Double parton scattering: A study of the effective cross section within a Light-Front quark model

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Double

parton

scattering:

A

study

of

the

effective

cross

section

within

a

Light-Front

quark

model

Matteo Rinaldi

a,b

,

Sergio Scopetta

a,b,

∗

,

Marco Traini

c,d

,

Vicente Vento

e

a_{DipartimentodiFisicaeGeologia,UniversitàdegliStudidiPerugia,ViaA.Pascoli,I-06123Italy} b_{IstitutoNazionalediFisicaNucleare,SezionediPerugia,Perugia,Italy}

c_{DipartimentodiFisica,UniversitàdegliStudidiTrento,ViaSommarive14,I-38123Povo(Trento),Italy} d_{IstitutoNazionalediFisicaNucleare–TIFPA,Trento,Italy}

e_{DepartamentdeFisicaTeòrica,UniversitatdeValènciaandInstitutdeFisicaCorpuscular,ConsejoSuperiordeInvestigacionesCientíficas,46100Burjassot} (València),Spain

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received19June2015

Receivedinrevisedform6October2015 Accepted11November2015

Availableonline14November2015 Editor:J.-P.Blaizot

Keywords: Partons

Multipartoninteractions Partoncorrelations

We presentacalculationoftheeffectivecrosssection

σ

eff,animportant ingredientinthedescription

ofdoublepartonscatteringinproton–protoncollisions.Ourtheoreticalapproachmakesuseofa Light-Frontquarkmodelasaframework tocalculatethedoublepartondistributionfunctionsatlow-resolution scale. QCD evolutionis implemented toreach the experimental scale. The obtainedvalues of

σ

eff in

thevalenceregionareconsistentwiththepresentexperimentalscenario,inparticularwiththesetsof datawhichincludethesamekinematicalrange.Howevertheresultofthecompletecalculationshowsa dependenceof

σ

eff onxi,afeaturenoteasilyseenintheavailabledata,probablybecauseoftheirlow

accuracy. Measurements of

σ

eff in restrictedxi regions are addressedto obtainindicationsondouble

partoncorrelations,anovelandinterestingaspectofthethreedimensionalstructureofthenucleon. ©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

MultiPartonInteractions(MPI),occurringwhenmorethanone parton scattering takes place in the same hadron–hadron colli-sion, have been discussed in the literature since long time ago

[1]and are presently attracting considerable attention, thanks to the possibilities offered by the Large Hadron Collider (LHC) (see Refs. [2–6] forrecent reports).In particular, thecross section for doublepartonscattering(DPS),thesimplestMPIprocess,depends on specific non-perturbative quantities, the double parton distri-bution functions (dPDFs), describing the number density of two partonswithgivenlongitudinal momentum fractionsandlocated atagiventransverseseparationincoordinatespace.dPDFsare nat-urallyrelatedto partoncorrelationsandtothethree-dimensional (3D)nucleonstructure,asdiscussedalsointhepast

[7]

.

No data are available for dPDFs and their calculation using non perturbative methods is cumbersome. A few model calcula-tionshavebeenperformed, tograspthemostrelevantfeaturesof

*

Correspondingauthor at:DipartimentodiFisica e Geologia,Universitàdegli StudidiPerugia,ViaA.Pascoli,I-06123Italy.

E-mailaddress:sergio.scopetta@pg.infn.it(S. Scopetta).

dPDFs

[8–10]

.Inparticular,inRef. [10]aLight-Front(LF)Poincaré covariant approach, naturally reproducing theessential sumrules ofdPDFs,hasbeendescribed.Althoughithasnotyetbeenpossible toextract dPDFsfromdata,a signatureofDPShasbeenobserved and measured in severalexperiments [11–16]: the so called “ef-fective cross section”,

σ

eff. Despiteoflarge errorbars,thepresent experimental scenarioisconsistentwiththeideathat

σ

eff is con-stantw.r.t.thecenter-of-massenergyofthecollision.

In this letter we present a predictive study of

σ

eff which makes use of the LF quark model approach to dPDFs developed inRef.[10].

