Elastic scattering in geometrical model

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Elastic

scattering

in

geometrical

model

Zbigniew Plebaniak

a,

∗

,

Tadeusz Wibig

a,b

a_{NationalCentreforNuclearResearch,AstrophysicsDivision,CosmicRayLaboratory,ul.28PułkuStrzelcówKaniowskich69,90-558Łód´z,Poland} b_{FacultyofPhysicsandAppliedInformatics,UniversityofŁód´z,ul.Pomorska149/153,90-236Łód´z,Poland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received20May2016

Receivedinrevisedform31August2016 Accepted31August2016

Availableonline6September2016 Editor:L.Rolandi Keywords: Elasticscattering Opticalmodel Crosssection Cosmicrays

Theexperimentaldataonproton–protonelasticandinelasticscatteringemergingfromthemeasurements attheLargeHadronCollider,callsforanefficientmodeltofitthedata.Wehaveexaminedtheoptical, geometricalpicture and wehave foundthe simplest,lineardependence ofthismodel parameterson thelogarithmoftheinteractionenergywiththesignificantchangeoftherespectiveslopesatonepoint corresponding totheenergyofabout300GeV.Thelogarithmicdependenceobservedathighenergies allows one to extrapolate the proton–proton elastic, total(and inelastic) cross sections to ultra high energiesseenincosmicrayseventswhichmakesasolidjustificationoftheextrapolationtoveryhigh energydomainofcosmicraysandcouldhelpustointerpretthedatafromanastrophysicalandahigh energyphysicspointofview.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

The process of elastic scattering of hadronshas been studied experimentallyinawideenergyregionformorethanhalfa cen-tury.Inthe1960’swiththeavailablecenterofmass(c.m.s.) ener-giesof

√

s

=

4–6GeV itwasfoundthat theconventional “diffrac-tion cone” mechanism failed what was clearly visible at larger transferredmomenta.Additionaldataattheenergies of

√

s

=

19, 20,23, 28, 31, 45, 53, 62 GeV were published in the middle of 70’s. At the end of the previous millennium the range of avail-ableenergiesends around2TeV. Onlyrecentlytheresultsofthe TOTEMcollaborationattheLHConelasticppscatteringprocesses at

√

s

=

7TeV werepublished[9,12].

Themeasurements attheLHC at7TeV c.m.s.collisionenergy setthenextpoint onanenergyscalewheretheoptical modelof hadrons can be examined. The observed so far evolution of the protonshadowprofileandthe energydependenceofthe param-eters describing its shape could be extended towards the limit of the ultra high-energy cosmic rays (UHECR), where important questionsofphysicsandastrophysicsarestillunanswered.Itis ex-pectedthat theanswerscould be linked (also)to some extentto thevalueoftheproton–protoncrosssectionsataround1020_{eV of} laboratoryenergy.

*

Correspondingauthor.

E-mailaddress:zp@zpk.u.lodz.pl(Z. Plebaniak).

Manyphenomenologicalmodelsofprotonhavebeenproposed. As itissaid by DremlininRef. [39] [....]“Mostofthemaspiretobe ‘aphenomenologyofeverything’relatedtoelasticscatteringofhadrons inawideenergyrange.Doingsointheabsenceofapplicablelawsand methodsofthefundamentaltheory,theyhavetousealargenumber ofadjustableparameters.Thefreeparametershavebeendeterminedby fittingthemodelresultstotheavailableexperimentaldata.” [...] Indepen-dentoftheirsuccessandfailure,wearesurethat,“inthelongrun,the physicalpicturemaybeexpectedtobemuchmoreimportantthanmost ofthedetailedcomputations”. (thelastcitationisfromthe1969 pa-perbyChengandWupublishedinthefirstvolumeofPhys.Rev. D

[36]).

