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Physics
Letters
B
www.elsevier.com/locate/physletb
Transverse
parton
momenta
in
single
inclusive
hadron
production
in
e
+
e
−
annihilation
processes
M. Boglione
a,
b,
∗
,
J.O. Gonzalez-Hernandez
a,
b,
c,
R. Taghavi
d aDipartimentodiFisica,UniversitàdiTorino,ViaP.Giuria1,10125Torino,ItalybINFN–SezioneTorino,ViaP.Giuria1,10125Torino,Italy
cDepartmentofPhysics,OldDominionUniversity,Norfolk,VA23529,USA dDepartmentofPhysics,YazdUniversity,Yazd,Iran
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory:
Received2May2017
Receivedinrevisedform6June2017 Accepted12June2017
Availableonline17June2017 Editor:A.Ringwald
Keywords:
Non-perturbativeQCD
Transversemomentumdependent fragmentationfunctions TMDevolution
We study the transverse momentum distributions of single inclusive hadron production in e+e−
annihilation processes. Although the only available experimental data are scarce and quite old, we find that the fundamental features of transverse momentum dependent (TMD) evolution, historically addressedinDrell–Yanprocessesand,morerecently,inSemi-inclusivedeepinelasticscatteringprocesses, arevisibleine+e−annihilationsaswell.Interestingeffectsrelatedtoitsnon-perturbativeregimecanbe observed.
Wetesttwodifferentparameterizationsforthep⊥dependenceofthecrosssection:theusualGaussian distribution and a power-law model. We find the latter to be more appropriate in describing this particularsetofexperimentaldata,overarelativelylargerangeofp⊥values.Weusethismodeltomap someofthefeaturesofthedatawithintheframeworkofTMDevolution,anddiscussthecaveatsofthis and otherpossible interpretations,relatedtotheone-dimensionalnatureofthe availableexperimental data.
©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Transversemomentum dependent distribution and fragmenta-tion functions (TMDs) are fundamental tools to understand the structure of nucleons in terms of their elementary constituents, quarks and gluons. TMDs are non-perturbative quantities which embedimportantcorrelationsamongpartonicandhadronic intrin-sic properties,like spin or orbital angularmomentum, and their internaltransversemotion.
TMD partondistribution functionscanbe interpreted,at lead-ing twist, as number densities of partons carrying a light-cone momentumfractionx oftheparentnucleonmomentum P . Unpo-larizedandpolarizedTMD distributionfunctionshaveextensively been studied in Drell–Yanprocesses and semi-inclusive deep in-elastic scattering (SIDIS) in the past; new-generation, dedicated experimentsare currentlyrunning (likeDrell–YanatCOMPASS or atRHIC)orarebeingplanned(liketheElectron–IonColliderinthe USandtheAFTERproposalatCERN-LHC).
*
Correspondingauthor.E-mailaddresses:boglione@to.infn.it(M. Boglione),jogh@jlab.org (J.O. Gonzalez-Hernandez),r.taghavi@stu.yazd.ac.ir(R. Taghavi).
Of equal importance are the TMD fragmentation functions, whichembedfundamental informationonthehadronization pro-cess,whereahadronh,carryingalight-conefractionz ofthe frag-menting parentparton,isproduced.TMD fragmentationfunctions canbemeasured insingle- ordouble-inclusivehadronproduction ine+e−annihilationprocessesor,inamoreinvolvedway,inSIDIS, wheretheynecessarilycoupletoaTMDdistributionfunction.Even withthebestSIDISdatapresentlyavailable,severalcomplications remaintobesolved.Extensiverecentstudiescanbefound,for ex-ample,inRefs.[1–5].
In e+e− collisions, at c.m. energies below the Z0 mass, the electron and positron predominantly annihilate to form a single virtual photon, which can subsequently produce a qq pair.
¯
The quarkandanti-quarkwillthenconvertintohadrons.Atsufficiently high energies, these multi-hadronic events are expected to form two back-to-back jets(due to the limitedtransverse momentum alongtheoriginalquarkdirection).Singleinclusivedistributionsin variables relativetothejetdirection, whichisexpectedtobe the quarkdirection,willthereforegiveinformationaboutthe fragmen-tationofquarksintohadrons.Inparticular,thedependenceofthe inclusivedistributionsinmomentumtransversetothejetaxiswill http://dx.doi.org/10.1016/j.physletb.2017.06.0340370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
provide the goldenchannel toward the phenomenological extrac-tionofTMDFFs.AnillustrationofthisprocessisgiveninFig. 1.
While much effort has been put into measuring and extract-ingunpolarizedandevenpolarizedTMDPDFs(theSiversfunction is a well-known example), little or no experimental information onTMD FFsispresently available.BELLEandBaBarCollaborations have recently presented new multi-dimensional data analyses of the Collins asymmetry in e+e−
→
h1h2X processes, which haveallowedafirstglancetotheintrinsictransversemotionofpartons insidehadronsthroughtheextractionoftheCollinsfunction,a po-larized,chirally-oddTMD fragmentationfunction.Relevant recent literatureonthisandothere+e−relatedsubjectsarepresentedin Refs.[6–9].However,nomodern measurementsexistofthe unpo-larized TMDFFs, althougha thorough knowledgeof thisfunction wouldbeoffundamentalimportanceforanyTMDstudy.
