Proton-proton forward scattering at the LHC

UNIVERSIDADE ESTADUAL DE CAMPINAS

SISTEMA DE BIBLIOTECAS DA UNICAMP

REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP

Versão do arquivo anexado / Version of attached file:

Versão do Editor / Published Version

Mais informações no site da editora / Further information on publisher's website:

https://www.sciencedirect.com/science/article/pii/S0370269319307695

DOI: 10.1016/j.physletb.2019.135047

Direitos autorais / Publisher's copyright statement:

©2019

by Elsevier. All rights reserved.

DIRETORIA DE TRATAMENTO DA INFORMAÇÃO

Cidade Universitária Zeferino Vaz Barão Geraldo

CEP 13083-970 – Campinas SP

Fone: (19) 3521-6493

http://www.repositorio.unicamp.br

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Proton-proton

forward

scattering

at

the

LHC

M. Broilo

a

,

b

,

D.A. Fagundes

c

,

E.G.S. Luna

b

,

d

,

∗

,

M.J. Menon

e

a_{InstitutodeFísicaeMatemática,UniversidadeFederaldePelotas,96010-900,Pelotas,RS,Brazil}

b_{InstitutodeFísica,UniversidadeFederaldoRioGrandedoSul,CaixaPostal15051,91501-970,PortoAlegre,RS,Brazil}

c_{DepartamentodeCiênciasExataseEducação,UniversidadeFederaldeSantaCatarina- CampusBlumenau,89065-300,Blumenau,SC,Brazil} d_{InstitutodeFísica,FacultaddeIngeniería,UniversidaddelaRepública,J.H.yReissig565,11000Montevideo,Uruguay}

e_{InstitutodeFísicaGlebWataghin,UniversidadeEstadualdeCampinas,13083-859,Campinas,SP,Brazil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received26June2019

Receivedinrevisedform8October2019 Accepted14October2019

Availableonline23October2019 Editor:B.Grinstein

Keywords:

Othernonperturbativecalculations Elasticscattering

Totalcross-sections

RecentlytheTOTEMexperimentattheLHChasreleased measurementsat√s=13 TeVofthe proton-proton total cross section,

σ

tot, and the ratio of the real to imaginary parts of the forward elastic amplitude,

ρ

.SincethenanintensedebateontheC -parityasymptoticnatureofthescatteringamplitude wasinitiated. Weexaminetheproton-protonand theantiproton-protonforwarddata above10GeVin the contextof aneikonal QCD-basedmodel, wherenonperturbative effects are readily includedviaa QCDeffective charge.We show that,despitean overallsatisfactory descriptionoftheforward data is obtainedbyamodelinwhichthescatteringamplitudeisdominatedbyonlycrossing-evenelasticterms, there isevidence that theintroduction ofacrossing-odd termmay improve theagreement with the measurementsof

ρ

at√s=13 TeV.IntheReggelanguagethedominanteven(odd)-under-crossingobject isthesocalledPomeron(Odderon).

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Theriseofthetotalcrosssectionwithenergyinhadron-hadron collisionswastheoreticallypredictedmanyyearsago[1].Thisrise, as indicated by the data accumulated from collider and cosmic ray experimentsover the last severaldecades,is dictatedby jets withtransverse energy ET muchsmaller than the totalenergy s

available in the hadronic collision [2]. From the QCD viewpoint, theseminijets arisefromsemihard scatteringsofelementary par-tons, defined as hard scatterings of partons carrying very small fractionsofmomentaoftheir parenthadrons[3,4].Inthispicture the high-energy behavior of the cross sections is driven mainly bysemihardprocessesinvolvinggluons,sincetheygivethe domi-nantcontributionatsmallx [5–7].Inaccordancewithverygeneral predictionsbasedonaxiomatic field theory,all modelsusingthis QCD-based formalism have assumedover the years that at high energies the scattering amplitude is dominated by only a single crossing-evenelasticamplitude.In theReggelanguage this domi-nanteven-under-crossingobjectisnamedPomeron[8];intheQCD languagethePomeronis,initssimplestconfiguration,acolor sin-gletmadeupoftwogluons[9,10].

*

Correspondingauthor.

E-mailaddress:luna@if.ufrgs.br(E.G.S. Luna).

Very recentlythe TOTEMexperimentat theLHC has released measurements at

√

s

=

13 TeV of the proton-proton (pp) total crosssection,

σ

tot

=

110

.

6

±

3

.

