Pomeron loop and running coupling effects in high

arXiv:0706.2540v1[hep-ph]18Jun2007PomeronloopandrunningcouplingeffectsinhighenergyQCDevolutionA.DumitruInstitutf¨urTheoretischePhysik,J.W.GoetheUniversit¨at,Max-von-LaueStrasse1,D-60438FrankfurtamMain,GermanyE-mail:[e-mail]∗,L.Portugal†ServicedePhysiqueTh´eoriquedeSaclay,CEA/DSM/SPhT,F-1191Gif-sur-Yvette,FranceE-mail:[e-mail],[e-mail]‡PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USAE-mail:[e-mail]∗,VillaTambosi,StradadelleTabarelle286,I-38050Villazzano(TN),ItalyE-mail:[e-mail]:Withintheframeworkofa(1+1)–dimensionalmodelwhichmimicsevolutionandscatteringinQCDathighenergy,westudytheinfluenceoftherunningofthecouplingonthehigh–energydynamicswithPomeronloops.Wefindthattheparticlenumberfluctuationsarestronglysuppressedbytherunningofthecoupling,byatleastoneorderofmagnitudeascomparedtothecaseofafixedcoupling,foralltherapiditiesthatwehaveinvestigated,uptoY=200.Thisreflectstheslowingdownoftheevolutionbyrunningcouplingeffects,inparticular,thelargerapidityevolutionwhichisrequiredfortheformationofthesaturationfrontviadiffusion.Weconcludethat,forallenergiesofinterest,processeslikedeepinelasticscatteringorforwardparticleproductioncanbereliablystudiedwithintheframeworkofamean–fieldapproximation(liketheBalitsky–Kovchegovequation)whichincludesrunningcouplingeffects.∗MembreduCentreNationaldelaRechercheScientifique(CNRS),France.†OnleavefromInstitutodeF´ısica,UniversidadeFederaldoRiodeJaneiro,Brazil‡OnleavefromtheFundamentalTheoreticalPhysicsgroupoftheUniversityofLi`ege,Belgium1.IntroductionIntherecentyears,onehasassistedattwomajordirectionsofprogressinourunderstandingoftheQCDdynamicsathighenergy,bothaimingatimprovingoverthepreviousstatusofthetheory,asencodedintheBalitsky–Kovchegov[1,2](BK)andJalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner(JIMWLK)equations[3–5].Oneofthesedirectionsstartedwiththerecognition[6–8]ofthefundamentalroleplayedbygluon–numberfluctuations(or‘Pomeronloops’)intheQCDevolutionwithincreasingenergy,whichledtomorecompleteevolutionequations(the‘Pomeronloopequations’)[8–14]andtoasurprisinglinktoproblemsinstatisticalphysics[7],whichisrichinconsequences[8,15–19].Theotherlineofresearchconcentratedontheinclusionofnext–to–leadingorder(NLO)effectsinthenon–linearBKequation[20–27],andveryrecentlyculminatedinanexplicitcalculationoftherunningcouplingeffectsassociatedwithfermionloopswithinperturbativeQCD[28–33].Bothtypesofeffects—PomeronloopsandNLOcorrections—turnouttobeeffectsofO(1),whichmodifyinadramaticwayourpreviousexpectationsbasedontheBK–JIMWLKequations.Yet,sofar,thesetwotypesofeffectshaveneverbeenconsideredtogetherinaunifiedframework(inparticular,theirmutualinfluencehasneverbeenaddressed),mainlybecauseofthecomplexityofthePomeronloopequations,whicharequitedifficulttodealwithalreadyatleading–order.Inthispaper,weshallforthefirsttimestudytheeffectsoftherunningofthecouplingonthehigh–energyevolutionwithPomeronloops,withinaone–dimensionalmodelproposedinRef.[34]whichcapturesinasimplifiedformtherelevantdynamicsinQCD.Forthismodel,thatweshallextendheretothecaseofarunningcoupling(insuchawaytomimictheone–looprunningcouplingofQCD),weshallprovidearatherdetailednumericalanalysisofthe‘dilute–dense’scatteringprocesssimilartodeepinelasticscattering(DIS)inQCD.Ourresultspointouttowardsaveryinteresting,andratherunexpected,conclusion:The‘Pomeronloop’effects(i.e.,theeffectsofparticle–numberfluctuations)arestronglysuppressedbytherunningofthecoupling—byatleastoneorderofmagnitudeascomparedtothecaseofafixedcoupling—,insuchawaythattheyremainnegligibleuptothehighestenergythatwehaveinvestigated,whichcorrespondstoarapidityY=200.Thisfindinghasacorollaryofgreatpracticalinterest:itimpliesthat,forstudiesofthehigh–energydynamics(say,intheenergyrangeatLHC)onecanreliablyresortonappropriatemean–fieldapproximations,soliketheBKequationproperlygeneralizedtoincludeNLO(or,atleast,runningcoupling)effects.Ournumericalanalysisisfurthercorroboratedbyanalyticestimateswhichhelpclarifyingthemechanismresponsibleforthesuppressionoffluctuations:thisisnotthefactthat,becauseoftherunning,thevalueofthecouplingiseffectivelysmaller,asonemightnaivelythink;rather,thisisrelatedtothefactthattherunningofthecouplingdrasticallyslowsdowntheevolution.Witharunningcoupling,notonlythesaturationmomentumgrowsmuchslowerwiththeenergy,butthesameisalsotrueforthe“BFKLdiffusion”[35],whichisthemechanismthroughwhichthesaturationfront(say,forthegluonoccupationnumber,orforthedipolescatteringamplitudeinDIS)evolvestowardsitsasymptotic,‘geometricscaling’,shapeathighenergy.Namely,thediffusiveradiusgrowswithYlikeY1/6(ratherthantheusualY1/2behavioratfixedcoupling),andsuchariseistooslowforthefronttoreachitsasymptoticshapewithintheenergyrangeunderconsideration.Rather,thefrontpreservesapre–asymptoticshape,whichisnotfavorableforthegrowthoffluctuations.Hence,duringthiswhole‘pre–asymptotic’evolution(thatwe–1–foundtoextenduptoY=200atleast),theDISdynamicsisthesameaspredictedbythecorrespondingmean–fieldapproximation(withru