Whilewaitingforup-to-date,highstatisticsand(possibly) mul-tidimensionalresultson the p⊥ distributionofe+e− unpolarized crosssections(theBELLECollaborationhasalreadypresentedsome oftheir preliminary Monte Carlosimulations at SPIN2016[10]), we concentrate on a set of rather old measurements of sin-gle inclusive hadron production in e+e− annihilation processes, e+e−
→
hX , by the TASSO Collaboration at PETRA (DESY) [11, 12]: p⊥ distributions were provided for four different c.m. en-ergies between 14 and 44 GeV, corresponding to charged parti-cleproduction summed over all charges andall particle species, withnoflavorseparation.Cross sectionsare givenasfunctionsof p⊥, integratedover the energy fraction zh=
2Eh/
√
s of the de-tectedhadronh.Notethat,uptocorrectionsoforderp2⊥
/
Q2(hereQ2
≡
s), zh coincides with the light-cone momentum fraction z. Althoughnoinformationisofferedaboutpossiblecutsappliedto zh,averagevaluesare providedforeach c.m. energyset,as sum-marizedinTable 1:they correspondtoratherlow values,ranging from
zh=
0.
13 at 14 GeV to zh=
0.
08 at 44 GeV. Together withTASSOdata,wealsoconsidertheanalogousMARKII Collabo-rationmeasurements [13],collected attheSLAC storageringPEP, ata fixedc.m. energyof29GeV,andPLUTO dataontheaverage transversemomentumsquare[14],collectedatPETRA(DESY).Crucial to all these data is the correct determination of the jet axis, to which the p⊥ distributions are most sensitive, be-sidepropertreatmentofgeometricacceptanceeffects,triggerbias, kinematics cuts and radiative corrections. These corrections, ob-tained by comparison to Monte Carlo simulations, are somehow modeldependent.Clearlyalltheseissuesintroduceverylarge un-certaintieswhich,accordingtomodernstandards,werelargely un-derestimated.
Although these data are affected by several limitations (no hadron separation,limited coverage, zh integration,low
zh val-ues,difficultiesinreconstructing thejet axis,etc),they represent an extremelyinteresting exampleofa directmeasurement of in-trinsic transverse momenta. In fact, as mentioned above, the jet axisresultingfromeache+e−scatteringidentifiesthedirectionof thefragmenting qq pair¯
(q andq should¯
be back to back in the e+e− c.m.frame ifradiative effectsare appropriately subtracted), andthedetected p⊥representsadirectmeasurementofthe trans-versemomentumofthefinalhadronwithrespecttothe fragment-ingparentparton,seeFig. 1.The purpose of this article is two-folded: first, we will de-viseand test an appropriate functional formto describe the p⊥ distributionsmeasured by TASSOandMARKII, achievingasmuch informationaswepossiblycanontheTMD unpolarizedFF;then, acarefulinterpretationofourresultswillbeprovidedfocusingon thefeaturesrelatedto TMDfactorizationwithin aTMD evolution scheme,inthenon-perturbativeregime.
Asthesemeasurementsofferquitelimitedinformation,wewill not be able to perform a detailed extraction of the TMD
frag-Fig. 1. Illustrationofatypicalhadroniceventfromthee+e− annihilationprocess, showingthereconstructedjetaxis,thehadronmomentumPh anditstransverse
componentp⊥,perpendiculartothejetaxis.
Table 1
Upper and central panels: center of mass energies andcorresponding zh meanvaluesfor theTASSO and
MARK IIcrosssections.Lowerpanel:centerofmass en-ergiescorrespondingtoPLUTOmeasurementsofp2
⊥. Experiment c.m. energy zh TASSO 14 GeV 0.13 22 GeV 0.11 35 GeV 0.09 44 GeV 0.08 MARK II 29 GeV 0.09 PLUTO 7.7 GeV – 9.4 GeV – 12.0 GeV – 13.0 GeV – 17.0 GeV – 22.0 GeV – 27.6 GeV –
mentationfunctionsasdoneinthepast[2,8,15–17].However,we will observe that interesting signatures of TMD factorization in the non-perturbative regime can be detected in these data sets. Inparticular,wewillfindindicationsthat apower-law p⊥ behav-ior,differentfromtheGaussianparametrizationoftheTMDFFswe usuallyusedinourpreviousanalyses,mightreproducethesedata moresuccessfully,especially as p⊥ grows largerthen afew hun-dredMeVandweentertheregioninwhichTMDevolutioneffects starttobecomemorevisible.
2. Formalism
Similarlyto thecollinear case[18],the e+e−
→
h X cross sec-tioncanbecastedinthefollowingform:d
σ
h dz d2p ⊥=
LμνW μν=
4π α
2 3s z F h 1(
z,
p⊥;
Q2) .
(1)Within TMD-factorization, up to power suppressed terms, the hadronictensorW canbeexpressedas
Wμν
=
WμνT M D+
Wcollμν.
(2)The first term on the right hand side of Eq. (2), WT M Dμν , corre-spondstotheregionofsmalltransversemomenta,whilethe sec-ond,Wcollμν,iscalculablewithincollinearfactorizationandcontains correctionsthat become important atlarger valuesof p⊥.In the
caseathand,asthereisonlyone observedfinal hadron,onemay writefortheTMDterm:
WμνT M D
∝
f|H
f(
Q;
μ
)
|
μνDh/f(
z,
p⊥;
μ
, ζ
D),
(3) where Q isthe hardscale of theprocess, z and p⊥ are the ob-servedhadronicvariables,μ
istherenormalizationscale,whileζ
D is a regulator for the light-cone divergences that arise in TMD-factorization.ThehardfactorH
f canbecalculatedinperturbation theory,whiletheTMDFFisdefinedbytherelationsDh/f
(
z,
zk⊥;
μ
, ζ
D)
≡
1(
2π
)
2 d2b⊥e−ik⊥·b⊥Dh˜
/f(
z,
b⊥;
μ
, ζ
D) ,
(4)˜
Dh/f(
z,
b⊥;
μ
, ζ
D)
≡
j˜
Cj/f⊗
dh/j(
z;
μ
b)/
z2×
expμ μb d
μ
˜
˜
μ
γ
D(
α
s(
μ
˜
)
;
1)
−
γ
K(
α
s(
μ
˜
))
log√
ζ
D˜
μ
×
exp˜
K(
b∗;
μ
b)
log√
ζ
Dμ
b×
exp gh/j(
z,
b⊥)
+
gK(
b⊥)
logζ
Dζ
D(0).
