Matching factorization theorems with an inverse-error weighting

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Physics

Letters

B

www.elsevier.com/locate/physletb

Matching

factorization

theorems

with

an

inverse-error

weighting

Miguel

G. Echevarria

a

,

∗

,

Tomas Kasemets

b

,

Jean-Philippe Lansberg

c

,

Cristian Pisano

d

,

Andrea Signori

e

a_{INFN,SezionediPavia,ViaBassi6,27100 Pavia,Italy}

b_{PRISMAClusterofExcellence&MainzInstituteforTheoreticalPhysics,JohannesGutenbergUniversity,55099 Mainz,Germany} c_{IPNO,CNRS-IN2P3,Univ.Paris-Sud,UniversitéParis-Saclay,91406 OrsayCedex,France}

d_{DipartimentodiFisica,UniversitàdiCagliariandINFN,SezionediCagliari,CittadellaUniversitaria,I-09042Monserrato(CA),Italy} e_{TheoryCenter,ThomasJeffersonNationalAcceleratorFacility,12000JeffersonAvenue,NewportNews,VA 23606,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12January2018

Receivedinrevisedform27March2018 Accepted28March2018

Availableonline3April2018 Editor:A.Ringwald

We propose a new fast method to match factorization theorems applicable in different kinematical regions, such as the transverse-momentum-dependent and the collinear factorization theorems in QuantumChromodynamics. At variancewithwell-knownapproaches relying ontheirsimple addition andsubsequentsubtractionofdouble-countedcontributions,ourssimplybuildsontheirweightingusing thetheoryuncertaintiesdeducedfromthefactorizationtheoremsthemselves.Thisallowsustoestimate the unknowncompletematched cross sectionfrom aninverse-error-weighted average.The methodis simpleandprovidesanevaluationofthetheoreticaluncertaintyofthematchedcrosssectionassociated withtheuncertaintiesfromthepowercorrectionstothefactorizationtheorems(additionaluncertainties, suchas thenonperturbative ones,should beaddedforapropercomparison withexperimentaldata). Its usageisillustratedwith severalbasicexamples, suchas Z boson, W boson, H0 bosonandDrell– Yanlepton-pairproductioninhadroniccollisions,and comparedtothestate-of-the-artCollins–Soper– Sterman subtraction scheme. It is also not limited to the transverse-momentum spectrum, and can straightforwardly beextended tomatch any(un)polarized cross sectiondifferentialinothervariables, includingmulti-differentialmeasurements.

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

1. Motivation

Inprocesses with a hard scale Q and a measured transverse momentumqT,forinstancethemassandthetransverse

momen-tumofanelectroweakbosonproducedinproton–protoncollisions, theqT-differentialcrosssectioncanbeexpressedthroughtwo

dif-ferent factorization theorems. For small qT



Q , the

transverse-momentum-dependent (TMD) factorization applies and the cross section is factorized in terms of TMD parton distribution/frag-mentationfunctions(TMDsthereafter) [1–3].The evolutionofthe TMDsresumsthe largelogarithms of Q

/q

T [4–6]. Forlarge qT

∼

Q



m, withm a hadronic massofthe order of1 GeV, thereis onlyone hardscale intheprocess andthecollinearfactorization is the appropriate framework. The cross section is then written intermsof(collinear)partondistribution/fragmentationfunctions

*

Correspondingauthor.

E-mailaddress:[email protected](M.G. Echevarria).

(PDFs/FFs).InordertodescribethefullqT spectrum,theTMDand

collinearfactorizationtheoremsmust properlybematchedin the intermediateregion.

Many recent works on TMD phenomenology and extractions of TMDsfromdata did not take into account thematching with fixed-ordercollinearcalculationsforincreasingtransverse momen-tum(seee.g.Refs. [7,8]).Suchamatchingisoneofthecompelling milestonesforthenextgenerationofTMDanalysesandmore gen-erally for a thorough understanding of TMD observables [9]. In addition,it hasrecentlybeenshownthat the preciselymeasured transverse-momentum spectrumof Z boson atthe LHCdoesnot completelyagree withcollinear-based NNLOcomputations,1 hint-ing at possible higher-twist contributions at the per-cent level. Thushavingareliableestimationofthematchinguncertaintyfrom powercorrectionsisveryopportune.

1 _{See https://gsalam.web.cern.ch/gsalam/talks/repo/2016-03-SB+SLAC-SLAC}

-precision.pdf.

https://doi.org/10.1016/j.physletb.2018.03.075

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

Thiswork contributes to thiseffort by introducing a new ap-proach,whosemainfeaturesareitssimplicityanditseasyandfast implementationin phenomenological analyses (fitsand/or Monte Carloeventgenerators).In addition,this schemeprovides an au-tomatic estimate of the theoretical uncertainty associated to the matchingprocedure. All theseare crucial features in light ofthe computationaldemandsofglobalTMDanalysesandevent genera-tionforthenextgenerationofexperiments [10–13].

Aswewillshow,ityieldscompatibleresultswithother main-streamapproachesintheliterature,suchastheimprovedCollins– Soper–Sterman (CSS) scheme [14] (see also Ref. [15]),which re-finesthe originalCSSsubtractionapproach [16–19].Thelatter, in simpleterms,isbasedon addingtheTMD-basedresummed (

_W

) and collinear-based fixed-order (

Z

) results, and then subtracting the double-counted contributions (

A

). The improved CSS (iCSS) approachenforces the necessarycancellations forthe subtraction methodtowork.

Othermethodshavebeenintroducedintheframeworkof soft-collinear effective theory by using profile functions for the re-summation scales in order to obtain analogous cancellations to those in the iCSS method, see e.g. Refs. [20–23]. One can also findother schemes to matchTMD andcollinear frameworks,e.g. Refs. [24–26].

Intheschemeweintroduce,nocancellationbetweenthe TMD-based resummed contribution,

W

,and thecollinear-based fixed-order contribution,

Z

, is needed. We simply avoid the double counting(andtherewith thesubtractionof

A

) byweighting both contributionstothematchedcrosssection,withtheconditionthat theweightsadd uptounity. Thisrendersthecomputationofthe matchedcrosssectionveryeasytoimplement.Clearly,theweights cannot be arbitrary and should ensure that, in their respective domainsofapplicability,thepredictionsofbothfactorization the-oremsarerecovered.

Both factorized expressions can be seen asapproximationsof theunknown,true theory,uptocorrectionsexpressedasratiosof the relevantscales (power corrections, inthe following). In TMD factorization the power corrections scale as a power of qT

/

Q ,

whereas incollinearfactorizationthey scaleasa powerofm/qT,

up to further suppressed nonperturbative contributions [1]. We simplyimplementan estimate oftheseuncertaintiesinthe well-knownformulaofaninverse-errorweighting–orinverse-variance weighted average– oftwo measurements toobtain ourmatched predictions.Assuch,italsoautomaticallyreturnsanevaluationof thecorrespondingmatchinguncertainty.

The methodwe propose canstraightforwardly be extendedto match any (un)polarized cross section differential in other vari-ables,includingforinstanceeventshapes, multi-differential mea-surementsordoublepartonscatteringwithameasuredtransverse momentum [27].