2. Phenomenology of the scattering process

The elasticscatteringamplitude F

(

s

,

t

)

describingthe proton– protonscattering

d

σ

el

d

|

t

|

=

π

|F

(

t

)

|

2

_,

₍₁₎

couldbeparameterizedinmanywaysstartingfromthesimple ex-ponential exp

(

Bt

)

proposedalreadyin1964by OrearinRef.[54]. Newdataallowsformoresophisticatedform. Itwas proposedby BargerandPhillips[56]in1973intheform

F

(

s

,

t

)

=

i



A

(

s

)

e12B(s)t

+



_C

₍

_s

₎

_eφ(s)_e12D(s)t



,

(2)

whichcanbeusedfor7TeVLHCscatteringdataexplicitly[45,50], ormodified,asproposed,e.g.,inRef.[42]

http://dx.doi.org/10.1016/j.physletb.2016.08.064

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Table 1

Theextrapolatedcrosssectionsinmbathigherenergies.

Energy (√s) 14 TeV 24 TeV 30 TeV 57 TeV 95 TeV

Fagundes et al.[41] 108.6±1.2

Bourelly et al.[27] 103.63±1.0

Petrov et al.[55] 106.73

Block, Halzen et al.[26] 107.30

Islam et al.[48] 110.00 Jenkovszky et al.[50] 111.00 Block[22] 133.40±1.6 AKENO[53] 104±26 124±34 Fly’s Eye[18] 120±15 AUGER[3] 133.20±13 Telescope Array[1] 170.00±50 This work 105.56 115.8 120.33 132.66 143.09 F

(

s

,

t

)

=

i



A

(

s

)

e12B(s)t_G

₍

_s

_,

_t

₎

+



C

(

s

)

eφ(s)e12D(s)t



,

(3)

orinthe numberofpossibilities inspected byKhoze, Martinand RyskininRef.[51].

Adifferentmodificationwas proposed by Menonand collabo-ratorsinRef. [35]who considertheparameterizationofthe scat-teringamplitudeasasumofOrearexponentials[54]:

F

(

s

,

t

)

=

n



i=1

α

ieβit

.

(4)

Theyobtained,withthesummationofuptosixcomponents, per-fect fits to the ISR data from 19.4 GeV to 62.4 GeV [40]. Their ‘model-independent’ analysis of elastic proton–proton scattering data [16,17,40]was extended to higher energies andthe param-eters

α

i and

β

iwereexpressedasfunctionsoftheavailablec.m.s.

energy.PredictionsforLHCweregiventhereandarelisted inthe

Table 1.

On the other handthe absorption processes can be naturally studiedinageometricalframework.Thecorrespondencebetween interactiongeometryandthemomentumtransferspaceisdefined withthe Fourier transform withthe help of the profile function

(

s

,

b

)

(ortheeikonal



)

F

(

s

,

t

)

=

i ∞



0 J0



b

√

−t



(

s

,

b

)

b db

=

=

i ∞



0 J0



b

√

−

t



{

1

−

exp [

−(

s

,

b

)

]

}

b db

.

(5)

Thisgivesthepossibilitytoapplytheform-factorformalismtothe hadroninteraction

(

s

,

t

)

=

C

(

s

)

Gp

(

t

)

Gp

(

t

)

whereC

(

s

)

worksfor

theabsorptioncoefficient.

(

s

,

b

)

= (

1

−

i

α

)

∞



0 J0

(

qb

)

G2p,E

(

t

)

f

(

t

)

f

(

0

)

q dq

,

(6)

(q

=

√

−

t). This formalism has beenproposed anddeveloped by Bourrely,SofferandWusincelate70’s[28–30]using

Gp,E

(

t

)

=

1



1

−

t

/

m2₁

 

1

−

t

/

m2₂

 ,

f

(

t

)

=

f

(

0

)

a 2

₊

_t a2

₋

_t

.

(7)

The initial simple model withsix free parameters (at high ener-gies) becomes atthe LHC energies much more complicated[31].

Theasymptoticformhasbeeneventuallyestimatedandcompared withthenumericalresultsinRef.[27].