Thedefinitionof

σ

eff isreviewedinthenextsection,wherean operative expression, suitable for microscopic studies and model calculations, is derived andthe presentexperimental situation is summarized.Then theresultsofourapproacharepresented crit-ically, discussingthedynamicaldependenceof

σ

eff inview of fu-tureexperiments.Conclusionsaredrawninthelastsection.

2. Theeffectivecrosssection

Theeffectivecrosssection,

σ

eff,isdefinedthroughthesocalled “pocketformula”,whichreads,iffinalstates A and B are produced inaDPSprocess(see,e.g.,

[5]

):

http://dx.doi.org/10.1016/j.physletb.2015.11.031

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

σ

eff

=

m 2

σ

_App

σ

_Bpp

σ

_doublepp

.

(1)

m is a process-dependent combinatorial factor: m

=

1 if A and B are identical and

m

=

2 if they are different.

σ

_App_(B₎ is the dif-ferential cross section for the inclusiveprocess pp

→

A

(

B

)

+

X ,

naturallydefinedas:

σ

_App

(

x1

,

x1

,

μ

1

)

=



i,k F_ip

(

x1

,

μ

1

)

Fp  k

(

x1

,

μ

1

)

σ

ˆ

ikA

(

x1

,

x1

,

μ

1

),

(2)

σ

_Bpp

(

x2

,

x2

,

μ

2

)

=



j,l Fp_j

(

x2

,

μ

2

)

Fp  l

(

x2

,

μ

2

)

σ

ˆ

jlB

(

x2

,

x2

,

μ

2

),

(3) where F_ip_(j) isaone-bodypartondistributionfunction(PDF) with

i

,

j

,

k

,

l

= {

q

,

q

¯

,

g

}

,

μ

1(2) is the factorizationscale for the process

A

(

B

)

,

σ

_doublepp ,the remaining ingredient in Eq.(1), appears inthe naturaldefinitionofthecrosssectionfordoublepartonscattering:

σ

d

=



σ

_doublepp

(

x1

,

x1

,

x2

,

x2

,

μ

1

,

μ

2

)

dx1dx1dx2dx2

,

(4) andreads:

σ

_doublepp

(

x1

,

x1

,

x2

,

x2

,

μ

1

,

μ

2

)

=

m 2



i,j,k,l



Di j

(

x1

,

x2

;

k⊥

,

μ

1

,

μ

2

)

σ

ˆ

ikA

(

x1

,

x1

,

μ

1

)

×

Dkl

(

x1

,

x2

; −

k⊥

,

μ

1

,

μ

2

)

σ

ˆ

jlB

(

x2

,

x2

,

μ

2

)

dk_⊥

(

2

π

)

2

.

(5) Intheaboveequation,k_⊥(

−

k_⊥)isthetransversemomentum un-balanceoftheparton1(2),conjugatedtotherelativedistancer_⊥

(thereadershouldnotconfusek_⊥withtheintrinsicmomentumof theparton,argumentoftransverse momentumdependentparton distributions).Thequantity

D

i j

(

x1

,

x2

;

k⊥

)

,calledsometimes “dou-blegeneralizedpartondistributions” (2GPDS) [17,18],istherefore theFouriertransformofthesocalleddoubledistributionfunction,

Di j

(

x1

,

x2

;

r⊥

)

, which represents the number density of partons pairs i, j with longitudinal momentum fractions x1, x2, respec-tively, at a transverse separation r_⊥ in coordinate space. dPDFs, describingsoftPhysics,arenonperturbativequantities.

Twomain assumptions areusually madefortheevaluationof dPDFs:

a)factorizationofthetransverseseparationandthemomentum fractiondependence:

Di j

(

x1

,

x2

;

k⊥

,

μ

)

=

Di j

(

x1

,

x2

,

μ

)

T

(

k⊥

,

μ

)

;

(6)

b)factorizedformalsoforthe

x

1,

x

2dependence:

Di j

(

x1

,

x2

,

μ

)

=

Fi

(

x1

,

μ

)

Fj

(

x2

,

μ

) θ (

1

−

x1

−

x2

)(

1

−

x1

−

x2

)

n

.