4 mb[11],andinasubsequentwork

anotherindependentmeasurement,

σ

tot

=

110

.

3

±

3

.

5 mb,together

withmeasurementsoftheratioreal-to-imaginarypartsofthe for-wardamplitude,

ρ

=

0

.

09

±

0

.

01 and

ρ

=

0

.

10

±

0

.

01 [12].These measurements can be simultaneously well-described by models that includetheodd-under-crossing partner ofthePomeron:The Odderon [13,14]. The Odderon concept is physically reasonable sinceits presencedoesnotviolateasymptotictheoremsbased on analyticity, unitarity and crossing. Also from the QCD viewpoint theOdderonis,initssimplestconfiguration,acolorsingletmade upofthreegluons[15].Morespecifically,inperturbative QCDthe Odderon can be associated to a colorless C -odd t-channel state with an intercept at or nearone. The TOTEM results havesince then triggered an intense debate on the question of whether or notthecombinedbehaviorof

σ

tot and

ρ

athighenergies is

actu-allyamanifestationoftheOdderon[14,16].

This Letter revisits this issueby investigating the behavior of the forward quantities

σ

tot and

ρ

, from pp and p p (antiproton-

¯

proton) scattering, through an eikonal QCD-based model with a scattering amplitude dominated asymptotically by only single crossing-evenamplitudes[7,17,18].Weconsideramodelinwhich the bridge betweentheparton dynamics,described by QCD, and thedynamicsofobservablehadronsystemsisprovidedbytheQCD

https://doi.org/10.1016/j.physletb.2019.135047

0370-2693/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

2 M. Broilo et al. / Physics Letters B 799 (2019) 135047

partonmodelusing updated setsofquark and gluondistribution functions,standardQCDcrosssectionsforparton-partonprocesses andphysicallymotivatedcutoffsthat restrict theparton-level dy-namics to the semihard sector. The nonperturbative dynamics of the QCD is treated in the context of a well-established infrared effectivecharge dependent on thedynamical gluon mass.Such a model,involvingonlyeven-under-crossingamplitudesdominantat veryhighenergies,providesasatisfactory(fromastatistical view-point) global description of

σ

tot and

ρ

data over a wide range

ofenergies,andfurthermore,thelarger

ρ

value predictedbythe model at

√

s

=

13 TeV suggests that a crossing-odd elastic term mayplayanimportantroleinthesoftandsemihardinteractions. 2. Themodel

Theunitarityofthematrix S requiresthat theabsorptivepart oftheelasticscatteringamplitudereceivescontributionsfromboth elasticandinelasticchannels.Inimpactparameter(b) representa-tiontheunitarityconditionimpliestherelation

2Re h

(

s

,

b

)

= |

h

(

s

,

b

)

|

2

+

Ginel

(

s

,

b

),

(1)

whereh

(

s

,

b

)

istheprofilefunctionandGinel

(

s

,

b

)

istheinelastic

overlap function,i.e. the contributionfrom all inelastic channels. Theprofilefunctionh

(

s

,

b

)

isrelatedtothe elasticscattering am-plitude

A(

s

,

t

)

by

A

(

s

,

t

)

=

i ∞



0 b db J0

(

b

√

−

t

)

h

(

s

,

b

),

(2) wheret istheusualMandelstamvariable.Inordertosatisfy uni-tarity constraints, the profile function is conveniently written as h

(

s

,

b

)

=

1

−

e−χ(s,b)_{,wheretheeikonalfunction}

_χ

₍

_s

_,

_b

₎

_isa com-plex generalizedphase-shift.Interms oftheeikonal functionthe elasticamplitudereads

A

(

s

,

t

)

=

i ∞



0 b db J0

(

b

√

−

t

)



1

−

e−χ(s,b)



,

(3) where

χ

(

s

,

b

)

=

Re

χ

(

s

,

b

)

+

iIm

χ

(

s

,

b

)

≡

χ

R

(

s

,

b

)

+

i

χ

I

(

s

,

b

)

.

Hence the scattering amplitude is completely determined once the eikonal function is known. The eikonal is written in terms of even and odd parts connected by crossing symmetry. In the caseof pp and p p channels,

¯

thiscombination reads

χ

ppp p¯

(

s

,

b

)

=

χ

+

(

s

,

b

)

±

χ

−

(

s

,

b

)

.Twoobservablesthatplayakeyroletounravel thestructure ofthe elasticscatteringamplitude athighenergies, namely

σ

tot and

ρ

,canbewrittenintermsof

χ

(

s

,

b

)

.Weseefrom

theopticaltheoremthatthetotalcrosssectionissuchthat

σ

tot

(

s

)

=

4

π

Im

A

(

s

,

t

=

0

),

(4)

thus,from(3) wehave

σ

tot

(

s

)

=

4

π

∞



0 b db

[

1

−

e−χR(s,b)_cos

_χ

I

(

s

,

b

)

].