Thispaperisorganized asfollows:inSec. 2we describeboth factorizationtheorems forlow andhightransversemomenta,and howtheyare combinedwiththeinverse-error-weightingmethod. InSec.3weshowthroughseveralexamples( Z , W ,H0 _andDrell– Yanlepton-pairproduction) howthemethodworks.InSec. 4we comparethenumericalresultstotheiCSS subtractionscheme. Fi-nally,Sec. 5gatherstheconclusions andbrieflydiscusses the ap-plicabilityofourmethodtootherprocesses.

2. Theinverse-errorweightingmethod

The mainidea behind the schemewe are proposingis touse thepowercorrectionstotheinvolvedfactorizationtheoremsin or-dertodirectlydeterminetowhichextenttheapproximationscan betrustedindifferentkinematicregions,andtousethisinorder to bridge the intermediate region obtaining the complete

spec-trum. In this context, an inverse-error weighting is conceptually thesimplestmethodonecouldthinkof.

LetushaveacloserlookattheTMDandcollinearfactorization theoremsandtheirregions ofvalidity, byconsideringacross sec-tiond

σ

differentialinatleastthetransversemomentumqT ofan

observed particle.ForqT



Q ,the TMDfactorizationcan reliably

be appliedandtheqT-differentialcrosssectioncangenericallybe

writtenas d

σ

(

qT

,

Q

)





qTQ

=

W

(

qT

,

Q

)

+



O



qT Q



a

+

O



m Q



a



d

σ

(

qT

,

Q

) ,

(1)

where

W

is theTMDapproximation ofthecrosssection d

σ

,the scale m is a hadronic mass scale on the order of 1 GeV and Q isthehardscaleintheprocess,forinstancetheinvariantmassof theproducedparticle.AsqT increases,theaccuracyoftheTMD

ap-proximationdecreases andthepowercorrectionsareincreasingly relevantuntiltheexpansionbreaksdownasqT approachesQ .

On the contrary, for large qT

∼

Q



m, the collinear

factor-ization theorem applies andthe qT-differential cross section can

genericallybewrittenas d

σ

(

qT

,

Q

)





qT∼Qm

=

Z

(

qT

,

Q

)

+

O



m qT



b d

σ

(

qT

,

Q

) ,

(2)

where

Z

isthecollinearapproximationofthefullcrosssection d

σ

.

Z

iscalculatedatafixed-orderinthestrongcouplingconstant

α

s.

For qT

∼

Q



m,

Z

is a good approximation of the full cross

section,butasqT decreasestheaccuracyofthecollinear

approxi-mationdiminishes,whichfinallybreaksdownasqT approaches m.

Armed withboth thesefactorizationtheorems, validin differ-entand(sometimes)overlappingregions,thefullqT spectrumcan

be constructed througha matchingscheme.Such a schememust make surethat the resultagrees with

W

inthe smallqT region

and with

Z

in the large qT region, and that there is a smooth

transitionintheintermediateregion.

As announced, inthispaper we introducea new scheme,the inverse-errorweighting (InEW for short),where thepower correc-tions tothefactorizationtheoremsareusedtoquantifythe trust-worthiness associated to the respective contributions, and thus employed tobuild aweightedaverage.Theresultingmatched dif-ferentialcrosssectionoverthefullrangeinqT isgivenby

d

σ

(

qT

,

Q

)

=

ω1W

(

qT

,

Q

)

+

ω2Z

(

qT

,

Q

) ,

(3)

wherethenormalizedweightsforeachofthetwotermsare

ω1

=



W

−2



W

−2

_{+ }

_Z

−2

,

ω2

=



Z

−2



W

−2

_{+ }

_Z

−2

,

(4) with



W

and



Z

being the uncertainties of both factorization theoremsgeneratedbytheirpowercorrections.Theuncertaintyon thematchedcrosssectionsimplyfollowsfromthepropagationof these(uncorrelated)theoryuncertainties:



d

σ

=

√

1



W

−2

_{+ }

_Z

−2

=



_W



_Z





2_W

+ 

2_Z d

σ

≈

_



W



Z



2_W

+ 

2_Z d

σ

,

(5)

where

{

W

,



Z

}

= {

W

,



Z

}

d

σ

, and in the last step we have replaced the unknown true cross section d

σ

by its estimated

value d

σ

. We emphasize that the uncertainty on the matched cross-section,

d

σ

,isdueonly tothematchingprocedure,which in the InEW method comes from the power-corrections. In any phenomenological application one should also include, once the matchedcross-sectionisobtained,allothersourcesofuncertainty, i.e.the onesrelatedtohigherperturbative ordersand nonpertur-bativecontributions.

FollowingEqs. (1) and (2), wenumerically implementthe un-certainties



_{W and}



Z as



_W

=



qT Q



a

+



m Q



a

,



Z

=



m qT



b



1

+

ln2



mT qT



.

(6) As an Ansatz, we have taken a

=

a andwill discuss the impact of this choice at the end of Sec. 3. In the region where qT

be-comesdifferentfromQ ,largelogarithmswillreducetheaccuracy ofthepowercountingwhichwasdone intheqT

∼

Q region.We

havethusincludedaln2

(m

T

/q

T

)

in



Z ,wherethetransversemass isdefinedasmT

=



Q2

₊

_q2

T,whichistheexpectedtypical

lead-inglogarithm inthe fixed-ordercalculations. Thislogarithm then allowsustohaveamorereliableestimationofthepower correc-tionstothecollinearresultinthewholeqT range,andnotonlyat

qT

∼

Q .

Thevaluesoftheexponentsa andb aregivenbythestrength of the power corrections and depend on the details of the pro-cess and its factorization. In the case of unpolarized processes, the smallestvalues allowed by Lorentz symmetry are a

=

2 and b

=

2, since qT is the only transverse vector that explicitly

ap-pearsin thefactorization theorems.This is consistent withwhat isfoundinRefs. [28,29] fortheTMDfactorizationtheorem,andin Refs. [30,31] for theqT-integratedcollinear factorizationtheorem,

whichshouldalsoapplyfortheqT-unintegratedwhenqT

∼

Q .We

thustakea

=

2 andb

=

2 asthedefaultchoice forthenumerical implementations.

Inordertoobtainamoreconservativeestimationofthepower corrections in the presence of large logarithmic corrections, the values of a andb could be reduced (see Sec. 13.12 in Ref. [1]). Moreover,smallervaluesareexpectedforspin-asymmetry observ-ables,whereqT isnottheonlyexplicitvector,butalsoatransverse

spinvector ST contributestothecrosssection.Eventhougha

=

1

andb

=

1 might be an extremechoice, wehave consideredit to getfirstindications onthematching uncertaintyinthepolarized cases,whichweplantostudyinmoredetailinforthcoming pub-lications.

Summarizing, we obtain the differential cross section for the fullqT spectrum asthe weightedaverage, Eq. (3), oftheTMD and

collinearapproximations

W

and

Z

withtheir weights calculated astheinverse ofthesquare ofthepower correctionsto the fac-torizedexpressions,asinEq. (6). Theuncertaintyofthematched resultautomaticallyfollowsfromEq. (5).

Letusnotethatthederivationofthepowercorrectionsinboth factorizationtheoremsisonlyvalidinandaroundtheirregionsof validity. For example, forqT

>

Q the power counting leading to

thepowercorrectionsfortheTMD crosssection breaks down.In thisregion,however,thecollinear-factorizationtheoremfully dom-inatestheresultandthematched resultcorrectlyreproduces the

Z

-termandthereby thecrosssection (ananalogouslogic applies tosmallqT).