The pure geometrical picture of protonscattering andthe re-lation of the scatteringamplitude to the transmissioncoefficient (

||

)appearsalreadyin1968inthepaperbyChouandYang[37]. The main point there is to find the (mean) opaqueness, which maybe,ingeneral,acomplex-valuedfunction,forthegivenvalue of the impact parameter. It is quite natural to assume that the hadron hastheinternal structuredefinedby thedensityfunction

ρ

(

x

,

y

,

z

)

.Taking z asacollisionaxiswecandefineahadron pro-file D

(

b

)

=

∞



−∞

ρ

(

x

,

y

,

z

)

dz

,

(8)

andfortwocollidinghadronstheconvolutionis

(

b

)

= (

b

)

=

i Kpp ∞



−∞



D

(

b

−

b

)

D

(

b

)

d2b

.

(9)

Any particular model could be fully characterized by the gener-alized opacity: the eikonal function



(in the impact parameter space) as itis written in Eq.(5).Its particular shape can be ob-tainedusingdipoleelectromagneticformfactorslikeitisdone,for example,inRef.[37]similartotheonegiveninEq.(7).

Another interesting way of introducing



is to use the evo-lution ofthe imaginarypart ofthe profilefunction

(

s

,

b

)

=

1

−

exp [

−(

s

,

b

)

] whichcouldbe, accordingto Ref.[34],determined using the nonlinear differential logistic equation. The concept is that it includes, in a natural way, saturation effects expected as energygrows.Thisassumptionleadsto

(

s

,

b

)

=

1

e(b−b0)/γ

+

₁

,

(10)

where b0 and

γ

areproton radial scaleparameters whichdefine the cross section scaling properties. A very similar profile func-tion was foundasa specialcaseofthemodel ofRybczy ´nskiand Włodarczykwhere shapesofcollidingprotons aredefinedby the event-by-eventfluctuationsoftheradiusoftheprotoninthe‘black disk’ picture [57]. Ifthefluctuationsare negligiblethe blackdisk limitisretained,whileforthecrosssectionfluctuationsdescribed bythegammadistribution,anotherextremeisobtained:the Gaus-sianprotonprofile.

Introducing new scaling variable b

ˆ

=

b

/

b0

(

s

)

to Eq. (10) the protonprofilesatisfiesthe(modified)geometricalscaling(if

γ

/

b0 isconstant) d

σ

el dt

∼

b 2 0



f

|t|

b2₀



2

,

(11)

andeventuallythescatteringpicturetendstotheblackdisklimit when the energy goes to infinity (b0

→ ∞

):

σ

tot.

∼

σ

el.

∼

b20 (

σ

el.

/

σ

tot.

=

1

/

2).

InthepaperbyIslam,LuddyandProkudin[49]theprofile func-tion

(

s

,

b

)

waschosenarbitrarily[47]

(

b

,

s

)

=

g

(

s

)



1 1

+

e(b−b0)/γ

+

1 1

+

e−(b+b0)/γ

−

1



.

(12) ThecomparisonofEq.(12)withEq.(10)showsaninteresting sim-ilarity. The results of the Islam model agree withthe measured data above ISR energies quite well [49], however, at 7 TeV the agreementisnotasperfect[44].

Itisknownfora longtime,that thegeometricalscalingholds belowISR energies(

√

s

<

20GeV).TheanalysisbyBrogueira and DiasdeDeus[34]showsthatstartingfromthehighestISRenergy theprotonappearstobegettingblackerandedgieralready,inSPS at

√

s

=

200GeV itbecomesquiteclearandthistendency contin-uouslybecomesmorevisibleastheenergygrows.

Inthe series of papers ofBlock and co-workers there is pro-posedthe “Aspenmodel” [21,25].The eikonal function



in this modelisasumoffourseparatecomponentsrelatedtoindividual

qq gg, gq interactionsandtheoddeonexchange

(

s

,

b

)

=

i

σqq

(

s

)

A

(

b

,

μqq

)

+

σgg

(

s

)

A

(

b

,

μgg

)

+

+

σqg

(

s

)

A

(

b

,

μqg

)

+

σodd

(

s

)

A

(

b

,

μodd

)



,

(13) with A

(

b

,

μ

)

∼

K3

(

μ

b

)

. The model is used mainly for the esti-mation andextrapolationof thetotal (elastic andinelastic) cross sectionsto extremelyhighenergies. Its agreementwith high en-ergyscatteringdataisnot perfect,asitisshownin[24] forLHC 7 TeV.ThemodifiedBesselfunctionappearsintheAspenmodelas aresultof convolutionsofthe hadrondensities distributedagain

[37]inthewaywhichleadstodipoleelectromagneticformfactor fromsimilartotheoneshowninEq.(7).