(5)

Theparameter

ρ

,thereal-to-imaginaryratiooftheforwardelastic scatteringamplitude,

ρ

=

Re

A

(

s

,

t

=

0

)

Im

A

(

s

,

t

=

0

)

,

(6)

can,inturn,bewrittenusing(3) as

ρ

(

s

)

=

−



_∞ 0 b db e−χR(s,b)sin

χ

I

(

s

,

b

)



_∞ 0 b db

[

1

−

e−χR(s,b)cos

χ

I

(

s

,

b

)

]

.

(7)

Since athigh energies thesoft andthe semihardcomponents ofthescatteringamplitudearecloselyrelated,wecanassumethat theeikonalfunctionisadditive withrespecttothesoftand semi-hard (SH) partoninteractions, andwrite the eikonal as

χ

(

s

,

b

)

=

χ

sof t

(

s

,

b

)

+

χ

S H

(

s

,

b

)

[5,17,18].Weassumethattheoddsemihard

eikonal is nulland the even contribution is obtained asfollows. FromconsiderationsbasedontheQCDpartonmodelandthe uni-taritycondition (1),theprobabilitythatneithernucleonisbroken upinacollisionatimpactparameterb isgivenbye−2χR(s,b)_[5,6].

Itfollowsthat theevenpartofthesemihardeikonalcontribution factorizes[6,7,17,18],

χ

_{S H}+

(

s

,

b

)

=

1

2

σ

Q C D

(

s

)

WS H

(

s

,

b

),

(8) whereWS H

(

s

,

b

)

isanoverlapdensityforthepartonsatb ands,

WS H

(

s

,

b

)

=

1 2

π

∞



0 dk_⊥k_⊥ J0

(

k⊥b

)

G2

(

s

,

k⊥

),

(9)

and

σ

Q C D

(

s

)

istheusualQCDcrosssection

σ

Q C D

(

s

)

=



i j 1 1

+ δ

i j 1



0 dx1 1



0 dx2 ∞



Q2 min d

|ˆ

t

|

d

σ

ˆ

i j d

|ˆ

t

|

(

ˆ

s

, ˆ

t

)

×

fi/A

(

x1

,

|ˆ

t

|)

fj/B

(

x2

,

|ˆ

t

|) 



ˆ

s 2

− |ˆ

t

|



,

(10)

where

ˆ

s andt are

ˆ

theMandelstaminvariantsfortheparton-parton collision, with

|ˆ

t

|

≡

Q2 and x1x2s

>

2

|ˆ

t

|

>

2Qmin2 , where Qmin2 is

theminimal momentumtransferinthe semihardscattering[5–7, 19]. In the above expression x1 and x2 are the fractions of the

momentaoftheparenthadrons A andB carriedbythepartonsi and j,withi

,

j

=

q

,

q

¯

,

g,d

σ

ˆ

i j

/

d

|ˆ

t

|

isthe differentialcrosssection

for i j scattering,and fi/A

(

x1

,

|ˆ

t

|)

( fj/B

(

x2

,

|ˆ

t

|)

) isthe partoni ( j)

distributioninthehadron A (B).

Since thegluon distributionbecomes very largeas x

→

0,the parton-parton scattering processes used in the computation of

χ

S H

(

s

,

b

)

mustcontainatleastonegluonintheinitialstate.Thus,

in the calculation of

σ

Q C D

(

s

)

, we selectthe processes gg

→

gg,

qg

→

qg, qg

¯

→ ¯

qg, and gg

→ ¯

qq. In fact, at

√

s

=

7 TeV and Qmin

=

1

.

3 GeV, their relative contributionto

σ

Q C D

(

s

)

is around

98.8% for the post-LHC fine-tuned parton distribution functions (PDFs)suchasCT14[20].