3. Illustrationofthemethod

Inthe following,we illustrate how the methodworks forthe computationoftheqT distributionofdifferentelectroweakbosons

producedinproton–protoncollisionsattheLHCat

√

s

=

8 TeV.In particular,wewillconsiderthefollowingprocesses:

•

Z

/W boson

production(Sec.3.1)

•

Drell–Yan(DY)lepton-pairproduction(Sec.3.2)

•

H0bosonproduction(Sec.3.3).

These processesare sensitive to either quark TMDs( Z

/W boson

and DY production) or gluon TMDs (H0 boson production), and allowustoillustratetheimplementationofthematchingscheme fromlowtohighvaluesofthehardscale.

The cross sections differential with respect to the transverse momentumqT of Z/W bosonandDrell–Yanproductionhavebeen

computedusingthepubliccode DYqT2 [32,33].ForH0 boson pro-ductionwehaveusedthepubliccode HqT3_[34].

We have worked with the highest perturbative accuracy im-plemented in DYqT and HqT: NNLL(next-to-next-to-leading log-arithmic) accuracyintheresummed contribution

W

(i.e.



cusp

∼

O

(

α

3

s

))

andNLO(next-to-leadingorder)corrections(i.e.

O

(

α

s2

))

at

largeqT forthefixed-ordercontribution

Z

.ForcollinearPDFswe

haveusedtheNNPDF3.0setatNNLOwith

α

s

(M

Z

)

=

0.118 [35].

The treatment of the different bT regions (where bT is the

Fourier-conjugated variable to the observed transverse momen-tum qT) is identical in both HqT and DYqT. The large bT region

istreatedwiththe so-calledcomplexbT (or minimal)prescription,

which avoids the Landau pole in the coupling constant by de-formingtheintegrationcontourinthecomplexplane [36,37].The smallbT region,instead,istreatedbyreplacingthelog(Q2b2T

)

with

log(Q2b2T

+

1)[38,39],avoidingunjustified higher-order

contribu-tions. This is analogous to introducing a lower cutoff bmin in bT

space [14,40–42].Wenotethatthiscutoffiscrucialinorderto re-covertheintegratedcollinearfactorizationresultupon integration overthetransversemomentum.

In DYqT, the nonperturbative TMD part in the resummed term is implemented as a simple Gaussian smearing factor in bT space [32,33] exp(SN P

)

=

exp(

−

gN Pb2T

).

Since we are

inter-ested in processes at different energy scales, we have included a logarithmic dependence of gN P on the invariant mass Q of

the produced state (see e.g. Ref. [43]) to mimic more realistic values: gN P

(Q

)

=

g0N Pln(Q2

/

Q02

)

with Q0

=

1 GeV. Thus, we can write gN P

(Q

)

=

gN P

(M

Z

)ln

Q2



_/ln

_M2 Z



. In HqT an analo-gous smearing factor was introduced. For the gluon TMDs there is significantly lessexperimental input andthus phenomenologi-cal information(seehoweverRef. [44])andwe haverescaled the nonperturbative parameter for quark TMDs by a Casimir scaling factor CA

/C

F (see Sec. 3.3), where CA

=

Nc, CF

= (

Nc2

−

1)/2Nc

andNc

=

3 isthenumberofcolors.Letushowevernotethatsuch

nonperturbative factors, which would be essential for a proper comparison with data, are not involved in the matching proce-dure.

DYqT_{and HqT allowedustoseparatelycomputethecross} sec-tion atlow qT (

W

) andathighqT (

Z

) fromwhichwe have

im-plementedthematchingfollowingourInEW method.Thesecodes alsoallowedustocomputetheasymptoticlimit [14,18,19] ofthe resummedcontribution(

A

)whichwewilluseforthecomparison withiCSS method.

Theuncertaintiesinthefollowingsectionswillpurely befrom theInEW matchingscheme,namelyinducedby theestimationof thepowercorrections.Additionaluncertaintiesduetoscale varia-tions, collinear-partondistributions andTMDnonperturbative un-certainties should be added fora fair comparisonwith data.We

2 _{http://pcteserver.mi.infn.it/~ferrera/research.html.} 3 _{http://theory.fi.infn.it/grazzini/codes.html.}

Fig. 1. TheresummedtermW(yellowcurve),thefixed-ordertermZ(greencurve),andthematchedcrosssectionintheInEW approach(blueband)forZ bosonproduction (top–left),W+ bosonproduction(top–center),H0 _{bosonproduction(top–right),Drell–Yanlepton-pairproductionwith Q=4 GeV (bottom–left), Q=12 GeV (bottom–}

center),andQ=20 GeV (bottom–right).Allprocessesareinitiatedbyproton–protoncollisionswith√s=8 TeV.Theuncertaintyonthematchedcrosssectionisonlydue tothematchingscheme,i.e.includingpower-correctionuncertainties,andnoothereffectsareadded,suchastheperturbative-scalevariationsandthenonperturbative con-tributions.LowerpanelsquantifythedeviationoftheW- andZ-termswithrespecttothematchedcrosssection,aswellasitsmatchinguncertainty.(Forinterpretationof thecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle)

stress that this remarkwould apply to any(un)matched compu-tations. We leave for a future publication the phenomenological studyofthe InEW scheme, wherethe uncertainties on the func-tionalformandtheparametersofthenonperturbativecontribution willbeconsidered(seee.g.Refs. [7,8] forrecentphenomenological works).

3.1. Z

/W boson

production

Inthis section we study Z/W bosonproduction. We work in thenarrow widthapproximation andincludethe branching ratio intotwoleptons [32,33].

In Fig. 1 (top–left) we show the full transverse-momentum spectrum calculated with our InEW matching of the

W

and

Z

-terms for Z -boson production. The nonperturbative parameter we used is gN P

(M

Z

)

=

0.8 GeV2 [37]. The central curve

corre-spondstod

σ

andthebandtoits variationby

±

d

σ

(seeEq. (3) and Eq. (5)). We also show the

W

- and

Z

-terms individually. Thelower panelsinFig.1quantifythe deviation4 ofthe

W

- and

Z

-termswithrespecttothematchedcross section,aswellasits matchinguncertainty.

4 _{Bydeviation wemeanthepercentagedifferencebetweentwogivencurves.For}

theW-termweplot100· (W−dσ)/dσ,andsimilarlyfortherest.

The

Z

-termis ill behaved towards small values of the trans-verse momentum due to the presence of the large logarithms in Q

/q

T, while the

W

-term tends towards negative values for

large qT. There is a quite broad intermediate region where both

results aresimilar, andwhereboth factorizationtheoremsare on relativelystableground.Thismakesthematchingbetweenthetwo theoremsparticularlysimple,andwellbehaved.

Thecrosssection matchedintheInEW schemefollowsthe re-summed

W

-term up to qT

∼

15 GeV and then approaches the

fixed-order

Z

-term.Theuncertaintyfromthepowercorrectionsis smallinthelarge andvery smallqT regions,butincreasesinthe

region around the value ofqT where



_W

= 

Z (i.e. where both weightsarecloseto 1₂).