(7) The expression

θ (

1

−

x1

−

x2

)(

1

−

x1

−

x2

)

n, where n

>

0 is a parametertobe fixed phenomenologically,introduces thenatural kinematicalconstraint x1

+

x2

≤

1 (in Eqs. (6) and(7) the same scale

μ

=

μ

1

,

μ

2isassumed,forbrevity).

Onecomment aboutthe physical meaning of

σ

eff isin order. In Eq. (1), if the occurrence of the process B were not biased somehowby that of the process A, instead of the ratio

σ

B

/

σ

eff onewouldread

σ

B

/

σ

inel,representingtheprobability tohavethe process

B once A has

takenplaceassumingrarehardmultiple col-lisions. The difference between

σ

eff and

σ

inel measures therefore correlationsbetweentheinteractingpartons inthecolliding pro-ton.

Fig. 1. Center-of-massenergydependenceof

σ

effmeasuredbydifferentexperiments

usingdifferentprocesses[11–16].Thefigureistakenfrom[16].

Let usdiscuss now the dynamical dependenceof

σ

eff on the fractional momenta x1, x1, x2, x2. By inserting Eqs. (2)–(5) in Eq. (1), and omitting the dependenceon the factorization scales forsimplicity,onegetsthefollowingexpressionfor

σ

eff:

σ

eff

(

x1

,

x1

,

x2

,

x2

)

=

 i,kF p i(x1)Fp  k (x1)σˆikA(x1,x1) j,lF p j(x2)Fp  l (x2)σˆjlB(x2,x2)   i, j,k,lσˆikA(x1,x1)σˆjlB(x2,x2)  Di j(x1,x2;k⊥)Dkl(x1,x2; −k⊥) dk⊥ (2π)2

.

(8)

Eq.(8) clearly showsthe dynamical origin ofthe dependence of

σ

eff on thefractional momenta x1, x₁, x2,x₂. Evenwithin the “zerorapidity region”,( y

=

0),where

x

1

=

x₁,

x

2

=

x₂,sucha de-pendence,althoughsimplified,isstilleffective.

Assuming that heavy flavors are not relevant in the process, the dependence on the “partontype”, i

=

q

,

q

¯

,

g,

of the elemen-tarycrosssectionisbasically

[19]

:

ˆ

σ

i j

(

x

,

x

)

=

Ci j

σ

¯

(

x

,

x

),

(9)

where

σ

¯

(

x

,

x

)

is a universal function, and Ci j are color factors whichstayintheratio:

Cgg

:

Cqg

:

Cqq

=

1

: (

4

/

9

)

: (

4

/

9

)

2

.

(10)

UsingEq.(9),Eq.(8)simplifiesconsiderably:

σ

eff

(

x1

,

x1

,

x2

,

x2

)

=



i,k,j,lFi

(

x1

)

Fk

(

x1

)

Fj

(

x2

)

Fl

(

x2

)

CikCjl



i,j,k,lCikCjl



Di j

(

x1

,

x2

;

k⊥

)

Dkl

(

x₁

,

x₂

; −

k⊥

)

dk⊥ (2π)2

.

(11)

The presentexperimental scenario isillustrated in Fig. 1.The experiments

[11–16]

,atdifferentvaluesofthecenter-of-mass en-ergy,

√

s, andwithdifferentfinalstates,explore differentregions of xi. Experiments at high

√

s access low xi regions, in general. The old AFS data[11] are in thevalence region (0

.

2

≤

xi

≤

0

.

3), the Tevatron data [13,14] are in the range 0

.

01

≤

xi

≤

0

.

4 while therecentLHCdata

[15,16]

coveraloweraverage

x

i rangeandare dominatedbythegluedistribution.