It is well knownthat these elementaryprocesses are plagued by infrared divergences.Nevertheless, they can be regularized by considering an effective charge whose finite infrared behavior is constrainedby agluondynamical-massscale [21].Thedynamical massisintrinsicallyrelatedtoaninfraredfinitestrongcoupling

α

¯

s,

andits existence, based onthe fact that thenonperturbative dy-namics of QCD may generate an effectivemomentum-dependent mass Mg

(

Q2

)

forthe gluons(while preserving thelocal SU

(

3

)

C

invariance) [22],isstronglysupportedby QCDlattice results[23]. More specifically,lattice simulations reveal that thegluon propa-gatorisfiniteintheinfrared region[24] andthisbehavior corre-sponds,fromtheSchwinger-Dyson formalism,to amassivegluon [25]. In this Letter we adopt the functional forms of

α

¯

s and

Mg

(

Q2

)

obtainedthrough theuse of the pinch technique in

or-der toderive agaugeinvariant Schwinger-Dysonequation forthe gluonpropagatorandthetriplegluonvertex[21]:

¯

α

s

(

Q2

)

=

4

π

β

0ln

Q2

₊

_4M2 g

(

Q2

)



/

2

,

(11) M2_g

(

Q2

)

=

m2_g

ln

Q2

+

4m2_g



/

2



ln

4m2g

/

2





₋12 11

,

(12)

where

istheQCD scaleparameter,

β

0

=

11

−

2nf

/

3 (nf isthe

number of flavors) and mg is the gluon mass scale. From

lat-ticesimulationsandphenomenologicalresultsitsvalueistypically foundtobeoftheordermg

=

500

±

200 MeV[7,17,18,24,25].Note

thatinthelimit Q2

→

0 theeffectivecharge

α

¯

s

(

Q2

)

havean

in-fraredfixedpoint,i.e.thedynamicalmasstamestheLandaupole. We considera semihard overlap density WS H

(

s

,

b

)

that takes

intoaccounta“broadening”ofthespatialdistribution[17,26], WS H

(

s

,

b

;

ν

S H

)

=

ν

2 S H 96

π

(

ν

S Hb

)

3_K 3

(

ν

S Hb

),

(13)

where

ν

S H

=

ν

1

−

ν

2ln

(

s

/

s0

)

,with

√

s0

≡

5 GeV,andK3

(

x

)

isthe

modifiedBessel functionofthe second kind.Here,

ν

1 and

ν

2 are

constantstobe fitted.Theabove expressionisobtainedassuming a dipole form factor G

(

s

,

k_⊥

;

ν

S H

)

=

ν

2

S H

/(

k2⊥

+

ν

2S H

)



2

in equa-tion (9). The energydependence of WS H

(

s

,

b

)

is suggestive of a

partonpicturewherethesemihardinteractions aredominatedby gluonswhereassoftinteractionsaremainlyrelatedtointeractions amongvalencequarks[17].Henceevenandoddsoftcontributions basedonRegge-Gribovformalism[8,10,27,28] areverywellsuited forourpurposes:

χ

_{sof t}+

(

˜

s

,

b

)

=

1 2Wsof t

(

b

;

μ

+ sof t

)



A

+

√

B

˜

s

+

C

˜

s λ



,

(14)

χ

_{sof t}−

(

˜

s

,

b

)

=

1 2Wsof t

(

b

;

μ

− sof t

)

D

√

˜

se −iπ/4

_,

₍₁₅₎

where

˜

s

≡

s

/

s0,

μ

−_{sof t}

≡

0

.

5 GeV,

λ

=

0

.

12, and A, B, C , D and

μ

+_{sof t} are fittingparameters. Theterm C

˜

sλ _{in (14) representsthe} contributionofasoftPomeron, whosebehavioris welldescribed bya power closeto

˜

s0.12 _{[28]. Inorder to studythesoft}

contri-bution to the total cross sections at highenergies we have also attemptedaFroissaron[14] termC ln2

(

˜

s

)

intheplaceofthepower, butwe havenot observedanysignificant difference betweenthe twocases.Thesoftformfactorsaresimply“static”versionsof(13),

Wsof t

(

b

;

μ

+,−_{sof t}

)

=

(

μ

+,−_{sof t}

)

2 96

π

(

μ

+,− sof tb

)

3_K 3

(

μ

+,−_{sof t}b

).