TheresultsforW+productionareshowninFig.1(top–center). The scale-dependent nonperturbative parameter is modified to gN P

(M

W

)

0.78 GeV2bythechangeofthehardscaletothemass

oftheW boson(MW

=

80.385 GeV).Theresultsforthematched

cross sectioncloselyresemble thoseforthe Z boson,which isto be expected since both processes have a similar hard scale and probe quark andantiquark distributions. Thetransition point be-tween the

W

-termandthe

Z

-term hasmoved down to slightly lower qT,andtheuncertaintyisa littlelarger.TheresultforW−

production isvery similar,withjusta differentnormalizationfor thedifferentialcrosssection.

3.2.Drell–Yanprocess

In this section we study Drell–Yanlepton-pair production,or more precisely, virtual-photon (

γ

_{) production. The}

nonpertur-bative parameters are now given by gN P

(4 GeV)

0.25 GeV2,

gN P

(12 GeV)

0.44 GeV2 and gN P

(20 GeV)

0.53 GeV2.The

re-sultsforthe matchedcross section forDY production areshown fortheinvariantmassesQ

=

4,12,20 GeV inFig.1(bottom).The valuesarechosentocomplementtheresultsfortheheavy vector-bosonand H0 boson crosssections,andtodemonstratehow the methodperformsatdifferentscales.

Letusstart ourdiscussion fromthe lowestscale, Q

=

4 GeV. Thisvalueischosentodemonstratewhathappenswhenthehard scaleis verylow,andwhen theintermediate region,where both TMDandcollinearfactorizationsarevalid,collapses.Thematched crosssection followsthe TMD resultup to larger fractions of Q thanitdidforheavyvector-bosonproduction,startingtotend to-wardsthe collinearresult aroundqT

∼

Q

/2.

Forsuch low scales,

powercorrectionsareofcourselikelytobelarge.Thisisnicely re-flectedbytheuncertaintybandoftheInEW matchedresultwhich reachesmaximumvaluesofaround30%.Wenotethatsignificantly loweringthecenterofmassenergydoesnotchangethequalitative discussionofthematchingmethod.

Increasingtheinvariantmassoftheproducedboson,the uncer-tainty ofthe InEW schemedecreases and thetransitionbetween thetwofactorizationtheoremsmovestowardssmallerfractionsof qT

/Q .

Theregionwheretheresultsofboththeoremsarerelevant

alsooccupiesasmallerandsmallerportionoftheqT spectrum.At

Q

=

12 GeV, the maximal uncertainty has decreased below 20% and,at Q

=

20 GeV,islessthan10%.

3.3. H0_{bosonproduction}

Inthis section we study H0 bosonproduction. The heavy-top effectivetheoryisusedtointegrateoutthetopquark,resultingin adirectcouplingbetweengluonsandtheH0 boson.

Unlikethe previous processes, H0 production directly probes gluon TMDs (see e.g. Refs. [44–57]). There is much less phe-nomenologyandthereforeknowledgeaboutgluonTMDsthanfor quarks.As alreadymentionedin Sec.3, inorderto obtaina rea-sonable value for the nonperturbative parameter we use Casimir scaling.This results in gg_{N P}

(125 GeV)

= (

CA

/C

F

)g

N P

(125 GeV)

1.93 GeV2.

Fig.1(top–right)showsthematchedcrosssectionintheInEW scheme.It followsthe

W

-termup to qT

∼

15 GeV andthen

ap-proaches the

Z

-term. The uncertainty band is narrow, as power correctionsarestronglysuppressedintheentirespectrum.

The small size of the power corrections in combinationwith thelargedifferencebetweenthetwofactorizedapproximationsof thecrosssectionisachallengeforthematchinginthe intermedi-ateregion.AtqT

∼

15 GeV,thepowercorrections



_{W and}



Z are bothbelow0.05,butthe

Z

-termis50%largerthanthe

W

.Thisis, however,no longersurprisingwhentakingintoaccount thelarge uncertainty associated to the H0 boson transverse-momentum spectrum coming from the scale variations [23]. It is therefore likely that higher-order corrections will bring the collinear and TMDresultsclosertoeachother,resultinginasmoothermatching. Finally,let usnote thatwe did notobserve anyrelevant vari-ations of the central value of the matched cross section when loweringtheexponentsfroma

=

b

=

2 toa

=

b

=

1.However, as expected,thematchinguncertaintysignificantlygrows.ForZ ,W+ andH0_{bosonproductioncases,theuncertaintyatitsmaximumis} inflated7–8times,reaching

∼

15% atqT

∼

15 GeV andremaining

largerthan5% fromroughly4 to40 GeV.FortheDrell–Yancase, whosetransverse-spin-asymmetrystudyis ahot topicwithin the

TMDcommunity,theuncertaintyratherinflatesbyafactorof2to 3dependingonthelepton-pairmass.

On the other hand, the matching is quite stable under varia-tionsoftheexponentacomparedtoa.Fora

=

a,theuncertainty associated witha dominates down to very low qT

∼

1 GeV, and

therefore dominates the region where both TMD and collinear results are relevant. Lowering a leads to a slightly larger uncer-taintyinthelowqT regionandcan,forlow Q Drell–Yan,shiftthe

transitionbetweentheTMD andcollinearresultstowards slightly lower qT.However,wewouldliketonoteherethattheexponents

a,a andb canbefixedforagivenprocessbytheorderatwhich thedifferentpowercorrectionscontribute.

4. ComparisontoCSSsubtraction

In thissection, we compare the matched-cross-section results in the InEW scheme with the results in the iCSS subtraction scheme of Ref. [14]. We therefore briefly introduce the features oftheiCSS methodwhichareofrelevanceforourcomparison,and refertoRef. [14] foramoredetaileddiscussion.

ThewidelyusedCSSmethod [16–19] allowsforamatchingof theTMD result(

_W

) andthefixed-orderresult(

_Z

) inanadditive way.Doublecountingisavoidedbythesubtractionofthe asymp-totic term(

A

), i.e.the fixed-order expansion of the perturbative resultof

W

.For applicationsof the method inprocesses witha lowhard-scale,see,e.g.,Ref. [58] forSemi-InclusiveDeep-Inelastic Scattering (SIDIS) and Chap. 8 in Ref. [40] for

η

b production in

proton–proton collisions. Applicationsin processes with a higher hardscalecanbefoundin,e.g.,Refs. [40,59–61].

Themethod,althoughsuccessful,runsintodifficultiesatsmall qT,due to incomplete cancellations betweenthe fixed-order and

theasymptoticresults,andalsoatlargeqT,duetoincomplete

can-cellations betweenthe resummed and the asymptotic results.At low qT theproblemsare especiallymanifestwhen thehardscale

Qisnotlarge,namelywhenthereislittleornooverlapbetween the regions where the TMD and collinear factorizationtheorems arevalid [14,58].

Recently, a solution to these issues has been proposed in Ref. [14], the iCSS method.In order toenforce the required can-cellations, the different termsin the crosssection are multiplied bycutofffunctions,dampingthemoutsidetheir regionofvalidity. This solves the problem of the incomplete cancellations, but in-troducesa dependenceboth onthe functionalform ofthecutoff functionsandon thepoint in qT whereone switches on andoff

thedifferentcontributions.