3. The modification of the simplest model

The predictions for the simplest models of hadrons are well known (see, e.g., Ref. [23]). From the geometrical point of view, thegeneralpictureissuchthatprotonsbecomeblacker,edgierand larger(BEL)[46]. Oneofthesimplestandquiteobvioushadronic matterdistributions istheexponentialone

ρ

h

(

r

)

=

m3_h

8

π

e

−mh|r|

_.

₍₁₄₎

Thecomplexformoftheeikonalcouldbedefinedusingthe

λ

fac-tor which defines the ratio of the real to the imaginary part of the



λ(

s

)

=

 ((

b

,

s

))

((

b

,

s

))

.

(15)

Theenergydependenceof

λ

hasbeenknownquiteaccuratelyfor alongtime anditwassmoothlyparameterized,e.g.,byMenonin Refs.[40,52].Forthepresentcalculationwehaveslightlymodified thissolution.InFig. 1 ourdependency of

ρ

(

s

)

isshownin com-parisonwithselecteddata.

TheexponentialformofEq.(14)hasbeenusedinRef.[58]and the results of scatteringcross section were given there. In gen-eraltheagreementwiththedataisseenbelowandattheregion ofthefirstdip(

|

t

|

<

0

.

7 GeV2).The diffractive-likepictureofthe scatteringdifferentialcrosssectionisrathersatisfactorythere,but thedeficitofhighermomentatransfers(

|

t

|

>

1 GeV2_)isthe essen-tialdefectofthesimple modelingeneral. Toobtain thesolution closer to the high p_⊥ experimental distribution we have exam-inedaslightlymoresophisticatedhadronicmassdistribution: we

Fig. 1. Ratioofthe realandthe imaginarypartofthescatteringamplitude ρ=  (F(b,s)) / (F(b,s)).Solidcurveistheresultoftheparameterization ofλ(s)from Ref.[40,52]modifiedslightlybyususedinthepresentwork.Pointsrepresentdata from[6,8,11,14,43].

Fig. 2. Profilefunctionsforthreeenergies(√s=19GeV,546GeVand7TeV)used inthepaper.

haveusedinstead oftheone exponentialdistribution thesumof two withdistinct exponentsm1 andm2 anddifferent normaliza-tionfactorsc1 andc2.

ρ

h

(

r

)

=

1 8

π

c1m31e−m1|r|

+

c2m32e−m2|r|

.

(16)

ThefourparametersofthedistributionproposedinEq.(16)are thesubjectofthefittingprocedure. Thevaluesofm1 andm2 are notmuchdifferentfromeachother,aswellasthevaluesofc1and

c2andtheprofilesobtainedeventuallyfromEq.(14)andthese ad-justedusingEq.(16)arequitesimilar.Theprofilesobtainedinthe present work atthree specific and characteristic energies (low – 19GeV,high,themiddleSPS:546GeV,andrecentLHC7TeV)are shown,asexamplesinFig. 2.Presentedprofilefunctions

(

b

)

are describedbyfollowingequation:

(

b

)

=

1

−

e−(b)

,

(17)

where

(

b

)

iscalculatedusingEq.(9).

Multicomponentgeometrical modelsof highenergyscattering appearasaresultofthedecompositionoftheinteractingnucleon intoconstituentsofdifferentnature,whichcouldhave,thus, differ-entdistributionsontheimpactparameterplane.In“Aspenmodel” of Block and Halzen [26], interacting protons are compounds of quarks andgluons. This approachleads to the three (qq, gg and qg)differentpartsofthehighenergyeikonalprofilefunction,see Eq.(13).