(16)

Inthisscenariothesoftodd eikonal

χ

_{sof t}−

(

˜

s

,

b

)

is significantonly inthelow-energyregime.Weensurethecorrectanalyticity prop-ertiesoftheC -evenamplitudesfromthesubstitution s

→

se−iπ/2

throughoutEqs. (10), (13) and(14) [7,19], validforan even am-plitude and equivalent to the use of a dispersion relation. The Eq. (15) already have the correct analyticity structure. The pre-scriptions

→

se−iπ/2_{isappliedonEq. (10) takingintoaccountthe}

factthat

σ

Q C D

(

s

)

corresponds tothe realpartofa moregeneral

complexanalytic function F

(

−

is

)

.Thus

σ

Q C D

(

s

)

can becorrectly

reproduced by means of a novel flexible function F , and if F is aholomorphicfunctioninaregion

,theanalytical continuation ensurestheuniquenessoftherealpartof F

(

−

is

)

.Specifically,we consideracomplexanalyticparametrizationgivenby

F

(

−

is

)

=

b1

+

b2eb3[X(s)] 1.01 b4

+

b5eb6[X(s)] 1.05 b7

+

b8eb9[X(s)] 1.09 b10

,

(17) Table 1

Best fit parameters of the QCD-based model with CT14 PDFs. The dynami-cal gluon mass scale was set tomg= 400 MeV,whiletheminimalmomentum transferwassettoQmin=1.3 GeV.

ν1[GeV] 1.70±0.21 ν2[GeV] 0.015±0.014 A [GeV−2_{] 88} .2±9.6 B [GeV−2_{] 51} .7±9.7 C [GeV−2_{] 27.2±5.7} D [GeV−2_{] 24.2±1.4} μ+sof t[GeV] 0.90±0.16 ζ 166 χ2_{/ζ 1.30} P(χ2_{; ζ ) 5.4×10}−3

whereb1

,

...,

b10 are freefit parameters and X

(

s

)

=

ln ln

(

−

is

)

. In

thisproceduretheb1

,

...,

b10parameters arethenadjusted to

op-timallysatisfytherelationReF

(

−

is

)

=

σ

Q C D

(

s

)

withlessthan1%

error.It followsthat ImF

(

−

is

)

=

Im

σ

Q C D

(

s

)

.In thecaseofCT14,

the data reduction yields: b1

=

100.220 GeV−2, b2

=

0.434

×

10−1 _GeV−2_,b

3

=

1.274, b4

=

1.919,b5

=

0.122

×

10−7 GeV−2,

b6

=

14.050,b7

=

0.504,b8

=

3.699

×

103 GeV−2,b9

= −

80

.

280,

b10

= −

2

.

632.

3. Resultsandconclusions

Inouranalyseswecarryoutaglobalfittoforward(t

=

0) scat-teringdatafrom

√

smin

=

10 GeVtoLHCenergies.Weusedatasets

compiledandanalyzedbytheParticleDataGroup [29] aswellas the recentdataat LHCfromthe ATLAS [30] and theTOTEM [11, 12] Collaborations,withthe statisticandsystematicerrors added in quadrature.Specifically, we fit to the total crosssections,

σ

totpp

and

σ

tot¯p p, andtheratios ofthe realto imaginary partofthe

for-wardscatteringamplitude,

ρ

pp _and

_ρ

p p¯ _{.Indoingthisweutilizea}

χ

2 _{fittingprocedure,adoptingan interval}

_χ

2

₋

_χ

2

min

correspond-ing to the projection ofthe

χ

2 _{hypersurface containing 68.3% of}

probability,namely1

σ

.The value of

χ

2

min isdistributedasa

χ

2

distributionwith

ζ

degreesoffreedom. Asa convergencecriteria weconsideronlydatareductionswhichimplypositive-definite co-variance matrices, since theoretically the covariancematrix fora physicallymotivatedfunctionmustbe, atthe minimum, positive-definite. As tests of goodness-of-fitwe adopt the chi-square per degreeoffreedom

χ

2

_/ζ

_{andtheintegratedprobability P}

₍

_χ

2

_;

_{ζ )}

_.

In performing calculations using the QCD-based formalism it is necessary to make some choice out of the many PDFs avail-able.WehavechosentheCT14PDFsfromaglobalanalysisbythe CTEQ-TEAgroup [20].The CT14isanext generationofPDFsthat include data fromthe LHC forthe first time. The CT14 set,that also includes updated data from the HERA and Tevatron experi-ments, is available from LHAPDF [31]. The CTEQ-TEA team have incorporatedimportantenhancementsthatmadeCT14 instrumen-taltoouranalysis.Morespecifically,CT14employs,atinitialscale Q0,aflexibleparametrizationbasedontheuseofBernstein

poly-nomials,which permit a better captureof thePDFs variations in the DGLAP evolution; CT14 alsoincludes for the first time mea-surementsofinclusiveproductionofvectorbosons[32],aswellas ofjets[33],at7and8TeVasinput fortheglobalfits.The inclu-sion of single-inclusive jet production measurements at 7TeV is sufficienttoconstrainthegluondistributionfunctionandleadsto aclearimprovementinthegluonPDFuncertainty.