ThecrosssectionintheiCSS methodiswrittenas

d

σ

(

qT

,

Q

)

=

W

iCSS

(

qT

,

Q

)

+

Y

iCSS

(

qT

,

Q

) ,

(7)

where

W

iCSS

(

qT

,

Q

)

=

W

(

qT

,

Q

)

W

(

qT

,

Q

;

η

,

r

) ,

Y

iCSS

(

qT

,

Q

)

=

Z

iCSS

(

qT

,

Q

)

−

A

iCSS

(

qT

,

Q

) ,

Z

iCSS

(

qT

,

Q

)

=

Z

(

qT

,

Q

)

Z

(

qT

; λ,

s

) ,

A

iCSS

(

qT

,

Q

)

=

A

(

qT

,

Q

)

W

(

qT

,

Q

;

η

,

r

)

Z

(

qT

; λ,

s

) ,

(8)

withthecutofffunctions

W

(

qT

,

Q

;

η

,

r

)

=

exp



−



qT

η

Q



r

,

Z

(

qT

; λ,

s

)

=

1

−

exp



−



qT

λ



s

.

(9)

The parameters

{

η

,

λ

}

control the value ofqT around which the

Fig. 2. Fromlefttorightandtoptobottom,comparisonbetweentheInEW andtheiCSS schemesfor:Z bosonproduction,W+bosonproduction,H0_{bosonproductionand}

Drell–Yanlepton-pairproductionatQ=4,12,20 GeV.

thesecutoffs.5_{Insimpleterms,thedampingfunction}

_{W switches}

offboththe

W

-termandthe

A

-termatlargeqT,whilethe

damp-ingfunction

Z switches

offboth the

Z

-termandthe

A

-termat smallqT.ForintermediateqT,thethreetermsarekept.

The values for these four parameters given in Ref. [14] are

{

η

,

r

}

= {

1/3,8

}

and

{λ,

s

}

= {

2/3 GeV,4

}

.We havechosen a dif-ferentdefaultvaluefor

λ

(λ

=

1 GeV)forswitchingoffthe

Z

and

A

towardslowqT values,inorderforthecrosssectionnottostart

deviating fromthe

W

towards too low qT. The variations ofthe

parameterswe performhoweverincludealsothedefaultvalue of Ref. [14].

To be able to compare with the iCSS approach we need to constructawayto estimatethe matchinguncertaintyintheiCSS scheme,bothduetothepowercorrectionsandtotheparameters inthematchingscheme.Todoso,we notethat thecrosssection intheiCSS methodcanbewrittenas:

d

σ

(

qT

,

Q

)

=

⎧

⎪

⎨

⎪

⎩

W

+ 

Wd

σ

,

qT

 λ

W

+

Z

−

A

+ 

W



Zd

σ

,

λ



qT



η

Q

Z

+ 

Zd

σ

,

qT



η

Q

,

(10) sincethedampingfunctions

W and

Z are

devised as(almost) step functions. At small qT, since the cross section is effectively

givenbythe

W

-term, thepowercounting(relative)errorwillbe



_{W (see}Eq. (1)).AtlargeqT,thecrosssectioniseffectivelygiven

bythe

Z

-term,andthepowercounting(relative)errorwillbe



_Z (seeEq. (2)).Intheintermediateregionthecrosssection isgiven by thesubtractionof thedouble-counted contributions, andthus the powercounting (relative) erroris



_W



Z [14]. We therefore estimatetheerrorfromsubleadingpowersintheiCSS method(as afunctionofqT)as 1 d

σ



d

σ





iCSS

= 

W



1

−

Z



+ 

W



Z

W

Z

+ 

Z



1

−

W



.

(11)

5 _{TheauthorsofRef. [14] alsointroduceasmall-b cutoff(b}

min prescription)in

theW-term,whichhasaneffectaswellinthewaytheasymptoticA-termis cal-culated.

Inadditionto thisuncertaintyfromthepowercorrections, we needtoconsidertheuncertaintythatcomesfromthevariationof the matchingparameters in theiCSS approach.6 In particular, we takethedefaultvalues

η

=

1/3 and

λ

=

1 GeV (differentfromthe oneproposedinRef. [14])andvarythemby50%,i.e.

η

∈ [

1/6,1/2

]

and

λ

∈ [

0.5,1.5

]

GeV.Wekeeptheexponents

{

r,s

}

constant,since they haveto be large enough to give almost step functions, and thentheirvariationdoesnothaveanyrelevantimpact.

Intheintermediateregion,thismethodhasa potential advan-tageover theInEW in termsoftheformal powercounting uncer-tainty, i.e.



_W



Z

/(

2_W

+ 

Z2

)

1/2 forInEW compared to



W



Z for iCSS (where no variation of the matching parameters is in-cluded [14] though). This is of value, in particular, in high-scale processes suchas Z bosonproduction,where thereisan overlap region wheretheapproximationsinboth ofthetwofactorization theorems are appropriate. When the hard scale ofthe process is reduced, theoverlap ofthetwofactorizationtheorems decreases. As thishappens, the subtractionmethodno longerbenefits from the power counting advantage, since the uncertainty from the matchingparametersislarge,aswenowdemonstrate.

InFig.2,weshowthenumericaldifferencesbetweentheInEW andtheiCSS schemesfor Z bosonproduction,W+ boson produc-tion, H0 _{boson production,and Drell–Yanlepton-pair production} at Q

=

4,12,20 GeV.Thetotal uncertaintyfortheiCSS approach shown in Fig. 2 is obtained as the envelope of the uncertainty bandsd

σ

±

d

σ

,whereeachbandcorrespondstooneofthe men-tioned choicesofthe matchingparameters

{

η

,

λ

}

.We againnote that the uncertaintiesshown inFig.2 areonly duetothe differ-ent matching schemes, and do not include other effects such as the perturbative-scale variations and the nonperturbative contri-butions,whicharecommontoboth.

Starting withthe Z and W boson productionand comparing the InEW resultstothosein theiCSS scheme,we cannotice that wheretheuncertaintyintheInEW methodisthelargest,theiCSS scheme produces a significantly smaller uncertainty. This is pre-cisely dueto the reduction of thepower corrections obtainedin theintermediate regionwhensubtractingtheasymptoticterm

A

. At the scale of the Z boson mass,there is a significant overlap of the two regions where the two factorization theorems apply.

6 _{Theseshouldnotbeconfusedwiththeuncertaintiesfromtheperturbative-scale}

However, we can also see that as we approach the regions of thematchingpointsbetweenthelowandintermediatetransverse momentum,orbetweentheintermediateandhightransverse mo-mentum,thechoiceofthematchingparametershasalargeimpact on the results. Unlike the InEW scheme, the iCSS follows more closelythe

W

-term up to larger values of qT,but the extent to

whichthisholdstruehasastrongdependenceonthevalueofthe largestmatchingpoint. Thisis clearly reflectedin thesize ofthe uncertaintyinthisregionoftransversemomentum.Forboth pro-cesses, the uncertainty band for the InEW method is symmetric aroundthe centralvalue, whilethe estimationoftheuncertainty fortheiCSS isasymmetric,originatingmainlyfromthevariationof thematchingparameters.

