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Fluctuations, Saturation, and Diffractive Excitation in High Energy Collisions1 Christoffer Flensburg2 Dept.ofTheoreticalPhysics,Sölvegatan14A,S-22362Lund,Sweden 1 1 0 Abstract. Diffractive excitation is usually described by the Good–Walker formalism for low 2 masses, and by the triple-Regge formalism for high masses. In the Good–Walker formalism the n crosssection isdeterminedbythe fluctuationsin the interaction.By takingthe fluctuationsin the a BFKLladderintoaccount,itispossibletodescribebothlowandhighmassexcitationintheGood– J Walkerformalism.Inhighenergy ppcollisionsthefluctuationsarestronglysuppressedbysatura- 7 tion,whichimpliesthatpomeronexchangedoesnotfactorisebetweenDISand ppcollisions.The Dipole Cascade Model reproducesthe expected triple-Regge form for the bare pomeron,and the ] triple-pomeroncouplingisestimated. h p Keywords: Small-xphysics,Saturation,Diffraction,DipoleModel,DIS - PACS: 13.85.Hd,13.85.Lg p e h [ Introduction 1 v 4 Diffractive excitation represents large fractions of the cross sections in pp collisions 0 or DIS. In most analyses of pp collisions low mass excitation is described by the 4 Good–Walker formalism [1], while high mass excitation is described by a triple-Regge 1 . formula [2, 3]. In the Good–Walker formalism the fluctuations in the pomeron ladder 1 0 are normally not included, which is what limits the application to low masses. In the 1 dipolecascademodel[4,5,6,7,8]thefluctuationsintheladdersaretakenintoaccount, 1 allowingtheGood–Walkerformalismtodescribediffractiveexcitationat allmasses. : v It turns out that saturation plays a very important role in suppressing diffractive i X excitation. This means that pp collisions will have a much lower fraction of diffractive r excitation than without saturation, specially at higher energies. For DIS, saturation is a a smaller effect, and diffractive excitation can be expected to be stronger, which is confirmed by experiments. The impact parameter profile for diffractive excitation in high energy pp is found to be in the shape of a ring, due to the approaching black disc limitat lowb. Triple-Regge without saturation predicts powerlike growth with energy of the total, elasticanddiffractiveexcitationcrosssections.By removingsaturationeffects fromthe dipole cascade model, also that gives a powerlike energy growth, just like the triple- Reggemodels.Theintercept,slopeandtriple-Pomeroncouplingscanbeextractedfrom theenergy dependencies. 1 WorksupportedinpartbytheMarieCurieRTN“MCnet”(contractnumberMRTN-CT-2006-035606). 2 IncollaborationwithGöstaGustafsonandLeifLönnblad 100000 0.4 2.5 b=6 DIPSY 220 GeV DIPSY 2000 GeV DIPSY 2000 GeV 10000 AF-p + cutoff 0.35 AFpe-aF AFpe-aF 2 0.3 b=4 1000 0.25 1.5 F) 100 b=2 F) 0.2 b=0 T) b=6 b=3 P( P( P( 0.15 b=3 1 b=0 10 b=9 0.1 b=6 b=9 0.5 1 0.05 b=9 0.1 0 0 1e-05 0.0001 0.001 0.01 0.1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 F F T FIGURE1. ThedistributionofinteractionamplitudesforDIS(left),unsaturatedpp(mid)andsaturated pp(right). The dipole cascade model In the Good–Walker formalism the incoming mass eigenstates are not necessarily eigenstates of diffractive interaction. However, the mass eigenstates Y are linear com- k binationsof the diffraction eigenstates F with eigenvalues T and coefficients c . The n n kn diffractivecrosssectionoftheincomingmasseigenstateY canthenbewrittenbysum- 0 mingoveroutgoingmass eigenstates: ds /d2b=(cid:229) ( Y T Y )2 = Y T2 Y = T2 diff 0 k 0 0 h | | i h | | i h i k where the last average over T is with the weights from Y . By subtracting the elastic 0 part,onefinds thediffractiveexcitation: ds /d2b=ds /d2b ds /d2b= T2 T 2 V . diffex diff el T − h i−h i ≡ It turns out that it is the fluctuations in the interaction amplitude that gives the diffractiveexcitations. Inourmodel,theeigenstatesofinteractionarecascadesofcolourdipolesintransverse space. Mueller showed that the cascade was equivalent with leading logarithm BFKL [9, 10, 11] and has since been enhanced with several non-leading order effects such as energy-momentumconservation,confinement,runninga andimprovedsaturation.The s saturationintheinteractionisincludedthroughunitarisationT =1 e F,whereF isthe − − Bornamplitude.Inthecascade,asaturating2 2“dipoleswing”isincluded,providing → asaturationinthecascadeequivalentto theinteractionup toafew percent. This model has proven to describe a wide range of total, elastic and diffractive cross sectionsin both