The values of the fitted parameters are given in Table 1.The

χ

2

_/ζ

_{for the fit was obtained for 166 degrees of freedom. The}

resultsofthefits to

σ

tot and

ρ

forboth pp and p p channels

¯

are

4 M. Broilo et al. / Physics Letters B 799 (2019) 135047

Fig. 1. Total cross section for pp (solid curve) andp p (dashed curve) channels.¯

Fig. 2. Ratio oftherealtoimaginarypartoftheforwardscatteringamplitudefor

pp (solidcurve)andp p (dashed¯ curve)channels.

Fig.1we include, only asillustration,two estimatesof

σ

tot from

cosmic-ray experiments: theAuger resultat57 TeV[34] andthe TelescopeArrayresultat95TeV[35].

Notice that our QCD-based model allows us to describe in a satisfactory waythe forwardscattering observables

σ

totpp and

ρ

pp

from

√

s

=

10 GeV to 13TeV. Ourpredictions forthe totalcross sectionand

ρ

parameterat

√

s

=

13 TeVare

σ

totpp

=

105

.

6+

7.9 −6.3 mb

and

ρ

pp

₌

₀

_.

₁₁₈₂+0.0074

−0.0057,respectively.Theuncertaintiesinour

the-oretical predictions have been estimatedtaking into account the typical uncertainty of thegluon mass scale andthe PDFs uncer-taintiesontheproductioncrosssectionsattheLHC,asdiscussed in Ref. [17]. These resultsshow that ourmodel, whose formula-tioniscompatiblewithanalyticityandunitarityconstraints,iswell suited for predictions of forward observables to be measured at high-energyscales.Wearguethatthedistinctioninthemodel be-tween semihard gluons, which participate in hard parton-parton scattering, and soft gluons, emitted in any given parton-parton QCDradiationprocess,ishighly relevanttounderstanding our si-multaneous description of

σ

tot and

ρ

data over a wide range of

energies.TheintroductionofinfraredpropertiesofQCD,by consid-eringthat the nonperturbative dynamics ofQuantum Chromody-namicsgeneratean effectivegluon mass,isofcentralimportance since at highenergies the soft andthe semihard components of thescatteringamplitude arecloselyrelated[3].Mostimportantly, froma rigorousstatistical point ofview, ouranalysis showsthat the TOTEM measurements can be simultaneously described by a QCDscatteringamplitudedominatedby onlysinglecrossing-even

elasticterms.Crossing-evendominanceentailsthat

σ

totp p¯

−

σ

pp tot

→

0

and

ρ

totp p¯

−

ρ

pp

tot

→

0 ass

→ ∞

.

However, it is worth noting that the values of

ρ

measured by TOTEM at

√

s

=

13 TeV are lower than our prediction:

ρ

=

0

.

09

±

0

.

01 and

ρ

=

0

.

10

±

0

.

01 [12].Notethattheseexperimental valuesare measuredwithquiteextraordinary precision,andeven taking into account our theoretical uncertainties the model pre-diction is notcompatiblewiththe

ρ

values measuredby TOTEM at

√

s

=

13 TeV. These measurements,ifeventually confirmed by otherexperimentalgroups,suggestthatacrossing-oddelasticterm may play a central role in the soft andsemihard interactions at highenergies.Inotherwords,theTOTEM

ρ

datasuggestthe exis-tenceofanOdderon.

Acknowledgements

We are thankful to an anonymous referee for the impor-tant points raised on a previous version of this work. This re-search was partially supported by the Conselho Nacionalde De-senvolvimento Científico e Tecnológico (CNPq) under the grant 141496/2015-0. DAF acknowledges support by the project INCT-FNA (464898/2014-5). EGSL acknowledges the financial support fromtheANII-FCE-126412project.

References

[1]H.Cheng,T.T.Wu,Phys.Rev.Lett.24(1970)1456; C.Bourrely,J.Soffer,T.T.Wu,Phys.Rev.Lett.54(1985)757. [2]D.Cline,F.Halzen,J.Luthe,Phys.Rev.Lett.31(1973)491;

T.K.Gaisser,F.Halzen,Phys.Rev.Lett.54(1985)1754; T.K.Gaisser,T.Stanev,Phys.Lett.B219(1989)375;

A.Capella,J.VanTranThanh,J.Kwiecinski,Phys.Rev.Lett.58(1987)2015; I.Sarcevic,S.D.Ellis,P.Carruthers,Phys.Rev.D40(1989)1446.