ForDY at Q

=

4 GeV, the iCSS scheme runs into difficulties. Thereisnospaceleftfortheintermediateregion,andthe match-ingpoints

λ

and

η

Q areveryclosetoeach other.Thisleads toa verylarge uncertainty.This isnot surprisingconsideringthat the main advantage of the method is in the power counting uncer-taintyinthe intermediateregion.Moreover, forourchoiceofthe defaultvaluesfortheparameters,thecentralcurveintheiCSS lies farawayfromthecentralcurveintheInEW schemeatlowand in-termediateqT values.ThecentralcurveintheiCSS schememoves

fromtheresummedtothefixed-orderresultatalowertransverse momentumthanthecentralcurveintheInEW scheme,the oppo-sitetowhatwecouldseeinZ

/W boson

production.

Let us now compare the InEW and iCSS schemes at Q

=

12,20 GeV,wherethereismorespacefortheintermediateregion andtheuncertaintyintheiCSS schemeimproves.The iCSS uncer-taintyatthelargertransverse-momentumvaluesisdominatedby thevariationofthematchingpointandremainsofsimilarsize re-gardlessofthescale.Asmaller(larger)variationoftheassociated parameterwould ofcourse lead to a smaller (larger)estimate of theassociateduncertainty.

ForH0bosonproduction,theadvantageintheintermediate re-gionoftheiCSS schemeisclearlyvisible,withaverysmall uncer-taintybandforlowqT.Thelargerdependenceonthechoiceofthe

uppermatchingpoint ishoweverstill present.Bothschemes pro-duceresultswhichare clearlyoutsidetheir uncertaintybandsfor a large rangeof intermediate transverse momenta. At thispoint, weemphasizethatfor H0 _{productionthereisalargeuncertainty} coming from the scale variations [23]. Therefore, the difference betweenthetwomethodswillbedrownedintheother uncertain-ties, given the currently available perturbative accuracy. At very lowqT the iCSS rapidlystarts todeviate fromtheresummed

cal-culation,but thisis difficult to interpret. Changing the valuesof the matching parameter associated with the transition between thelowandintermediateregionwouldfixthisproblem.Adetailed optimizationoftheparameterchoicesintheiCSS schemeis, how-ever,obviouslyoutsidethescopeofthepresentwork.

5. Conclusions

The implementation of the matching between the TMD and collinear factorizationtheorems, together with a reliable estima-tionofits uncertaintyfrompowercorrections,isoneofthe com-pelling milestones for the next generation of phenomenological analysesofqT-spectra.Thisworkcontributestosuch aneffort by

introducing a new matching scheme: the inverse-errorweighting (InEW).

FromtheexpectedscalingofthepowercorrectionsfortheTMD andcollinearfactorizationtheorems,webuildamatchedcross sec-tion via a weighted average, where the normalized weights are givenbytheinverseofthe(squareofthe)powercorrections.

IntheInEW scheme,nocancellationofdouble-counted contri-butionsisneeded,sincetheresummedandfixed-orderresultsare

averaged, andnotsummed.Thismakes theimplementation ofthe cross-section matching in phenomenological analyses faster and more transparent, an important feature in light of the demands of globalTMD analyses. Moreover, theInEW scheme yields com-patibleresultswithothermainstreamapproachesintheliterature, suchastheimprovedCSSscheme.

We haveillustrated theapplication ofthe InEW method with the qT-spectra of Z boson, W boson, H0 boson and Drell–Yan

lepton-pairproductionattheLHC.However,theInEW schemecan be applied ina straightforward manner toany observable where a resummedanda fixed-orderfactorizationtheorems need tobe matched in order to describe the full spectrum of a given vari-able,suchastheqT-spectrawithpolarizedbeams,eventshapesor

multi-differentialobservables.Weleaveforthefuturethestudyof processessensitiveto(un)polarizedTMD fragmentationfunctions, such as e+e−

→

h1h2X and SIDIS, and low-scale processes sen-sitive to(un)polarized gluonTMDs,such aspseudoscalar quarko-nia producedata futurefixed-targetexperimentatthe LHC (AF-TER@LHC [12,49,62,63])orevenattheLHC [64–69],andthe pro-ductionofapairof J/ψ [44].

Acknowledgements

WethankA.Bacchetta,D.Boer,J.C.Collins,P.J.Mulders,J.Qiu, T. Rogers, L. Massacrier, H.-S. Shao, J.X. Wang for useful discus-sions. MGE issupported by the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 research and innova-tionprogram(grantagreementNo. 647981,3DSPIN).The workof JPL is supported in part by the French CNRS via the LIA FCPPL (Quarkonium4AFTER) and the IN2P3 project TMD@NLO and via the COPIN-IN2P3 agreement. AS acknowledges support from U.S. DepartmentofEnergycontractDE-AC05-06OR23177,underwhich JeffersonScience Associates, LLC,manages andoperates Jefferson Lab.TK acknowledges supportfromthe AlexandervonHumboldt Foundation andthe European Community underthe “Ideas” pro-gramQWORK(contract320389).

References

[1] J.Collins,FoundationsofPerturbativeQCD,CambridgeUniversityPress,2013,

http://www.cambridge.org/de/knowledge/isbn/item5756723.

[2]M.G.Echevarria, A.Idilbi,I.Scimemi,Factorizationtheoremfor Drell–Yanat lowqT andtransversemomentumdistributionson-the-light-cone,J.High

En-ergyPhys.07(2012)002,arXiv:1111.4996 [hep-ph].

[3]M.G.Echevarria,A.Idilbi,I.Scimemi,Softandcollinearfactorizationand trans-verse momentumdependentparton distributionfunctions,Phys. Lett.B726 (2013)795–801,arXiv:1211.1947 [hep-ph].

[4]S.M.Aybat,T.C.Rogers,TMDpartondistributionandfragmentationfunctions withQCDevolution,Phys.Rev.D83(2011)114042,arXiv:1101.5057 [hep-ph].

[5]M.G.Echevarria,A.Idilbi,A.Schaefer,I.Scimemi,Model-independentevolution oftransversemomentumdependentdistributionfunctions (TMDs)at NNLL, Eur.Phys.J.C73 (12)(2013)2636,arXiv:1208.1281 [hep-ph].

[6]M.G.Echevarria,A.Idilbi,I.Scimemi,UnifiedtreatmentoftheQCDevolutionof all(un-)polarizedtransversemomentumdependentfunctions:Collinsfunction asastudycase,Phys.Rev.D90 (1)(2014)014003,arXiv:1402.0869 [hep-ph].

[7]I.Scimemi,A.Vladimirov,AnalysisofvectorbosonproductionwithinTMD fac-torization,arXiv:1706.01473 [hep-ph].

[8]A.Bacchetta,F.Delcarro,C.Pisano,M.Radici,A.Signori,Extractionofpartonic transversemomentumdistributionsfromsemi-inclusivedeep-inelastic scatter-ing, Drell–Yanand Z-bosonproduction,J. HighEnergyPhys. 06(2017)081, arXiv:1703.10157 [hep-ph].

[9]R. Angeles-Martinez,etal., TransverseMomentumDependent(TMD) parton distributionfunctions:statusandprospects,ActaPhys.Pol.B46 (12)(2015) 2501–2534,arXiv:1507.05267 [hep-ph].

[10]J.Dudek,etal.,Physicsopportunitieswiththe12GeVupgradeatJeffersonLab, Eur.Phys.J.A48(2012)187,arXiv:1208.1244 [hep-ex].