(5) Noticetheconjugatevariabletob⊥isk⊥ratherthanp⊥(k⊥isthe component of the fragmenting parton momentum transverse to thefinalhadron,inareferenceframeinwhichthelatterispurely longitudinal).Details onthe variousingredients ofEq.(5) canbe foundinRefs. [18,19].Forourpurposesitsufficestonote thatin thefirst three factorsof Eq.(5)all the ingredients arecalculable withinperturbation theory,exceptforthecollinearfragmentation functiondh/j.NoticethattheWilsoncoefficients,C˜
j/f,theintegral ofthe anomalous dimensions,γ
D andγ
K, andthe Collins–Soper (CS)kernel,K ,˜
dependonb⊥onlythroughthequantityb∗,which issettoremainsmallerthansomemaximumbmax.Thelastfactor inEq.(5)correspondsessentiallytonon-perturbativeinformation. Inwhatfollowswewillconsiderthecaseinwhichthepurely perturbativeingredientsoftheTMDFF,namelyC˜
j/f,γ
K(D)andK˜
arecalculatedtoorder
α
s.ThereissomefreedominthedefinitionofEq.(5),encompassed inthearbitraryquantities
μ
,μ
b,b∗,ζ
D,ζ
D(0).Forouranalysiswe adoptthe usual choices [19],μ
→
Q ,ζ
D→
Q2,ζ
D(0)→
Q02 andμ
b=
2e−γE/
b∗,whereQ0issomeinitialscaleandγ
EistheEuler– Mascheroniconstant.WiththesechoicestheCSkernelK vanishes˜
atorderα
s,so thethird factorin Eq.(5)reducesto one. Tothis same order, the factor containing the anomalous dimensionsγ
D andγ
K canbe expressed in aclosed analytic form. Forour pur-poses,itwillbeusefultoclassifytheresultofthisintegralinterms ofitsdependencieson Q andb∗,namelyexp
μ μb d
μ
˜
˜
μ
γ
D(
α
s(
μ
˜
)
;
1)
−
γ
K(
α
s(
μ
˜
))
log√
ζ
D˜
μ
−→
N
(
Q)
f(
b∗,
Q0)
expλ
(
b∗)
log Q Q0,
(6)forwhichwe haveusedtheresultsoftheAppendicesinRef. [3]. The quantities
N
(
Q)
, f(
b∗,
Q0)
andλ
(
b∗)
are flavor inde-pendent functionsthat encode the most prominent perturbative effectsinthedefinitionoftheTMDFF.Thelastofthesethree func-tions,λ
(
b∗)
≡
32 27log log 2e −γEQ C Db∗
,
(7)is the most interesting since, being multiplied by log
(
Q/
Q0)
, itcorrelates b∗ withQ , whichmeansithastheeffectofmodifying the shape ofthe TMD FFunder evolution.With these considera-tions,onemaywritefortheTMDFF:
˜
Dh/f(
z,
b⊥;
μ
, ζ
D)
=
N
(
Q)
j˜
Cj/f⊗
dh/j(
z;
μ
b)/
z2 egh/j(z,b⊥) f(
b∗,
Q0)
×
expλ
(
b∗)
+
gK(
b⊥)
log Q Q0.
(8) Therefore,except fortheoverall normalizationfactorN
(
Q)
,theeffectsofevolutionatorder
α
scanbemappedtoeitherthe non-perturbative functiongK(
b⊥)
,ortheperturbativequantityλ
(
b∗)
.Wenotethatwiththechoice
μ
b=
2e−γE/
b∗,theorder-α
sWilson coefficientsC˜
j/f donotcontainanyQ2-dependence(seeappendix inRef.[19]).In the region where TMD effects dominate (see Eqs. (1)–(3)), flavorindependenceofgK and
λ
inEq.(8)impliesF
−1 dσ
h dz d2p ⊥∝
expλ
(
b∗)
+
gK(
b⊥)
log Q Q0b⊥→z b⊥
,
(9) where the symbolF
−1 indicates the two-dimensional inverse Fourier transform, from momentum to impact parameter space, and the transformation b⊥→
z b⊥ is neededto account for the extrafactorofz thatappearsinthedefinitionofEq.(3),compared to theTMD terminthe hadronictensor inEq.(2).Inthe follow-ingsectionwewilluserelations(7)–(9),validtoorderα
s,tomake an interpretationofourresults.Forthis, wewill modelthecross sectioninawayconsistentwithEq.(3).3. Datafittingandresults
A fullanalysiswithin aTMD-evolution schemeshould include all of the contributions in Eq. (2), as well as a matching pre-scription to interpolate between regions of small and large p⊥, asoriginally prescribed inRef. [20]. Itis importantto stress that all of these ingredients provide crucial constraints that any full analysisshouldinclude.Somerecentstudiesrelatedtothe compli-cations involvedinthe matchingprescriptions forSIDISprocesses are presented inRefs. [4,21].However, forsuch type of analyses, multidimensionaldatasetsaremostsuitable,whereonecan com-pletely disentangletheeffectsofdifferentkinematicsvariables.In the caseofthemeasurements ofRefs. [11–14],the large system-aticuncertaintiesinducedbyz-integrationcanonlyrenderlimited insightonTMD-effects.Moreover,very lowvaluesof
zhcan en-danger the applicability of factorizationtheorems. However, it is still interesting to investigate what information about evolution canbeextractedfromthesemeasurements.Infact,evenforthese z-integrated cross sections,one may expect the shape of the p⊥ distributionstobeaffectedbyTMD-evolutioneffects.Inorderto addressthisquestion itbecomesessential tomake anestimate ofwherethelargetransversemomentumcorrections startbecoming importantinthe TASSOandMARKII datasets.To doso,we startby consideringthe errorsoftheTMD approxima-tion,which are oforder
O(
k⊥/
Q)
=
O(
p⊥/
z Q)
.In general, one expectsthatat p⊥∼
z Q ,the crosssection shouldreceive contri-butionsfromthecollinearterminEq.(2).Usingtheaveragevalues ofTable 1,onemayestimatethatthiscontributionsshouldbe sig-nificant at p⊥∼
2GeV. We identify this asthe matching region. Inthis articlewe willattempt to extract information onlyabout thenon-perturbativeevolutioncarriedbygK inEq.(8),sowewill constrainouranalysistotheregionof p⊥<
1.