Anotherideawhichleadstothetwocomponentsystemisthe one proposed by Bialas andBzdak [20]. The proton is there de-composedintoapairofaquarkandadiquark.Theaverageradius

Fig. 3. Thedifferentialelasticcrosssectionsfromourmodelforc.m.senergiesof19 GeV,546GeVand7TeVshownasafunctionofthe(|t|×σtot)according tothe suggestionofEq.(11)comparedwiththemeasurements[4,9,12,19,33].

ofthe diquark distributionis significantly largerthan that ofthe remaining single quark constituent. Thissimple idea isimproved by the additionof the realpart tothe forward scattering ampli-tudeandisexaminedby CsorgoandNemesinRef. [38]whereit isshownthatitcanbesuccessfullyappliedtotheLHCenergiesof 7TeV.

The other model ofIslam, Luddyand Prokudin[49] describes thehighenergyproton–protonscatteringassumingthat nucleons havethehard innercoreandthediffractive,softoutercloud.The amplitude isthesumofthehard core–corescatteringdominated athigh

|

t

|

valuesandthesoft,low

|

t

|

scatteringoftheoverlapping clouds.

4. Results

Proton profiles shown in Fig. 2 were obtained with the ad-justmentprocedureusingthedataofthedifferentialelasticcross sections.Wepayourattentiontoreproducethemain characteris-ticsofthemeasureddistributions:

– theslopeatthelowmomentumtransfer, – thepositionofthefirstdiffractivepeak,

– thebehavior afterthepeakandtheslopeathighmomentum transfers.

Theaccuracyoftheobtaineddatadescriptionforthethree charac-teristicenergiesisshowninFig. 3.Wehavepresentedourmodel predictionsofthedifferentialcrosssectiondistributionasa func-tion of the product of the value of the momentum transfer and thetotalcrosssectionwhichfollowstheideaofRef.[34]givenby Eq.(11).

Fourparametersm1,2 andc1,2 were foundfortenenergydata sets starting from

√

s

=

20 GeV, through five energies of ISR (23–62GeV)anSPSpointat546GeV,twoTevatronmeasurements at0.6and2TeV andtheLHCdatasetmeasured atc.m.s.energy of7 TeV. Allthe parameter valuesadjusted individually for each energy are given in Figs. 4 (a) and (b) as solid symbols. Recent LHC data obtained at the energy of 8 TeV are not included for thefittingprocedureofourmodelbecauseofthelowmomentum transferrange

(

0

.

029

<

|

t

|

<

0

.

195

)

[13]covered.Anyway,wetried to usethem forthe slope ofthe differentialelastic cross section determinationandtheresultisgiveninFig. 5.

Comparingthevaluesobtainedfordifferentenergies weseea clearregularity.TheresultsinFig. 4suggestasimpleformofthe low and high energy asymptotics for all the four parameters. It seemsthatthedependencetendstogetlinearinlogarithmicscale forallfourparameter cases.The linesare showninFig. 4 bythe

Fig. 4. Thevaluesofadjustedparametersm1,2 (a)andc1,2(b)areshownbythe solidsymbols.Dashedlinesshowasymptoticlinearenergydependencies(forhigh andlowenergies)ofallparameters.Solidlinesarethesmoothoverallenergy de-pendencieseventuallyadoptedfortheresultspresentedinthiswork(Figs. 2,3,5 and6).

Fig. 5. Calculated slopesoftheelasticdifferentialcrosssectionsasafunctionof interactionenergy.Lineshowsourmodelpredictions,pointsarefitstothe experi-mentalresultsfrom[3,4,7,10,13,19,33]intherangeof0.1<|t|<0.3GeV2_. dotted lines. The relatively small number of high energy elastic scattering experiments providing differential elastic cross section distributions ofthequalityandrangegoodenoughforourmodel parameterestimationproceduredoesnotallowustomakestrong conclusions,buttheobtainedevidencelooksquiteimpressive. Ad-ditionally thegenuinecharacter oftheproposed parameterization is confirmed by the fact that the change from low to high en-ergy regime is located at approximately the same energy point. Thechancecoincidenceofsuchbehavior isratherunbearable.