[11]A.Accardi,etal.,ElectronIonCollider:thenextQCDfrontier,Eur.Phys.J.A 52 (9)(2016)268,arXiv:1212.1701 [nucl-ex].

[12]S.J.Brodsky,F.Fleuret,C.Hadjidakis,J.P.Lansberg,Physicsopportunitiesofa fixed-targetexperimentusingtheLHCbeams,Phys.Rep.522(2013)239–255, arXiv:1202.6585 [hep-ph].

[13]A.Blondel,F.Zimmermann,Ahighluminositye+e−colliderintheLHCtunnel tostudytheHiggsBoson,arXiv:1112.2518 [hep-ex].

[14]J.Collins,L.Gamberg,A.Prokudin,T.C.Rogers,N.Sato,B.Wang,Relating trans-versemomentumdependentandcollinearfactorizationtheoremsina general-izedformalism,Phys.Rev.D94 (3)(2016)034014,arXiv:1605.00671 [hep-ph].

[15]L. Gamberg, A. Metz, D. Pitonyak, A. Prokudin, Connections between collinear andtransverse-momentum-dependentpolarizedobservableswithin theCollins–Soper–Stermanformalism,arXiv:1712.08116 [hep-ph].

[16]J.C.Collins,D.E.Soper,Back-to-backjetsinQCD,Nucl.Phys.B193(1981)381; Erratum:Nucl.Phys.B213(1983)545.

[17]J.C.Collins,D.E.Soper,Partondistributionanddecayfunctions,Nucl.Phys.B 194(1982)445–492.

[18]J.C. Collins, D.E.Soper, G.F. Sterman, Transverse momentumdistribution in Drell–Yan pair and W and Z boson production, Nucl. Phys. B 250(1985) 199–224.

[19]P.B.Arnold,R.P.Kauffman,WandZproductionatnext-to-leadingorder:from largeq(t)tosmall,Nucl.Phys.B349(1991)381–413.

[20]Z.Ligeti,I.W.Stewart,F.J.Tackmann,Treatingtheb quarkdistributionfunction with reliableuncertainties,Phys. Rev.D78(2008)114014, arXiv:0807.1926 [hep-ph].

[21]R.Abbate,M.Fickinger,A.H.Hoang,V.Mateu,I.W.Stewart,ThrustatN3_LLwith

powercorrectionsandaprecisionglobalfitforαs(mZ),Phys.Rev.D83(2011)

074021,arXiv:1006.3080 [hep-ph].

[22]I.W.Stewart,F.J.Tackmann,J.R.Walsh,S.Zuberi,JetpT resummationinHiggs

productionatN N LL+N N L O ,Phys.Rev.D89 (5)(2014)054001,arXiv:1307. 1808 [hep-ph].

[23]D.Neill,I.Z.Rothstein,V.Vaidya,TheHiggstransversemomentumdistribution atNNLLanditstheoreticalerrors,J.HighEnergyPhys.12(2015)097,arXiv: 1503.00005 [hep-ph].

[24]E.L.Berger,J.-w.Qiu,Y.-l.Wang,Transversemomentumdistributionofυ pro-ductioninhadroniccollisions,Phys.Rev.D71(2005)034007,arXiv:hep-ph/ 0404158.

[25]P.M.Nadolsky,N.Kidonakis,F.I.Olness,C.P.Yuan,Resummationoftransverse momentumandmasslogarithmsinDISheavyquarkproduction,Phys.Rev.D 67(2003)074015,arXiv:hep-ph/0210082.

[26]P.M.Nadolsky,D.R.Stump,C.P.Yuan,SemiinclusivehadronproductionatHERA: theeffectofQCDgluonresummation,Phys.Rev.D61(2000)014003,arXiv: hep-ph/9906280;Erratum:Phys.Rev.D64(2001)059903.

[27]M.G.A.Buffing,M.Diehl,T.Kasemets,Transversemomentumindoubleparton scattering:factorisation,evolutionandmatching,arXiv:1708.03528 [hep-ph].

[28]I. Balitsky, A. Tarasov,Higher-twist corrections togluon TMD factorization, J. HighEnergyPhys.07(2017)095,arXiv:1706.01415 [hep-ph].

[29]I.Balitsky,A.Tarasov,PowercorrectionstoTMDfactorizationforZ-boson pro-duction,arXiv:1712.09389 [hep-ph].

[30]J.-w. Qiu,G.F. Sterman,Power corrections inhadronicscattering.1. Leading 1/Q2 _{corrections to the Drell–Yan cross-section,Nucl. Phys. B 353(1991)}

105–136.

[31]J.-w.Qiu,G.F.Sterman,Powercorrectionstohadronicscattering.2. Factoriza-tion,Nucl.Phys.B353(1991)137–164.

[32]G.Bozzi,S.Catani,G.Ferrera,D.deFlorian,M.Grazzini,Transverse-momentum resummation:aperturbativestudyofZproductionattheTevatron,Nucl.Phys. B815(2009)174–197,arXiv:0812.2862 [hep-ph].

[33]G.Bozzi,S.Catani,G.Ferrera,D.deFlorian,M.Grazzini,ProductionofDrell– Yanleptonpairsinhadroncollisions:transverse-momentumresummationat next-to-next-to-leadinglogarithmicaccuracy,Phys.Lett.B696(2011)207–213, arXiv:1007.2351 [hep-ph].

[34]D. deFlorian,G. Ferrera,M. Grazzini,D. Tommasini,Transverse-momentum resummation: Higgsbosonproduction atthe Tevatronandthe LHC,J. High EnergyPhys.11(2011)064,arXiv:1109.2109 [hep-ph].

[35]R.D.Ball,etal.,NNPDFCollaboration,PartondistributionsfortheLHCRunII, J. HighEnergyPhys.04(2015)040,arXiv:1410.8849 [hep-ph].

[36]E. Laenen, G.F. Sterman, W. Vogelsang, Higher order QCD corrections in promptphotonproduction,Phys.Rev.Lett.84(2000)4296–4299,arXiv:hep -ph/0002078.

[37]A.Kulesza,G.F.Sterman,W.Vogelsang,Jointresummationinelectroweak bo-sonproduction,Phys.Rev.D66(2002)014011,arXiv:hep-ph/0202251.

[38]G.Parisi,R.Petronzio,Smalltransversemomentumdistributionsinhard pro-cesses,Nucl.Phys.B154(1979)427–440.

[39]G.Bozzi,S.Catani,D.deFlorian,M.Grazzini,Transverse-momentum resumma-tionandthespectrumoftheHiggsbosonattheLHC,Nucl.Phys.B737(2006) 73–120,arXiv:hep-ph/0508068.

[40] A. Signori, Flavor and Evolution Effects in TMD Phenomenology,PhD the-sis,VrijeU.,Amsterdam,2016,https://userweb.jlab.org/~asignori/research/PhD_ thesis_Andrea.pdf.

[41]D.Boer,W.J.denDunnen,TMDevolutionandtheHiggstransversemomentum distribution,Nucl.Phys.B886(2014)421–435,arXiv:1404.6753 [hep-ph].

[42]D. Boer, Linearly polarized gluon effects in unpolarized collisions, PoS QCDEV2015(2015)023,arXiv:1510.05915 [hep-ph].

