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Date: January19,2016 The evolution of σγP with coherence length 6 1 0 2 n a J Allen Caldwell 8 1 Max Planck Institute for Physics (Werner-Heisenberg-institut) ] Munich, Germany h p - p e h [ 1 v 2 Abstract 7 4 4 Assumingthe formσγP ∝ lλeff atfixed Q2 forthe behaviorof thevirtual-photonprotonscat- 0 tering cross section, where l is the coherence length of the photon fluctuations, it is seen that the 1. extrapolatedvaluesofσγP fordifferentQ2 crossforl ≈ 108 fm. Itisarguedthatthisbehavioris 0 notphysical,andthatthebehaviorofthecrosssectionsmustchangebeforethiscoherencelengthl 6 isreached.Thiscouldsetthescalefortheonsetofsaturationofpartondensities. 1 : v i X r a 1 Introduction According to quantum field theory, the microscopic world is a dynamic environment where short-lived statesareconstantlybeingcreatedandannihilated. Increasingtheresolutionofaprobeusedtosensethis environment,weseeevermorestructureuntil,weanticipate,atsomesmallenoughdistancescalewesee universal distributions. It has already been observed that the energy dependence of the scattering cross sectionofhighenergyhadrons becomesuniversal[1]andthatthesizeofthecrosssectiononlydepends on the number of valence quarks in the scattering particles and the center-of-mass energy. We study in this paper the energy dependence of scattering cross sections for virtual photons on protons, where the interactioncrosssectionisdominatedbythestronginteraction. Weanticipatethatinahighenergylimit, thescatteringcrosssectionsofvirtualphotonswillalsoachieveauniversalbehaviorsincetheinteraction willhaveassourcethephotonfluctuating intoaquark-antiquark pair. Indeepinelastic scattering ofelectrons onprotonsatHERA,thestrongincrease intheprotonstruc- turefunction F withdecreasing x forfixed, large, Q2 isinterpreted asanincreasing density ofpartons 2 intheproton, providing morescattering targets fortheelectron. Thisinterpretation relies onchoosing a particularreference frametoviewthescattering -theBjorkenframe. Intheframewheretheprotonisat rest, it is the state of the photon or weak boson that differs with varying kinematic parameters. For the bulk of the electron-proton interactions, the scattering process involves a photon, and we can speak of different states of the photon scattering on a fixed proton target. What is seen is that the photon-proton crosssection risesquickly withphotonenergy ν forfixedvirtuality Q2 [2]. Weinterpret thisasfollows: astheenergy ofthephoton increases, timedilation allowsshorter livedfluctuations ofthephoton tobe- comeactiveinthescatteringprocess,therebyincreasingthescatteringcrosssection. Weusetheconcept of coherence length of the short-lived photon states to analyze the behavior of the photon-proton cross section. Thisanalysisisanupdateoftheworkreportedin [3]andhasbeeninspiredbythespace-timepicture ofGribov[4]. 2 Coherence Length TheHandconvention [5]isusedtodefinethephotonfluxyielding therelation: Q4(1−x) FP(x,Q2) = σγP (1) 2 4π2α(Q2+(2xM )2) P where FP is the proton structure function, α is the fine structure constant and M is the proton mass. 2 P Q2 andxarethestandard kinematicvariablesusedtodescribe deepinelastic scattering. The behavior of σγP is studied in the proton rest frame in terms of the coherence length [6] of the photon fluctuations, l, and the virtuality, Q2. The physical picture isgiven inFig. 1, where the electron acts as a source of photons, which in turn acts as asource of quarks, antiquarks and gluons. Weexpect thescattering crosssection toincrease withcoherence lengthsincethephoton hasmoretimetodevelop structure. In the proton rest frame, the proton is a common scattering target for the incoming partons independent fromthevaluesofQ2 andl. Werecallthedefinitionofthecoherence length, l[7]: ~c l ≡ ∆E 1 g e proton e Figure1: Photonfluctuations scattering ontheprotonintheprotonrestframe. where ∆E, the change in energy of the photon as it fluctuates into a system of quarks and gluons, is givenby m2+Q2 ∆E ≈ (2) 2ν whereν isthephotonenergy andmisthemassofthepartonic state. ForQ2 ≫ m2,wehave Q2 ∆E ≈ (3) 2ν and 2ν~c ~c l ≈ ≈ . (4) Q2 xM P Note that M only appears because of the definition of x and the value of the coherence length is not P dependent on the proton mass. The coherence length increases linearly with the photon energy and inversely proportional to the virtuality. In the limit Q2 → 0, the dependence of the coherence length willbe modified since the m2 term cannot be ignored. Wewilltherefore limit our study of the limiting behavior ofthecrosssection totheregimeQ2 ≥ 1GeV2 whereweexpectthatEq.3shouldbevalid. 