Fig. 6. Valuesoftheelastic, inelasticand totalcrosssectioncalculatedwithour modelasafunctionoftheinteractionenergycomparedwiththemeasurements. Solidlinerepresentstotalcrosssection,dashedinelasticanddottedlineelasticcross sectionpredictions.PointsareexperimentalresultsfromRefs.[1–3,5,6,8,10,11,15,18, 32,43,53].

Theasymptoticdependenciesandthesmooth connections be-tweenthem showninall fourcasesin Fig. 4by the solidcurves wereusedtoobtainourmodelpredictionspresentedinFigs. 2,3, 5 and6.

Usingthevalues oftheparameters givenby thesolid linesin

Fig. 4somedetailedcharacteristicsoftheproton–protonscattering canbe obtained. Oneofthem isthelow

|

t

|

slope. Itcan be eas-ilyobtainedfrom published dataand we presentthem withour modelpredictioninFig. 5.

Themost importantcharacteristics ofthe proton–proton scat-teringaretheintegratedcrosssections(total,elasticandinelastic). WithourmodelandtheparametersasdescribedinFig. 4theycan becalculated.

ResultsaregiveninFig. 6.Ourmodelpredictionsarecompared therewiththedatafromacceleratormeasurementsandwiththe resultsobtainedbythecosmicrayexperiments:Fly’sEye[18],PAO

[3]and Telescope Array [1]. As it can be seen, the extrapolated modelpredictions arein agreementwithhigh-energy cosmic ray data.

Thenumerical modelpredictions forthe energies higherthan measuredsofaronesstarting fromtheLHC14TeV uptoUHECR energies,are givenintheTable 1.Theyarecompared withsome valuesexistingintheliterature.

5. Summary

Wehavedevelopedamodifiedsimpleopticalmodelofproton– protonscatteringwithfourmodelparameters,whichallowsusto describethehadronicmatterdistributionofcollidingprotons. Val-uesof all the parameters were adjusted to the elastic scattering datainthe widerange ofenergies fromstationarytarget experi-mentbelowISR to the7TeV energyofLHC

|

t

|

distributiondata. Usingthe eikonal parameterizationthe correctdescription ofthe totalandelastic crosssections,theelastic slopeandthe differen-tialelasticcrosssectionatlargevaluesofmomentumtransferhave beenfound.

However,the satisfactorydescription oftheexisting scattering datawasnotthemainresultofourwork.Wehavefound addition-allyasmooth andvery simplebehavior ofall parameter’senergy dependence.

Inparticular,wefound thatforthelow andthehighenergies thereareasymptoticregimeswhichseemtobelinearinthe loga-rithmicscaleoftheavailableinteractionenergy

√

s.Moreover,the changeofthelowenergytothehighenergyregimeisfoundtobe locatedforallthemodelparametersataroundthesamepointon theenergyscale–

√

s

=

300 GeV.Its more accurateestimationis

impossiblebecauseofthelackofthedatainthisparticularenergy range.

Thesefactsconstitutethesolidbasefortheextrapolationofthe scatteringcrosssectionsabove energiesavailablefromaccelerator measurements atpresent, up to the ultra-highenergies observed incosmicrays.

In agreement with “BEL” behavior our model shows that the proton becomes blacker and larger as the energy increases. In-creasing of c1 parameter which represents normalization of in-ner core of the proton shows that blackening is faster above

√

s

=

300 GeV. Other three parameters indicate also a slow in-creaseofhadronicradius.Thanks tothe newparameterizationof thehadronicmatterdistributionweobtainbetteragreementwith scatteringdifferentialcrosssectionacceleratordataforhigher mo-mentatransfersforallavailableenergies.

Using our model, we calculate the predictions for the pp

to-talcrosssectionat

√

s

=

14TeV,and57TeV,andtheyare

σ

tot pp

=

105

.

56mb and132.66mb,respectively.Fortheinelasticcross sec-tionstherespectivevaluesare:

σ

inel

pp

=

79

.

71mb and98.69mb.

Appendix A. Supplementary material

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttp://dx.doi.org/10.1016/j.physletb.2016.08.064.

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