[3]L.V.Gribov,E.M.Levin,M.G.Ryskin,Phys.Rep.100(1983)1; E.M.Levin,M.G.Ryskin,Phys.Rep.189(1990)267; M.G.Ryskin,Surv.HighEnergyPhys.11(1997)115.

[4]G. Agakichiev, et al., CERES/NA45 Collaboration, Phys. Rev. Lett. 92 (2004) 032301.

[5]L.Durand,H.Pi,Phys.Rev.Lett.58(1987)303; L.Durand,H.Pi,Phys.Rev.D38(1988)78; L.Durand,H.Pi,Phys.Rev.D40(1989)1436.

[6]G.Pancheri,Y.N.Srivastava,Phys.Lett.B182(1986)199;

A.Corsetti,A.Grau,G.Pancheri,Y.N.Srivastava,Phys.Lett.B382(1996)282; D.A.Fagundes,A.Grau,G.Pancheri,O.Shekhovtsova,Y.N.Srivastava,Phys.Rev. D96(2017)054010.

[7]E.G.S.Luna,A.F.Martini,M.J.Menon,A.Mihara,A.A.Natale,Phys.Rev.D72 (2005)034019;

E.G.S.Luna,A.A.Natale,Phys.Rev.D73(2006)074019.

[8]V.Barone,E.Predazzi,High-EnergyParticleDiffraction,Springer-Verlag,Berlin, 2002.

[9]L.N.Lipatov,Sov.J.Nucl.Phys.23(1976)338;

V.S.Fadin,E.A.Kuraev,L.N.Lipatov,Sov.Phys.JETP44(1976)443; V.S.Fadin,E.A.Kuraev,L.N.Lipatov,Sov.Phys.JETP45(1977)199; Y.Y.Balitsky,L.N.Lipatov,Sov.J.Nucl.Phys.28(1978)822.

[10]J.R.Forshaw,D.A. Ross,QuantumChromodynamics and thePomeron, Cam-bridgeUniversityPress,Cambridge,1997.

[11]G.Antchev,etal.,TOTEMCollaboration,Eur.Phys.J.C79(2019)103. [12]G. Antchev, et al., TOTEM Collaboration, CERN-EP-2017-335v3, arXiv:1812.

04732 [hep-ex].

[13]L.Lukaszuk,B.Nicolescu,Lett.NuovoCimento8(1973)405;

D.Joynson,E.Leader,B.Nicolescu,C.Lopez,NuovoCimentoA30(1975)345. [14]E.Martynov,B.Nicolescu,Phys.Lett.B778(2018)414;

E.Martynov,B.Nicolescu,Phys.Lett.B786(2018)207. [15]C.Ewerz,arXiv:hep-ph/0511196;

C.Ewerz,arXiv:hep-ph/0306137; M.A.Braun,arXiv:hep-ph/9805394.

[16]V.A.Khoze,A.D.Martin,M.G.Ryskin,Phys.Lett.B780(2018)352; M.Broilo,E.G.S.Luna,M.J.Menon,Phys.Lett.B781(2018)616; S.M.Troshin,N.E.Tyurin,Mod.Phys.Lett.A33(2018)1850206; V.A.Khoze,A.D.Martin,M.G.Ryskin,Phys.Lett.B784(2018)192; V.A.Khoze,A.D.Martin,M.G.Ryskin,Phys.Rev.D97(2018)034019; P.Lebiedowicz,O.Nachtmann,A.Szczurek,Phys.Rev.D98(2018)014001; E.Gotsman,E.Levin,I.Potashnikova,Phys.Lett.B786(2018)472;

M.Broilo,E.G.S.Luna,M.J.Menon,Phys.Rev.D98(2018)074006;

W.Broniowski,L.Jenkovszky,E.RuizArriola,I.Szanyi,Phys.Rev.D98(2018) 074012;

T.Csörg ˝o,R.Pasechnik,A.Ster,Eur.Phys.J.C79(2019)62. [17]P.C.Beggio,E.G.S.Luna,Nucl.Phys.A929(2014)230;

C.A.S.Bahia,M.Broilo,E.G.S.Luna,Phys.Rev.D92(2015)074039. [18]E.G.S.Luna,Phys.Lett.B641(2006)171;

E.G.S.Luna,A.L.dosSantos,A.A.Natale,Phys.Lett.B698(2011)52; D.A.Fagundes,E.G.S.Luna,M.J.Menon,A.A.Natale,Nucl.Phys.A886(2012) 48.