[43]A.Bacchetta,M.G.Echevarria,P.J.G.Mulders,M.Radici,A.Signori,Effectsof TMDevolutionandpartonicflavorone+e−annihilationintohadrons,J.High EnergyPhys.11(2015)076,arXiv:1508.00402 [hep-ph].

[44]J.-P.Lansberg,C. Pisano,F.Scarpa, M. Schlegel, Pinningdownthe linearly-polarizedgluonsinsideunpolarizedprotonsusingquarkonium-pairproduction attheLHC,arXiv:1710.01684 [hep-ph].

[45]D.Boer,S.J.Brodsky,P.J.Mulders,C.Pisano,Directprobesoflinearlypolarized gluonsinsideunpolarizedhadrons,Phys.Rev.Lett.106(2011)132001,arXiv: 1011.4225 [hep-ph].

[46]J.-W.Qiu,M.Schlegel,W.Vogelsang,Probinggluonicspin–orbitcorrelationsin photonpairproduction,Phys.Rev.Lett.107(2011)062001,arXiv:1103.3861 [hep-ph].

[47]D.Boer,W.J.denDunnen,C.Pisano,M.Schlegel,W.Vogelsang,Linearly polar-izedgluonsandtheHiggstransversemomentumdistribution,Phys.Rev.Lett. 108(2012)032002,arXiv:1109.1444 [hep-ph].

[48]J.P.Ma,J.X.Wang,S.Zhao,Transversemomentumdependentfactorizationfor quarkoniumproduction at low transverse momentum, Phys. Rev.D 88 (1) (2013)014027,arXiv:1211.7144 [hep-ph].

[49]D.Boer,C.Pisano,Polarizedgluonstudieswithcharmoniumandbottomonium atLHCbandAFTER,Phys.Rev.D86(2012)094007,arXiv:1208.3642 [hep-ph].

[50]D.Boer,C.Pisano,ImpactofgluonpolarizationonHiggsbosonplusjet produc-tionattheLHC,Phys.Rev.D91 (7)(2015)074024,arXiv:1412.5556 [hep-ph].

[51]W.J.denDunnen,J.P.Lansberg,C.Pisano,M.Schlegel,Accessingthetransverse dynamicsandpolarizationofgluonsinsidetheprotonattheLHC,Phys.Rev. Lett.112(2014)212001,arXiv:1401.7611 [hep-ph].

[52]M.G.Echevarria,T.Kasemets,P.J.Mulders,C.Pisano,QCDevolutionof (un)po-larizedgluonTMDPDFs and the HiggsqT-distribution,J. HighEnergy Phys.

07(2015)158,arXiv:1502.05354 [hep-ph];Erratum:J.HighEnergyPhys.05 (2017)073.

[53]A.Dumitru,T.Lappi,V.Skokov,Distributionoflinearlypolarizedgluonsand el-lipticazimuthalanisotropyindeepinelasticscatteringdijetproductionathigh energy,Phys.Rev.Lett.115 (25)(2015)252301,arXiv:1508.04438 [hep-ph].

[54]D.Boer,S.Cotogno,T.vanDaal,P.J.Mulders,A.Signori,Y.-J.Zhou,Gluonand WilsonloopTMDsforhadronsofspin≤1,J.HighEnergyPhys.10(2016)013, arXiv:1607.01654 [hep-ph].

[55]D.Boer,GluonTMDsinquarkoniumproduction,Few-BodySyst.58 (2)(2017) 32,arXiv:1611.06089 [hep-ph].

[56]D.Boer,P.J.Mulders,C.Pisano,J.Zhou,Asymmetriesinheavyquarkpairand dijetproductionat anEIC, J.HighEnergy Phys. 08(2016)001,arXiv:1605. 07934[hep-ph].

[57]J.-P.Lansberg,C.Pisano,M.Schlegel,Associatedproductionofadileptonand aϒ(J/ψ)attheLHCasaprobeofgluontransversemomentumdependent distributions,Nucl.Phys.B920(2017)192–210,arXiv:1702.00305 [hep-ph].

[58]M.Boglione,J.O.GonzalezHernandez,S.Melis,A.Prokudin,Astudyonthe in-terplaybetweenperturbativeQCDandCSS/TMDformalisminSIDISprocesses, J.HighEnergyPhys.02(2015)095,arXiv:1412.1383 [hep-ph].

[59]S. Catani, D. de Florian, G. Ferrera, M. Grazzini, Vector boson production athadroncolliders:transverse-momentumresummationand leptonicdecay, J. HighEnergyPhys.12(2015)047,arXiv:1507.06937 [hep-ph].

[60]T.Becher, M.Neubert, D. Wilhelm, Electroweakgauge-bosonproduction at smallqT:infraredsafetyfromthecollinearanomaly,J.HighEnergyPhys.02

(2012)124,arXiv:1109.6027 [hep-ph].

[61]T.Becher,M.Neubert,D.Wilhelm,Higgs-bosonproductionatsmalltransverse momentum,J.HighEnergyPhys.05(2013)110,arXiv:1212.2621 [hep-ph].

[62]J.P.Lansberg,S.J. Brodsky,F.Fleuret,C.Hadjidakis,Quarkoniumphysicsat a fixed-targetexperimentusingtheLHCbeams,Few-BodySyst.53(2012)11–25, arXiv:1204.5793 [hep-ph].

[63]D.Kikola,M.G.Echevarria,C.Hadjidakis,J.-P.Lansberg,C.Lorcé,L.Massacrier, C.M.Quintans,A.Signori,B.Trzeciak,Feasibilitystudiesforsingle transverse-spinasymmetrymeasurementsat a fixed-targetexperimentusingthe LHC protonandleadbeams(AFTER@LHC),Few-BodySyst.58 (4)(2017)139,arXiv: 1702.01546 [hep-ex].

[64]R.Aaij,etal.,LHCbCollaboration,Measurementoftheηc(1S)production

cross-sectioninproton–protoncollisionsviathedecayηc(1S)→p¯p,Eur.Phys.J.C

75 (7)(2015)311,arXiv:1409.3612 [hep-ex].

[65]M.Butenschoen,Z.-G. He,B.A.Kniehl,ηc production at theLHCchallenges

nonrelativistic-QCDfactorization,Phys.Rev.Lett.114 (9)(2015)092004,arXiv: 1411.5287 [hep-ph].

[66]H.Han,Y.-Q.Ma,C.Meng,H.-S.Shao,K.-T.Chao,ηc productionatLHCand

indicationsontheunderstandingof J/ψ production,Phys.Rev.Lett.114 (9) (2015)092005,arXiv:1411.7350 [hep-ph].

[67]H.-F.Zhang,Z.Sun,W.-L.Sang,R.Li,Impact ofηc hadroproductiondataon

charmoniumproductionandpolarizationwithinNRQCDframework,Phys.Rev. Lett.114 (9)(2015)092006,arXiv:1412.0508 [hep-ph].

[68]Y.Feng,J.-P.Lansberg,J.-X.Wang,Energydependenceofdirect-quarkonium productioninpp collisionsfromfixed-targettoLHCenergies:complete one-loopanalysis,Eur.Phys.J.C75 (7)(2015)313,arXiv:1504.00317 [hep-ph].

[69]J.-P.Lansberg,H.-S.Shao,H.-F.Zhang,ηc hadroproductionat next-to-leading