0GeV.Giventhe particular kinematics ofthe TASSO experiment and thelowvaluesof zh,onemaywonder whetherfactorization the-oremscanbe applied.Infact,itispossiblethatthe experimental datareceivecontributionsfromnon-TMDeffects.However,asthe dataareintegratedoverzh andthereisnopossibilityforusto im-posecutsover zh,wewillproceedundertheassumptionthatthe mainfeaturesoftheanalyzed data aregeneratedby TMDeffects, especiallyasfarastheirchangesinp⊥-shapewithQ isconcerned. Aswe willsee lateron, the p⊥ distributionsshow a broadening, consistentwiththeexpectedTMDbehavior.
Notethat wealso imposea lower cuton p⊥,such that p⊥
>
0.
03 GeV; this amounts to excluding the first data point in the TASSOdatasets.In order to relate the shape of the data to possible TMD-evolutioneffects,oneneedsamodelthatcanreproducethe trans-versemomentumdistributionofthefinalhadronh.Then,by look-ingattheb⊥-space,onemayconnecttheparametersofthemodel tosome oftheinformationcontainedinthe definitionofEq.(5), asdiscussedinRef.[3].
Atleasttwofunctionalformshavebeenshowntoappropriately describetransversemomentumdistributions[2,3].
•
Gaussian form:itisthemostcommonlyusedparametrization for phenomenological studies, it has been shown to repro-duce experimental data very successfully in both Drell–Yan andSIDISprocesses.Ithastheadvantageofbeingeasyto in-tegrateanalytically.•
Power-law: it isvery flexible,even withalimitednumber of free parameters. Notonly can it appropriately reproduce the behavior ofthe crosssection atsmall p⊥,butit canalso in-corporateitstailatlarger p⊥values.We will consider both of these functional forms in order to de-scribethelow-p⊥ regionof thedata.Ouraimis tofocus onthe kinematicsranges whereTMD-effects aredominant. As discussed above,one mayestimatethatperturbativeeffectswillstartto be-comeimportantroughly around p⊥
∼
2GeV,butcould infact be non-negligibleatevensmallervaluesoftransversemomentum, es-peciallyasthedataweconsiderareintegratedoverz.Ourworking hypothesisisthat for p⊥<
1GeV theTMD-terminEq.(2)isthe largestcontributiontothecrosssection.WemodelthestructurefunctionF1 inEq.(1)sothat
d
σ
h dz d2p ⊥ model=
4π α
2 3s q eq2Dhq(
z,
p⊥;
Q2)
=
4π α
2 3s q eq2Dhq(
z,
Q2)
h(
p⊥) ,
(10) whichextendstheleadingorderexpressionforthecollinearcross section. In Eq. (10), the sum runs over all q and q flavors,¯
and we have assumed that the TMD fragmentation function may be writtenasDhq
(
z,
p⊥)
=
Dhq(
z)
h(
p⊥) ,
(11)where Dh
q
(
z)
isthecollinear,unpolarizedFF(whichwe takefrom Ref. [22]). The function h(
p⊥)
incorporates all of the p⊥ depen-denceoftheTMD FF,itisflavorindependentanditisnormalized sothatitintegratestounity.The TASSO collaboration providescross sectionsdifferential in p⊥,normalizedtothefullyinclusivecrosssection,whichat lead-ingorderread
σ
totL O=
σ
0=
4π α
2 3s q eq2,
(12)sothatourfitswillinvolvetheexpression 1
σ
0 dσ
h dp⊥ model=
2π
p⊥N dz qeq2Dhq(
z;
Q2)
qeq2 h(
p⊥) .
