingPHOTOSandusethesameinput-parameterscheme
forα (seeRef.[2]fordetails).
InFigs.5and6wecompareourbestprediction(3.1)forthefactorizableinitial–finalO(α α)
s
correctionstothecombinationofNLOQCDcorrectionswiththeapproximateFSRobtainedfrom
the structure-function approach and PHOTOS for the case of W+ production and Z production,
10