Remarkablythe experimentalevidences are compatiblewitha constantvalueof

σ

eff inEq.(1),the

x

i-dependencebeingprobably hiddenwithin theexperimental uncertainties. In fact one should stress that the knowledge of the xi-dependence of

σ

eff would opentheaccesstoinformationonthe

x

i-dependenceofthedPDFs

Di j

(

x1

,

x2

;

r⊥

)

,entering the definitionof

σ

eff:a directwayto ac-cess the 3D nucleon structure [7]. Nowadays, the aspects of the 3Dnucleonstructurerelatedtothetransversepositionofpartons areinvestigatedthroughhard-exclusiveelectromagneticprocesses, such asdeeply virtual Compton scattering (DVCS), extractingthe Generalized Parton Distributions (GPDs) (see Ref. [20] for recent results). The information encoded in DPS, dPDFs and in

σ

eff, in itsfull

x

i dependence,areanywaydifferentandcomplementaryto thoseprovidedbyGPDsinimpactparameterspace.Whilethe lat-terquantities are one-bodydensities, depending on the distance oftheinteractingpartonwithgiven x from thetransverse center ofthe target, in DPS one is sensitive to the

relative distance

be-tween two partons withgiven longitudinal momentumfractions. Inotherwords,theinvestigationofdPDFsfromDPS,isrelevantto know,forexample,theaveragetransversedistanceoftwofast par-tonsortwoslowpartons:averyinterestingdynamicalfeature,not accessiblethroughGPDs.

3. Light-Frontquarkmodelcalculationoftheeffectivecross

section

dPDFshavea non-perturbativenature,and, atpresent, cannot be calculated in QCD. However they can be explicitly calculated, atalow resolutionscale, Q0

∼ 

QCD,usingquark models,as ex-tensivelydonefortheusualPDFs.Theresultsofthesecalculations shouldbethenevolvedusingperturbativeQCD(pQCD)inorderto matchdatatakenatamomentumscale Q

>

Q0.Theprocedureis nowadayswell established (see,e.g., Ref. [21] andthe references therein).TheQCDevolutionofthe

x

i-dependencesofdPDFs(from

Q0 to Q

>

Q0)isknown[22,23],andcurrentlyimplementedina systematicway(seeRef.[3,24]andthereferences therein).

Thefirst modelcalculationsofdPDFsinthe valenceregion,at thehadronicscale

Q

0,havebeenpresentedinabagmodel frame-work[8],andinaconstituentquark model(CQM),[9].Ofcourse CQM have the specific advantage of including correlations in a wayconsistentwiththequarkdynamics,fromtheverybeginning, a propertythatthebagmodelcannotfulfill.

InparticularthefullyPoincarécovariantLight-Frontmodel ap-proach we developed in Ref. [10] respects relevant symmetries, broken in the descriptions of Refs. [8,9], allowing for a correct evaluationoftheMellinmomentsofthedistributions and, conse-quently,foraprecisepQCDevolutiontohighmomentumtransfer. Inthiswayourmodelcalculationscanberelevantfortheanalysis ofhigh-energydata.The model,extensivelyappliedto the evalu-ationofdifferentpartondistributions,(see,e.g.,Refs. [25–27]and thereferences therein),isagoodcandidatetograspthemost rele-vantfeaturesofdPDFs.Forthepresentstudyitisenoughtorecall that the protonstate isgiven by a spatial wave function and an SU(6)symmetric spin-isospin part (see Ref. [25] for details).The spatial part is numerical solution ofa relativistic Mass equation, dynamically responsible forthepresence ofcorrelations between thetwo quarksinthe CQMwave function(anon-relativistic ver-sion of the model was introduced in Ref. [28]). The Light-Front calculationsof Di j

(

x1

,

x2

;

k⊥

,

μ

)

,inRef. [10], showsthatthe fac-torizationofEq.(6)isbasicallyvalid,butthecommonassumption ofEq.(7)isstronglyviolated.Besides,thestrongcorrelationeffects presentatthescaleofthemodelarestillsizable,inthevalence re-gion,attheexperimentalscale,i.e.afterQCDevolution.Atthelow valuesof

x,

presently studiedattheLHC,thecorrelationsbecome lessrelevant,althoughtheireffectsarestillimportantforthe spin-dependentcontributionstounpolarizedprotonscattering.

We haveexplicitly calculated single anddouble parton distri-butions entering Eq. (11), and then

σ

eff relying on the natural assumption Eq. (10)only. We adhere,in addition, to the simpli-fying choice of a single factorization scale

μ

1

=

μ

2

=

μ

0, used

in, e.g., Refs. [2,4,29].