[19]J.Kwiecinski,Phys.Lett.B184(1987)386;

J.Kwiecinski,A.D.Martin,Phys.Rev.D43(1991)1560. [20]S.Dulat,etal.,Phys.Rev.D93(2016)033006. [21]J.M.Cornwall,Phys.Rev.D22(1980)1452;

J.M.Cornwall,Phys.Rev.D26(1982)1453.

[22]I.C.Cloet,C.D.Roberts,Prog.Part.Nucl.Phys.77(2014)1;

A.C.Aguilar,D.Binosi,J.Papavassiliou,Front.Phys.(Beijing)11(2016)111203; C.D.Roberts,Few-BodySyst.58(2017)5.

[23]A.Cucchieri,T.Mendes,Phys.Rev.D81(2010)016005;

A.Cucchieri, D. Dudal,T. Mendes,N. Vandersickel, Phys. Rev.D 85(2012) 094513;

A.Ayala,A.Bashir,D.Binosi,M.Cristoforetti,J.Rodriguez-Quintero,Phys.Rev. D86(2012)074512;

P.Bicudo,D.Binosi,N.Cardoso,O.Oliveira,P.J.Silva,Phys.Rev.D92(2015) 114514.

[24]A.Cucchieri,T.Mendes,Phys.Rev.Lett.100(2008)241601;

I.L.Bogolubsky,E.M.Ilgenfritz,M.Muller-Preussker,A.Sternbeck,Phys.Lett.B 676(2009)69;

O.Oliveira,P.J.Silva,PoSLAT2009(2009)226.

[25]L.vonSmekal,R.Alkofer,A.Hauck,Phys.Rev.Lett.79(1997)3591; C.S.Fischer,J.M.Pawlowski,Phys.Rev.D75(2007)025012;

A.C.Aguilar,D.Binosi,C.T.Figueiredo,J.Papavassiliou,Eur.Phys.J.C78(2018) 181;

C.S. Fischer, J.M. Pawlowski,A. Rothkopf,C.A. Welzbacher, Phys. Rev.D 98 (2018)014009.

[26]P.Lipari,M.Lusignoli,Eur.Phys.J.C73(2013)2630. [27]E.G.S.Luna,M.J.Menon,Phys.Lett.B565(2003)123.

[28]V.A.Khoze,A.D.Martin,M.G.Ryskin,Eur.Phys.J.C18(2000)167; E.G.S.Luna,V.A.Khoze,A.D.Martin,M.G.Ryskin,Eur.Phys.J.C59(2009)1; E.G.S.Luna,V.A.Khoze,A.D.Martin,M.G.Ryskin,Eur.Phys.J.C69(2010)95; V.A.Khoze,A.D.Martin,M.G.Ryskin,Eur.Phys.J.C74(2014)2756; V.A.Khoze,A.D.Martin,M.G.Ryskin,J.Phys.G42(2015)025003. [29]M.Tanabashi,etal.,ParticleDataGroup,Phys.Rev.D98(2018)030001. [30]G.Aad,etal.,ATLASCollaboration,Nucl.Phys.B889(2014)486;

M.Aaboud,etal.,ATLASCollaboration,Phys.Lett.B761(2016)158. [31]https://lhapdf.hepforge.org/.

[32]S.Chatrchyan,etal.,CMSCollaboration,Phys.Rev.Lett.109(2012)111806; R.Aaij,etal.,LHCbCollaboration,J.HighEnergyPhys.06(2012)058; G.Aad,etal.,ATLASCollaboration,Phys.Rev.D85(2012)072004; S.Chatrchyan,etal.,CMSCollaboration,Phys.Rev.D90(2014)032004. [33]G.Aad,etal.,ATLASCollaboration,Phys.Rev.D86(2012)014022;

S.Chatrchyan,etal.,CMSCollaboration,Phys.Rev.D87(2013)112002. [34]P.Abreu,etal.,PierreAugerCollaboration,Phys.Rev.Lett.109(2012)062002. [35]U.Abbasi,etal.,TelescopeArrayCollaboration,Phys.Rev.D92(2015)032007;