(13) Anote ofcaution isnecessaryatthispoint. TheTASSO distri-butionsin p⊥ atdifferentenergiescannotbedescribedby simply usingthemodelofEq.(10).Instead,onemustincorporatea treat-mentfortheir normalizationsatdifferentvaluesof Q .InEq.(13)thisisreflectedbytheparameterN.Whilethismaybeinconflict witha possibleprobabilistic interpretationofthe functionh
(
p⊥)
, itshouldnotaffectourconclusionsregardingTMDevolution,since forthat,wewillfocusonaspects ofthe p⊥ distributionsthat re-gard their shape,butnot theprecisevaluesof theirmaxima.We note that accounting for all the features of these data may be challenging even within a full TMD analysis, aslarge systematic errorsmaytranslateintoout-of-controlnormalizationswhen deal-ingwithz-integrateddata.The fits inthe next subsections are performed on the TASSO p⊥-distributionsonly.WewillusetheMARKIIp⊥-dependent nor-malized cross section and the TASSO measurements of
p2⊥
tocross-check ourresults.PLUTO datawill be shownonlyfor com-pleteness, although wewill not usethemin ouranalysisasthey are not fully compatible with the other experiments. When ap-propriate,errorbandscorrespondingtoa 2
σ
confidencelevelare provided,obtainedbygeneratingrandompoints inthe parameter space forwhichχ
2i
∈ [
χ
2 0, χ
2 0+
χ
2]
, whereχ
2 0 isthe minimalvaluegivenbythefitand
χ
2 dependsonthenumberofparam-etersofthemodel;therelevantcasesinourfitsinvolveeither6 or 7 free parameters,whichcorrespondto
χ
2 valuesof12.
85 and14
.
34,respectively.3.1. Gaussianshapeatlowp⊥
WestartbyapplyingaGaussianmodelwithaconstantwidth: h
(
p⊥)
=
e−p⊥2/p⊥2
π
p⊥2.
(14)As mentioned earlier, in order to describe the data we need an appropriatetreatmentforthenormalization.Unexpectedly,wefind thatusingdifferentmultiplicativeconstants,oneforeachenergy,is notenoughtoobtainagoodfit,eveninthelimitedregionofp⊥
<
0.
5 GeV(seethefirstentryinTable 2).Instead,byalsointroducing a Q -dependentshiftforthecrosssections,sothatonehas1
σ
0 dσ
h dp⊥ model−→
2π
p⊥N dz qe2qDhq(
z;
Q2)
qeq2 h(
p⊥)
+ δ
Q,
(15)Fig. 2. GaussiandescriptionoftheTASSOp⊥distributionsat4differentc.m. ener-gies[12].OntheupperpaneltheGaussianmodelwithconstantwidth,seeEqs.(14) and(15),uptop⊥=0.5 GeV.OnthelowerpaneltheGaussianmodelwiththeQ dependentwidthofEq.(16),uptop⊥=1.0 GeV.
one can obtain a good description of the data, as seen in the second entry of Table 2 and on the upper panel of Fig. 2. This unconventionalprescriptiontodeal withthenormalization,while leaving little room for a partonic interpretation for the function h
(
p⊥)
, allows usto verifyquantitatively that asfar asthe width ofthe p⊥ distributionisconcerned, no significantchangecan be observed with growing Q. One may not conclude, however, that TMD-effectsdonotappearinthesetransversemomentumranges, butratherthatthedataanalyzeddonothavethenecessary accu-racytoshowpossiblewidthchangesinthisregion.Thus,onemust trytoextendthedescriptionofthedatatolargervaluesof p⊥.Forp⊥
>
0.
5GeV,anoticeabledependenceofthedistributions on the c.m energy suggests that a Q -dependent width may be appropriate.However, evenwiththisextension ofthemodel, de-scribingthedatapastthispointturnsouttobeextremelydifficult. Toillustratethis,weconsiderthefunctionalform p2⊥=
2 g1+
2 g2z2log Q 2 Q0,
(16)whichallows forsomecomparisonto previousphenomenological studies[23,24]where,within aCSSevolutionscheme,aGaussian behavior ofthe non-perturbativeSudakov factor in theb⊥ space wasassumed.Thistime,weuseamultiplicativeconstantforeach valueofQ (
δ
=
0 inEq.(15)),andfix Q0=
1.
6GeV.ThelastentryofTable 2 showstheresults ofthe fit for p⊥
<
1GeV,usingthe extendedGaussianmodelcorrespondingto Eq.(16).Theobtained minimalχ
2 valuepointstoaratherlowqualityofthedescriptionofthe data.The corresponding plot,inthe lower panel ofFig. 2, showsthatthereissometensionbetweenasuccessfuldescription ofthepeak atlow p⊥andanequallygooddescriptionofthetail atlarger p⊥ values.As we willshow in thenext subsection,the power-lawcan infactaccommodate forboth ofthesefeatures of thedata.
3.2. Power-lawshapeatlowandmoderatep⊥
So far,we havetried theGaussian ansatzforthe p⊥ distribu-tions. We have found that the data favors a constant width, for values up to p⊥
=
0.
5GeV. However, this can only be achieved by introducing a Q -dependent shift, which cannot be easily in-terpreted within apartonic picture.TheGaussian class ofmodels doesnotseemtobeappropriateforlargervaluesofp⊥.In order to describe the data up to p⊥
=
1GeV, we test a power-lawparametrization,givenbyh
(
p⊥)
=
2(
α
−
1)
M2(α−1) 1p2⊥
+
M2α,
(17) wherethefactor2
(
α
−
1)
M2(α−1)issetsothath(
p⊥)
integratesto unity.Thetwo-dimensionalinverseFouriertransformofthis func-tion has an exponentially decaying asymptotic behavior, consis-tentwithwhatwouldbeexpectedfromgeneralargumentswithin quantum fieldtheory,asdiscussed inRefs. [18,25].Inimpact pa-rameterspacethepower-lawbecomesF
−1 1 p2⊥+
M2α=
1 2απ
(
α
)
b⊥ Mα−1 K1−α
(
b⊥M)
large b−→
⊥ 1 2απ
(
α
)
b⊥ Mα−1
π
2 e−b⊥M√
b⊥M 1+
O
1 b⊥M,
(18) where K1−α(
b⊥M)
is the modified Bessel function ofthe second kind. In what follows, we use an independent normalization for each value of Q . As discussed before, we are interested in the shape ofthedistributions, ratherthan ontheiroverall normaliza-tions.In the power-law parametrization of the p⊥-differential cross section,theparameterM2isrelatedtothepositionofitspeak,p0⊥, bytherelationM2
=
2α
−
1p0⊥2.Sincethestudieddistributionsreachtheirmaximumatroughlythesamevalueoftransverse mo-mentum, p⊥
≈
0.