μ

0 has to be interpreted, in the present approach, as the hadronic scale, where only valence quarks u

and

d are

present. Consideringthesymmetriesofourmodel,one has

u

(

x

,

μ

0

)

=

2d

(

x

,

μ

0

)

,

D

uu

(

x1

,

x2

,

k⊥

,

μ

0

)

=

2Dud

(

x1

,

x2

,

k⊥

,

μ

0

)

andEq.(11)simplifiesto

σ

eff

(

x1

,

x1

,

x2

,

x2

,

μ

0

)

=

81u

(

x1

,

μ

0

)

u

(

x1

,

μ

0

)

u

(

x2

,

μ

0

)

u

(

x2

,

μ

0

)

64



Duu

(

x1

,

x2

;

k⊥

,

μ

0

)

Duu

(

x₁

,

x₂

; −

k⊥

,

μ

0

)

₍dk_2π⊥₎2

.

(12)

In principle,

σ

eff depends therefore on four momentum frac-tions. For the seek of convenience, in order to clearly show the mainfeaturesofourresults,

σ

eff hasbeenevaluatedatzero rapid-ity( y

=

0),where

x

i

=

xi,sothatoneremainswith

σ

eff

(

x1

,

x2

,

μ

0

)

=

81u

(

x1

,

μ

0

)

2u

(

x2

,

μ

0

)

2

64



Duu

(

x1

,

x2

;

k⊥

,

μ

0

)

2 dk_(2π⊥₎2

.

(13)

In order to illustrate our results we will concentrate on the valenceregionwherethepresentmodelismorepredictive.In par-ticularweconcentrateonthekinematicsoftheoldAFSdata

[11]

, which means y

0 (x1

x₁, x2

x₂) and 0

.

2

≤

x1,2

≤

0

.

3. The average momentum scale,again assumedtobe the same forthe processesinitiated bythetwo differentcollisions, turnsouttobe

Q2

250 GeV2.Theresultsofthecalculationsareshownin

Fig. 2

, at the scale of the model,

μ

2

0

0

.

1 GeV2, and after non-singlet evolutionto

Q

2 _{(detailsonthefixingofthehadronicscaleandon} thecalculationoftheQCDevolutioncanbefoundinRef.[10]).

What is immediately seen is an x1,2 dependence of the re-sults, which changeup to 100% even in thisnarrow kinematical range.Such a dependenceisfound atboth theexperimental and themodelscale.Theslopeofthesurfaceintherightpanelof

Fig. 2

isinvertedw.r.t.thatintheleftpanel.Itisnotasurprisingfeature, duetothedifferentevolutionpropertiesofthenumeratorand de-nominatorinEq.(12)andconsistentwiththeevolutioncalculated inRef.[10].

It is worth to notice that the three old experimental extrac-tions of

σ

eff fromdata

[11–13]

,whichinvolve thevalenceregion (cf. Fig. 1), lie in the obtained range of values of

σ

eff, shown in Fig. 2. It is important to remark that, since the dPDF evolu-tion on k_⊥ is still an open challenge, only the x1,2 evolution at fixed k_⊥ has been taken into account in evolving the model ef-fective cross section to Q2

250 GeV2. Moreover, we show our results,forthemomentbeing, onlyinaregiondominatedby va-lence quarks,wherethenon-singletsectoroftheevolution equa-tionsplaysamajorrole.Wehavesolvedthereforeonlythissector, wheretheinhomogeneous partoftheevolutionequations, describ-ing the possibility that the quarksbelonging to thesame proton andparticipatingto theinteraction are originatedfromthesame parton, has no effect.At low x, in processes initiated by gluons, this contribution, sometimes called in the literature, for brevity, “1v2”, hasbeen foundto give an importantcontribution [18,33], although some doubtson its actual impact have been raised re-centlyinRef.[34].

This x dependence, found at the hadronic scale aswell asat high Q2,canbeattributedtodifferentdynamicalandkinematical properties:

1. boththenumeratorandthedenominator vanishquicklywith

xi through the valenceregion, but the lattervanishes faster, mainlyduetothekinematicconstraint

x

1

+

x2

≤

1 ofthedPDF, a quantity appearing only inthe denominator.The LF model correctlyreproducesuchakinematicalconstraint;

2. thecorrelationsintroduced by the LF dynamicsandeffective bothinthe

x

iand

k

⊥dependence;

Fig. 2.σeff(x1,x2,Q2)for the values of x1, x2measured in Ref.[11]. Left panel: hadronic scale; right panel: Q2=250 GeV2. 3. thecorrelationsinducedbythepQCDevolutioninthevalence

region.