pp collisionsand DIS. Thisis describedin detailin[6, 7, 8]. Fluctuations and saturation Inthissectionwewilluseourmodeltostudyhowsaturationaffectsthefluctuationsin theinteraction amplitude,and thus diffractiveexcitation.We want to study theeffect of saturation,andwillseparatebetweentheBorn-levelamplitudeF,andthefullysaturated 1 W = 100 GeV W = 2000 GeV<<TT>>2 W = 14000 GeV 0.8 VT 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 b FIGURE 2. Impactparameterdistributionsfromthe MC for T =(ds /d2b)/2, T 2 =ds /d2b, tot el andV =ds /d2bin ppcollisionsatW =100,2000,and14h00i0GeV.bisinunitsohfiGeV 1. T diffex − amplitude T. Both these are calculated from our model with a Monte Carlo simulation program called DIPSY. A large number of colliding dipole cascades are generated and collidedatfiximpactparameter,andthefrequenciesofinteractionamplitudes,P(F)and P(T), arestudied. In the left plot of figure 1 is the distribution of Born amplitudes for g ⋆p at W = 220GeV.ItisseenthatthedistributionbehavesroughlyasapowerofF,givingarather wide distribution and large fluctuations. This corresponds to a large cross section for diffractive excitation in DIS. Since F is well below unity, T F and saturation is a ≈ smalleffect. The distribution of the Born amplitudes of pp at √s = 2000 GeV is shown in the middle plot of figure 1. This distribution behaves as a gamma function, which very wide and corresponds to a very large cross section for diffractive excitation. However, since F is not smaller than 1, unitarity is important. The distribution in the saturated amplitudeT is shown in the rightmostplot. The shape for the large b distributionsdoes not change much, but the low b distributions that previously were very wide are now sharplypeakedjustbelowT =1. T approaching1correspondstotheblackdisclimit, h i andisseentostronglysuppressthefluctuations,andthusthediffractiveexcitations.This isclearlyseenintheimpactparameterprofileinfigure2wherethecentralcollisionsget suppressedfluctuationsasenergyincrease.Thus,thediffractiveexcitationsliveinaring whereT 0.5. ≈ Comparison between Good–Walker and triple-Regge In unsaturatedReggeformalism,thecross sectionsare s = b 2(0)sa (0) 1 s pp¯se , tot − ≡ 0 ds 1 el = b 4(t)s2(a (t) 1), dt 16p − M2 ds SD = 1 b 2(t)b (0)g (t) s 2(a (t)−1) M2 e . (1) Xdtd(M2) 16p 3P (cid:18)M2(cid:19) X X X (cid:0) (cid:1) 1000 b) m ( 100 s total elastic single diffractive 10 100 1000 10000 (cid:214) s (GeV) FIGURE3. Thetotal, elastic andsingle diffractivecrosssectionsin theone-pomeronapproximation. The crossesare fromthe dipole cascade modelwithoutsaturation,and the linesare froma tunedtriple Reggeparametrisation. Here a (t)=1+e +a t is the pomeron trajectory, and b (t) and g (t) are the proton- ′ 3P pomeron and triple-pomeron couplings respectively. These cross sections are in most modelsincreasingmuchfasterthanthemeasuredones,andsaturationindifferentforms are added to fit with experiments. To make a comparison between our model in the Good–Walker formalism and the Regge models, it is better to compare the unsaturated models,to avoidthemodeldependence inthesaturationscheme. Running the DIPSY Monte Carlo without saturation, it turns out (figure 3) that the energy dependence of the total, elastic and diffractive cross sections fit perfectly to the Reggeparametrisationswith a (0) = 1+e =1.21, a =0.2GeV 2, ′ − s pp¯ = b 2(0)=12.6mb, b =8GeV 2, g (t)=const.=0.3GeV 1. (2) 0 0,el − 3P − This is not a trivial result. For example without the logarithmic corrections in 1 the fit would have been significantly worse. Similarly, without the confinement, energy conservation and running a , that is just leading logarithm BFKL, the increase with s energy would have been too strong. The NLL corrections are necessary for the two approaches toagree thiswell. REFERENCES 1. M.L.Good,andW.D.Walker,Phys.Rev.120,1857–1860(1960). 2. A.H.Mueller,Phys.Rev.D2,2963–2968(1970). 3. C.E.DeTar,etal.,Phys.Rev.Lett.26,675–676(1971). 4. E.Avsar,G.Gustafson,andL.Lönnblad,JHEP07,062(2005),hep-ph/0503181. 5. E.Avsar,G.Gustafson,andL.Lonnblad,JHEP01,012(2007),hep-ph/0610157. 6. E.Avsar,G.Gustafson,andL.Lönnblad,JHEP12,012(2007),arXiv:0709.1368[hep-ph]. 7. C.Flensburg,G.Gustafson,andL.Lonnblad,Eur.Phys.J.C60,233–247(2009),0807.0325. 8. C.Flensburg,andG.Gustafson(2010),1004.5502. 9. A.H.Mueller,Nucl.Phys.B415,373–385(1994). 10. A.H.Mueller,andB.Patel,Nucl.Phys.B425,471–488(1994),hep-ph/9403256. 11. A.H.Mueller,Nucl.Phys.B437,107–126(1995),hep-ph/9408245.