212GeV,forallvaluesofthec.m.energy,inour mainanalysiswehaveimposedtheconditionsthatM2
=
2α
−
1p0⊥2p0⊥
=
0.
212 GeV.
(19)WehaveverifiedthatsettingM2 free,doesinfactsatisfyEqs.(19)
within errors. It is,however,usefulto reduce the numberof pa-rametersbydirectlyimposingtheserelations.
First,totestthatthepower-lawcanappropriately describethe data,we conductedsimpleindependentfits foreach value ofthe c.m. energy,which rendersone value of
α
foreach Q . Themost interesting aspect of this preliminary fit is that it shows a clear dependence of the parameterα
on Q , despite the use of inde-pendentnormalizations.Thetrendofthevaluesforα
isdisplayed in Fig. 3, which showsa decrease of the its optimal value with Q .Duetothelargeuncertaintiesinthedeterminationofα
,there are likelyseveralfunctionalformsthat canaccommodate the ob-servedbehavior.Inordertomakeaneducatedguessforasuitable Q -dependence inα
, weassume that the integrationover z does notalterthestructureoftherelationinEq.(9),namelyF
−1 dσ
h d2p ⊥∝
exp˜
g(
b⊥)
log Q Q0,
(20)forsomefunction g
˜
(
b⊥)
.Thus,onecanseethatalogarithmic be-haviorforα
maybeappropriatebylookingattheasymptoticlimit ofthepower-lawinb⊥-space,Eq.(18).First,forthevaluesα
0andTable 2
Fits of the TASSO four sets of cross sections, corresponding to the Gaussian parameterization of the p⊥ distributions. ParametrizationIreferstotheusualchoiceofEq.(14),ParametrizationIIreferstotheGaussiancorrectedbyaQ-dependent shift,seeEq.(15),whileParametrizationIIIcorrespondstoaGaussiandistributionwithaQdependentwidth,asinEq.(16).
Parametrization Normalization Gaussian width χ2
pt Gaussian – I N= {N14,N22,N35,N44} p2⊥ =constant p⊥∈ [0.03−0.50]GeV N14=2.3±0.2, N22=2.7±0.2 p⊥2 =0.118±0.004 GeV2 5.9 36 data points N35=3.1±0.1, N44=3.2±0.1 Gaussian – II N,δQ p2 ⊥ =constant p⊥∈ [0.03−0.50]GeV N=1.8±0.2 p2 ⊥ =0.098±0.005 GeV2 0.74 36 data points δ=0.22±0.03 GeV−2
Gaussian – III N= {N14,N22,N35,N44} p2⊥ =2g1+2g2z2log3Q.2
p⊥∈ [0.03−1.00]GeV N14=2.7±0.2, N22=3.3±0.3 g1=0.013±0.004 GeV2 2.7 56 data points N35=4.0±0.1, N44=4.3±0.2 g2=2.6±0.3 GeV2
Fig. 3. ValuesofαinEq.(17)thatbestdescribethedistributionsof[12].The trian-glesrepresenttheminimalvaluesofindependentfits,performedforeachvalueof thecenterofthec.menergy√s=Q .Theshadedbandsindicatethecorresponding uncertainty.
α
that describe the data attwo givenvalues Q0 and Q , if (20)holds,oneshouldhave bα0 ⊥ exp
˜
g(
b⊥)
log Q Q0∝
b⊥α,
(21)inthelarge-b⊥ limit.Thiscanbeachievedif
˜
g
(
b⊥)
large b−→ ˜
⊥α
log(
ν
b⊥) ,
(22) forsomevaluesα
˜
andν
,whichinturnprovidestherelationα
=
α
0+ ˜
α
log Q Q0.