Let us compare our predictions for

σ

eff to other estimates, mostlyperformedinaveryfarkinematicalregime.At verylow-x gluonsarestronglydominating(thisisthehypothesisin

[17]

, par-tiallycorrectedin

[18]

),sothatitisenoughtoconsider

i

,

j

,

k

,

l

=

g

in Eq. (11). Assuming, in addition, a fully factorized approach,

Dgg

(

x

,

x

;

k⊥

)

=

Fg

(

x

)

Fg

(

x

)

g

(

k⊥

)

,

σ

eff becomes:

σ

eff

(

x1

,

x1

,

x2

,

x2

)

→

σ

eff

=

1



g2

₍

_k ⊥

)

₍dk_2π⊥₎2

.

(14)

AnequivalentschemeisusedinRef.[17],relyingonan uncor-relatedansatz in coordinatespace, leading, in momentum space, toadPDF givenby theproductoftwogluon GPDsatzero“plus” momentumtransfer.1 _{Writing inturneach GPDinan}

_{x, t}

_{= −}

_k2

⊥ factorizedform,

σ

eff isobtainedtobeabouttwicethe experimen-tal value. Later, the same authors have corrected this prediction takingintoaccount apossible contributioncomingfrom interact-ingpartonsoriginatedbythesameparton.This“1v2”mechanism wasfoundtobe abletoreconciletoalargeextenttheprediction withthe measuredvalues

[18]

.Obviously,thevalidity ofEq.(14)

isspoiledbycorrelation effectsandrestrictedtovery low-xi.The problems related to the uncorrelated ansatz are discussed in a number of papers (see, e.g., Ref. [2,30–32]). In particular, in the valenceregionwearediscussinghere,thisassumptionisnot sup-portedby modelcalculations

[8–10]

andit iscertainly untrue in pQCD, beingalso spoiledby QCD evolution. Inother words, sev-eralargumentsleadtotheconclusionthat,ingeneral,

σ

eff should be

x

i dependent,namely:breakingofthefactorizationansatz;the QCD evolution; contribution of more than one parton type (not onlygluonsasatverylow

x

i)totheDPScrosssection.

NoneoftheassumptionsleadingtoEq.(14),validatverylow

x,

isreliableinourcalculation.Nevertheless,toobtainasingle num-berto be qualitativelycompared withthepresent

x independent

valueswhich have been reportedby theexperimental collabora-tions so far, one can try to reduce the resultsof our calculation

1 _{Foragenericfourmomentuma}μ_{,weusethefollowingnotationforthe“plus”}

(“minus”)lightconecomponent:a+= (a0_+a3

)/√2 (a−= (a0_−a3 )/√2).

to a fullyfactorized approach todPDFs, following thehypothesis oftenassumed(cf. Eqs.(6),

(7)

):

Duu

(

x1

,

x2

;

k⊥

,

μ

0

)

=

u

(

x1

,

μ

0

)

u

(

x2

,

μ

0

)

fuu

(

k⊥

),

(15)

whereanaturaldefinitionforthe“effectiveformfactor”, fuu

(

k⊥

)

, inourapproach,is

fuu

(

k⊥

)

=

1

4



dx1dx2Duu

(

x1

,

x2

;

k⊥

,

μ

0

),

(16)

a quantity which turns out to be scale independent. Withinthis approximation,Eq.(12)yields:

σ

eff

(

x1

,

x1

,

x2

,

x2

,

μ

0

)

→

σ

eff

=

81 64



f2 uu

(

k⊥

)

₍dk_2π⊥₎2

10

.

9 mb

,

(17)

a value which turns out to be independent on the momentum

scale Q and on the longitudinal momentum fractions xi, x_i,and compares reasonably well withthe sets of data shownin Fig. 1, inparticularwiththoseofthethreeoldest experiments,atlower values of

√

s, strongly involving the valence region. There is no reason why the LHC experimental scenario should be described by our model,which is,for the moment being, a valence quark one.Thissimplifiedresult,restrictedtothevalenceregion,is con-nectedtotheabilityofthemodeltocapture(initswavefunction) thecorrectaveragedistancebetweenthevalencequarksin trans-versespace,relatedtothe“effectiveformfactor”, fuu

(

k⊥

)

,Eq.(16), whichcanbewrittenalsoasfollows:

fuu

(

k⊥

)

=

3

(

k1

,

σ

1

,

τ

1

;

k2

+

k⊥

,

σ

2

,

τ

2

;

k3

,

σ

3

,

τ

3

)

| δ(

k1

+

k2

+

k3

)

×

Pu

(

1

)

Pu

(

2

)

|(

k1

+

k⊥

,

σ

1

,

τ

1

;

k2

,

σ

2

,

τ

2

;

k3

,

σ

3

,

τ

3

)

,

(18)

usingdirectly theproton state

|

andthe u-flavor projector for theparticle

i, P

u

(

i

)

= (

1

+

τ

z

(

i

))/

2.LookingatEq.(18),onecan no-ticethat,intheansatzgivenbyEqs.(15)and

(16)

,themodel cor-relationsdescribedbythedPDFarestillinparteffectivelypresent through fuu

(

k⊥

)

.

The

x-dependence

wearediscussingdoesnotemergefromthe presentdata,probablynotaccurate enough.Ourstudypoints out therefore to an experimental scenario where more precise

mea-Fig. 3. σeff(x1,x2,Q2)for the valuesof x1, x2 measured inRef. [11], at Q2= 250 GeV2_{,consideringthecontributionofgluonsperturbativelygenerated.}

surementsinnarrow

x

iregionscouldshednewlightonthe struc-tureof theproton andonthe nature ofhard proton–proton col-lisions. If the x-dependence is seen, one will gain, through

σ

eff, a firstindicationofdoublepartoncorrelationsandafreshlookat the3Dprotonstructure.

Ourtreatmentso farhasbeendevelopedconsidering only va-lence quarks, andone could wonder whether,in the valence re-gion,the inclusion of other degreesof freedom (sea quarks, glu-ons),possiblyrelevantinspecificchannels, could changethis im-portant xi dependence. For the moment being, we are able to show,in

Fig. 3

,in thesamevalencewindowof Fig. 2andforan illustrativepurposeonly, the effectivecrosssection asresultofa singletevolutionwhichtakesintoaccountthecontributionof per-turbativelygeneratedgluons, at Q2

=

250 GeV2.Thisamounts at calculating,accordingtoEq.(11),thefollowingexpression:

σ

eff

(

x1

,

x2

,

μ

)

=

N

(

x1

,

x2

,

μ

)/

D

(

x1

,

x2

,

μ

),

(19) where N

(

x1

,

x2

,

μ

)

=

C2ggg21g22

+

3CggCqg

[

u1g1g22

+

u2g2g21

]

+

9 4CggCqq

[

u 2 1g22

+

u22g12

]

+

27 4 CgqCqq

[

u 2 1u2g2

+

u22u1g1

]

+

9C2_gqu1u2g1g2

+

81 16C 2 qqu21u22

,

(20) and D

(

x1

,

x2

,

μ

)

=



_dk ⊥

(

2

π

)

2

[

C 2 ggD2gg

+

8CggCgqDggDgu

+

8C2gqD2gu

+

4C2_gqDggDuu

+

8CggCqqD2gu

+

16CgqCqqDguDuu

+

4Cqq2 D2uu

].