(23)WehaveimplementedEq.(23)intoafitandconfirmedthatinfact itreproduceswelltheTASSOdata.Resultsareshownonthethird panelofTable 3,andinthetopplotinFig. 4,whichincludeserrors correspondingtoa2
σ
confidencelevel.Wecomparetheseresults totheMARKIIdatasetinthebottompanelofFig. 4.The argument presented above, leading to a logarithmic Q -dependenceforthe power
α
,has some caveats.First, it depends on whetherEq. (20) is approximately correct.Furthermore since it considers only the asymptotic large-b⊥ behavior of Eq. (18), Eq. (23) does not need to hold for values larger than p⊥∼
M. Therefore, even if one can describe the data, any interpretation ofthelogarithmicbehavior ofα
,intermsoftheingredients that define the TMD FF, Eq. (5), should be taken with great caution. Nonetheless,itisworthwhiletoexplorethispossibility.Notice that the function g acquires
˜
its behavior fromλ
andgK. Sincethe firstvariesslowly withb⊥,andin factdoesfreeze to a constant value at large enough b⊥ (see Eq. (7)), one may seeEq.(23)as themanifestation ofa logarithmic large-b⊥ trend forgk
(
b⊥)
,analogousto Eq.(22).A behaviorconsistentwith dis-cussions in Refs. [3,25]. However, we stress that one may wellreproduce the data by using different assumptions. As a counter example we consider a simple picture where the cross section takestheform
d
σ
dz d2p ⊥∝
1 p2⊥
+
zM˜
2 β1+ β2z,
(24)andwhereoneaccountsfortheintegrationoverz bysomeaverage value
z,leadingto dσ
d2p ⊥∝
1 p2⊥
+
z˜
M2 β1+ β2z,
(25)wheretheparameters
β
1,β
2 andM˜
aretobedeterminedbyafit.In thiscase, itis indeedpossible toobtain a gooddescription of thedatabyusingtheexperimentalaveragevaluesofz ofTable 1, sincetheyexhibitaseeminglylogarithmictrend.However,itis dif-ficult tomake aconnectionto TMD evolutionsincethe valuesin
Table 1are in generalaffected by correlationsbetween Q and z of differentorigin, possibly relatedto effects that go beyondthe scope of TMD factorization, giventhe low values of
zh. In fact, noticethat theconditionofEq.(19)impliesalogarithmic behav-ior for ourfit parameter M2. Itis possible that, forinstance, the effectsoftheTMD evolution,encodedin gK,resultinchanges in the powerα
that fit the data, while the changes in M2 are the resultofcorrelationsinduced bythe integrationover z.Itseems, however,that the lackofinformation aboutthe z-dependence of the TMD FF in the TASSO andMARK II measurements hindersa more solid conclusion aboutTMD evolution effects inthese data sets.Finally,it isusefulto testthe modelofEqs. (17)and(23)for larger values of p⊥. In fact, this is necessary to calculate
p2⊥
,sinceitimpliesintegrationoverthefullrangeofp⊥.Forthis,we keep thefirstofthe constraints(19),butfreethe parameter p0⊥.
First, we performa fit includingdata up to p⊥
=
2GeV. Asseen inFig. 5andthelast entryofTable 3,themodelcansuccessfully accommodate thisextended range of p⊥. Thisrange is,however, not enough to reproduce to a good accuracy the corresponding TASSO measurements ofp2⊥
. Thus,we furtherextendthe rangeanalysisofTASSOdatatovaluesupto p⊥
=
3GeV andusethe re-sultingminimalparameterstoestimatep2⊥.Fig. 6showsour es-timateandthedatafromTASSO.Therangeofintegrationused im-pliesthat averagevaluesoftransversemomentum receive impor-tant contributionsfromnon-TMDeffects.Forcompleteness, Fig. 6shows also
p2⊥ data by PLUTO. We note that PLUTO measure-ments for p2⊥
are systematically smaller than those by TASSO.Thisreflectsthefactthatdataselectionisnotcompatiblebetween experiments. Theydo however, seem to follow the same depen-denceon Q .
Table 3
Fits of the TASSO four cross sections, corresponding to the power-law parameterization of the p⊥ distributions. ParametrizationIrefersto4independentfits(oneforeachdatasetcorrespondingtoadifferentc.m.energy)usingthe functionalformofEq.(17),withconstantαandM2 parameters.ParametrizationIIreferstothesimultaneousfitofthe fourdatasets,usingthefunctionalformofEq.(17),withconstantαandM2
parameters.ParametrizationIIIrefersto thesimultaneousfitofthefourdatasets,usingthefunctionalformofEq.(17),withQ-dependentαandM2
parame-ters.ParametrizationIVreferstothesamecaseasIII,butnowthefitisperformedontheextendedrangep⊥<2GeV,for whichwefreetheparameterp0⊥.
Parametrization Normalization N= {N14,N22,N35,N44} Parameters χ2 pt Power-law – I N14=2.6±0.1 α= {α14,α22,α35,α44} χ142 =0.35 p⊥∈ [0.03−1.00]GeV N22=3.2±0.2 χ222 =0.30 14×4 data point N35=4.0±0.1 α14=3.3±0.4, α22=2.5±0.3 χ352 =0.88 N44=4.4±0.2 α35=2.2±0.1, α44=2.0±0.1 χ442 =0.84 Power-law – II N14=2.6±0.2 α= constant p⊥∈ [0.03−1.00]GeV N22=3.3±0.2 56 data points N35=4.0±0.1 α=2.2±0.1 2.87 N44=4.2±0.2
Power-law – III N14=2.6±0.2 α=α0+ ˜αlog(Q/Q0)
p⊥∈ [0.03−1.00]GeV N22=3.3±0.2 Q0=14 GeV 56 data points N35=4.0±0.1 0.66 N44=4.4±0.2 α0=3.1±0.4, α˜= −1.0±0.4 Power-law – IV N14=2.6±0.2 α=α0+ ˜αlog(Q/Q0) p⊥∈ [0.03−2.00] N22=3.2±0.3 Q0=14 GeV 76 data points N35=4.0±0.1 α0=3.5±0.3, α˜= −1.1±0.3 0.95 N44=4.3±0.2 p0⊥=0.219±0.005 GeV
Fig. 4. InthetoppanelweshowtheresultsfromourfittoTASSOexperimentaldata usingthepower-lawofEqs.(17)and(23),intherange0.03GeV<p⊥<1.0GeV. Toavoidoverlappingandprovideacleardisplayofthe2σ errorbands,weplot thedistributionsfor differentenergiesapplyinganarbitraryshift.Inthe bottom panelwe comparetheresults fromthe fit toTASSOdatatothe MARKIIcross section.Notethatadifferentnormalizationand itsuncertaintieshavetobe de-terminedindependentlyforthisdataset.Wedon’t displaythefirstbin,centeredat
p⊥=0.05 GeV. 4. Finalremarks
TMDFFs embedtheessence ofhadronization,oneofthemost importantmanifestations of QCD inthe non-perturbativeregime. Itisthereforeimportanttogatherasmuchinformationaspossible onthese softquantities,which cannot be computed, buthaveto beinferredfromexperiment.Overthelastfewyears,several
anal-Fig. 5. Results obtainedbyusingthepower-lawofEqs.(17)and (23)compared toTASSOp⊥-dependentdistributions[12],intherange0.03GeV<p⊥<2.0GeV. Errorbandsarecomputedusinga2σ-confidencelevel,asexplainedinSection3.
Fig. 6. Estimationofthetransversemomentummeanvalue,p2
⊥,obtainedbyusing theparametersextractedbyfittingtheTASSOp⊥distributionsupto3.0GeV.Empty squarescorrespondtoPLUTOdata[14]whilefilleddiamondscorrespondtoTASSO measurements[12].Theshadedarearepresentstheuncertaintyofourcalculation andiscomputedasexplainedinSection3.
yses havebeen performed to extract the polarized TMD FFs, like theCollinsfunction,usingthemeasurementsoftheCollins asym-metriesine+e−
→
h1h2X processesprovidedbyBELLEandBaBarCollaborations, whichdelivered multidimensionaldata (inbins of z1, z2, p⊥1, p⊥2) with impressive statistics and very high
preci-sion.Unfortunately,noanalogousdatahavebeenprovidedonthe p⊥-distributionsoftheunpolarized crosssections,ormultiplicities, to allow for the extractionof the unpolarized TMD FFs. The ab-senceofthesefundamental bricksencumbers theanalysisofany otherpolarized,aswellasunpolarized,process.
Atpresent,theonlyavailabledataaresomeold(andalmost for-gotten)measurements of e+e−
→
h X crosssections from TASSO and MARKII Collaborations. Although these are one-dimensional dataandare affected by large uncertainties,as discussedin Sec-tion1,theyhavetheuniqueadvantageofdeliveringmeasurements atdifferent c.m. energies; therefore,they can provide a valuable starting point not only to learn aboutthe unpolarized TMD FFs, butalsotostudythephysiognomyoftheirTMDevolution.In this article we assess the extent to which the effects of TMD evolution can be observed in these data. For this purpose, our main tool is an analysis based on a simple partonic picture in which the cross section is factorized, as in Refs. [2,8,15–17]. While these types of analysis typically use a Gaussian form, we alsotesta power-lawbehaviorto describethedata,assuggested inRef. [3]. We extendthis class ofmodels to the casein which theparameterizationissupplementedbysomeQdependence,and usetheseresultsto providean interpretationwithin a TMD evo-lutionframework,seeEq.(5),discussingthecaveatsrelatedtothe limitedamountofinformationprovidedbythesedata.
Westartby modelingthe p⊥ dependenceofthe crosssection by aGaussian shape andfit thefour sets ofTASSO cross section data(correspondingtofourdifferentc.m.energies)to extractthe corresponding free parameters, see Table 2. Our analysis shows that the Gaussian distribution can only describe the data up to p⊥
∼
0.
5 GeV,providedthecrosssectionsareadjustedwithan ad-hoc,additivetermδ
Q .InthisregionnoQ -evolutioneffectscanbe observedintheGaussian widthofthe p⊥-distributions.The diffi-cultiesrelatedtotheinterpretationoftheseresults,however,leads ustoconsideradifferentparameterization.Wethen focuson apower-law parametrizationofthe p⊥ de-pendenceofthecrosssection,whichshowstobemoreappropriate thantheGaussian modelandprovides asuccessfuldescriptionof theTASSOp⊥-distributionsoveramuchlargerrangeofp⊥values. Weperformedtwoconsistencychecks.First,we comparedthe re-sultsof our main fit onTASSO data,reported in the third panel ofTable 3,withtheMARK II data,forwhichwe founda reason-able agreement of our model. Second, we compared the results fromfittingTASSOp⊥-distributionstothereportedvaluesof
p2⊥. Wefoundthatthelattercanonlybereproducedbyextendingthe rangeoftransverse momentumto p⊥≤
3GeV.Finally,we providean argumentto explain that theQ depen-denceinthepower-lawcan be consistentwitha logarithmic be-havior,inthelargeb⊥limit,ofthefunction gK,whichencodesthe non-perturbativeevolutioneffectsinthedefinitionoftheTMDFF. Thenatureofthesedataforcesustobecautiouswiththe inter-pretationofourresults.In fact,it seems unlikelythat thesedata by themselves would allow to disentangle the TMD effects from other Q -dependence in the data. This is related to the integra-tionover z,which induces a degreeof ambiguityin thepossible interpretations.Thus, oneshould furthertestanyconjecture with multi-dimensionaldata.
Intheforeseeablefuture,unpolarizedsingle-hadronproduction at fixed energies by BELLE and BaBar Collaborations, differential inbothz and p⊥,mayindeedprovideenoughconstraintsonthe z-dependence ofthefragmentationprocess,allowing forthe pos-sibilityofafullTMDanalysiswhencombinedwiththeTASSOand MARKIIdata.
Acknowledgements
We thank Mauro Anselmino for his contributions during the initial stages of this work and for many useful discussions that followed.We are very gratefulto StefanoMelisforhelping usin findingreferencestotheexperimentaldatausedinthispaper,and forveryhelpfuldiscussionsandcross-checks.
The work of J.O.G.H was partially supported by JeffersonScience Associates,LLCunderU.S.DOEContract#DE-AC05-06OR23177and byU.S.DOEGrant#DE-FG02-97ER41028.
R.T. wishes to thank A. Mirjalili for his continuous support throughoutthisproject,andYazdUniversityforfinancingherstay attheUniversityofTurin,wheresheperformedpartofherwork.
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