(21) In Eqs. (20) and(21) we have used, forbrevity, the notation

ui

=

u

(

xi

,

μ

)

and Di j

=

Di j

(

xi

,

xj

;

k⊥

,

μ

)

. Looking at these equa-tions,oneshouldnoticethat:

i

)

ifthegluoncontributionsarenot inserted,Eq.(13)isrecoveredfromEq.(19);

ii

)

usehasbeenmade ofthesymmetriesofthemodel,givenaboveEq.(12),forthequark sector;

iii

)

thepossiblecontributionoftheseaquarksinthis kine-maticalwindowinspecificchannelshasnotbeenincluded.Indeed, withrespecttothegluonone,thelatterwill besuppressedbythe

colorfactor(cf.Eq.(10)) andithasbeenneglectedforthesakeof clarity.Asitiseasilyseenin

Fig. 3

,thenewresultdoesnotdiffer dramatically fromthat shownin

Fig. 2

,right panel.In particular, the

x

idependence,themostimportantfeatureoftheresultsofthe presentinvestigation, isfound to bevery similar, while a signifi-cantchangeinsizeisseen,uptoafactoroftwoinsomeregions. However, weobservethat the“1v2”contributionis notincluded, neither attheperturbative noratthenon perturbativelevel,and couldhave,accordingtosomestudies,animportantimpact,ofthe same order of the calculated effective cross section [18,33]. The studyofits possiblerole,aswell asthe analysisofthe low x

re-gion,arebeyondthescopeofthepresentpaperandwillbetreated elsewhere.

In closingthis section, let usqualitatively addressthe experi-mentalscenarioswhereourprediction,validforthemomentonly in thevalence region, can be tested. The presentrun ofthe LHC isvery promisingforthedetectionofDPSbutonlythelow x

re-gionwillbeaccessed.ItappearsthereforedifficulttodetectaDPS event generated by four valence quarks, as the one we are ad-dressing here. Nonetheless, valencequarks contributingto a DPS process together with seaquarks or gluonshave a chance to be observed.Thesecontributions couldbe selectedlookingatjetsof differentrapidity,sothatthe

x

1,

x

₁ and/or

x

2,

x

₂ valuesliein dif-ferent regions, namely, one in the valence region and the other atlower longitudinalmomentum. Thestudyofthe dPDFs associ-atedtotheseprocesseswouldrequiretheinclusionofperturbative and non-perturbative gluon and sea contributions in our analy-sis, andthe check ofa possiblerole ofthe “1v2” processes.This will be treated elsewhere. Concerning the possibility to observe our predictedx dependence of theeffective crosssection for va-lencequarks,hardlyaccessibleattheLHC,someinformationcould begot,inprinciple,inspectingcarefullytheoldTevatrondata.The latterhaveindeedbeentakenattypicalvaluesof

√

s and looking atobservedfinalstateswhichenterdeeplythevalenceregion.

4. Conclusions

An operativeexpressionof

σ

eff hasbeen obtainedintermsof dPDFsandstandardPDFs.Thankstothisnewresult,direct micro-scopic calculationsof

σ

eff canbeperformedindifferentscenarios and kinematicalframes. Inparticular, in thepresentanalysis, we have calculatedthe effectivecross section

σ

eff within a relativis-ticPoincaré covariantquark model.Extracted fromproton–proton scattering databy severalexperimental collaborations inthe last 30years,

σ

eff representsatool tounderstanddoubleparton scat-tering ina p–pcollision. Ourinvestigation predicts abehavior of

σ

eff which,whenaveragedoverthelongitudinalmomentum frac-tions xi, isconsistent with the presentexperimental scenario, in particularwiththesetsofdatawhichincludethevalenceregion. However, at the same time, an xi dependence of

σ

eff is found, a feature not easily read in the available data. This dependence isfoundwhenanonsingletevolutionofthevalencedistributions isperformed,aswellaswhenperturbativelygeneratedgluonsare includedintothescheme.

We conclude that the measurement of

σ

eff in restricted xi ranges would lead to a first indication ofdouble parton correla-tions inthe proton, addressing a novel and interesting aspect of the 3D structure of the nucleon. The analysis of particular pro-cesses where these effects could be mosteasily seen,as well as the extension ofthe model to obtain a better description of the low-x region,presentlystudiedatLHC,areinprogress.

Acknowledgements

This work was supported in part by the MINECO under

SEV-2014-0398.S.S. thankstheDepartmentofTheoreticalPhysics oftheUniversityofValenciaforwarmhospitalityandsupport.M.T. andV.V.thanktheINFN,sezionediPerugiaandtheDepartmentof PhysicsandGeologyoftheUniversity ofPerugiaforwarm hospi-talityandsupport.Severaldiscussions withL.Fanò aregratefully acknowledged.

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