3 Data Sets The analysis relies primarily on the recently released combination of HERA electron-proton neutral- current scattering data from the ZEUS and H1 experiments [8]. This is complemented by the small-x data from the E665 [9]and NMC[10] experiments. Weuse data in the kinematic range: 0.01 < Q2 < 500 GeV2, x < 0.01 and 0.01 < y < 0.8 for a global analysis and a more restricted range when studying the extrapolated behavior of the cross sections. The HERA data are presented in the form of reducedcrosssections xQ4 dσ2 y2 σ = ≈ F − F red 2πα2Y dxdQ2 (cid:20) 2 Y L(cid:21) + + where Y = 1+(1−y)2. The structure function F can be ignored in the (x,Q2) range selected for + 3 the analysis. We calculate F from the reduced cross section assuming R = F /(F − F ) = 0.25, 2 L 2 L andlimitthevalues ofy tobesmaller than 0.8tocontrol thepossible error duetothis assumption. The values ofxarerestricted tobeless than 0.01 toensure thatwearedealing withlarge coherence lengths compared to the proton size. The total experimental uncertainties are used (statistical and systematic added in quadrature on a point-by-point basis). The use of correlated systematics was investigated but foundnottogivesignificantdifferencesforthisanalysis. Inanycase,wearelookingmoreforqualitative behavior. − In total, 45 E665 data points, 13 NMC data points, 23 HERA e P data points, and (115, 160, 75, 276)HERAe+P datapointswithE = (460,575,820,920) GeVsatisfiedtheselection criteria. p 2 3.1 Bin-centering andfurther analysis Thedatafrom thedifferent experiments arereported insomecasesatfixedvalues ofxand varying Q2, inothercasesatfixedQ2 andvarying x,oratfixedQ2 andvarying y. Thisisnoproblem forthemodel fittingprocedure,butitdoesmakedatapresentationcumbersome. Thedataweretherefore‘bin-centered’ bymovingdatatofixedvaluesofQ2 usingaparametrization (seebelow)asfollows: Fpred(Q2,x)F (Q2,x) F (Q2,x) = 2 c 2 (5) 2 c Fpred(Q2,x) Sexpt 2 whereS isanormalizationfactorfortheexperimentinquestionthatresultedfromtheglobalfit. The expt followingQ2 valueshavebeenused: 0.25,0.4,0.65,1.2,2,3.5,6.5,10,15,22,35,45,90,120 GeV2. c 3.2 Functional form forglobalfit A global fit of all the data satisfying the selection criteria was performed to have a functional form for thebin-centering. Thefunctional formforthevirtual-photon protoncrosssectionwasconstructed using thefollowingreasoning. 3.2.1 Q2 dependence Forcompactphotonfluctuations,themaximumvalueofσγP isgivenbythesizeoftheprotonmultiplied by α, giving roughly 200 µbarn. However, pQCD calculations have pointed to the property of ‘color transparency’forsmalldipoles[11],indicatingthatatlargeQ2thecrosssectionshouldbehaveasσγP ∝ 1/Q2. I.e., the proton appears almost transparent for small dipoles with the cross section proportional to the size of the photon. The photon state will have a maximum size which is expected to be set by the mass of the lightest vector meson. An effective mass was used as a free parameter in the fits. We thereforehavethefollowingfactorinourparametrization: M2 σ 0Q2+M2 where σ is expected to be a typical hadronic cross section (multiplied by α). This term has two free 0 parameters(M2,σ ). 0 3.2.2 ldependence As discussed in the introduction, we expect (and already know from looking at previous data) that the cross sections will grow with increasing coherence length. We also know that at very small Q2, the energy dependence of the photon-proton cross section is close to that for hadron-hadron scattering [2], whilethecrosssectionrisesmoresteeplywithenergyatlargerQ2. Thetransitioninthebehaviorsetsin aroundQ2 = 1GeV2 [2,3]. Wetherefore usethefollowingfactortomodeltheldependence: l ǫeff (cid:18)l (cid:19) 0 3 with ǫ = ǫ Q2 ≤ Q2 eff 0 0 ǫ = ǫ +ǫ′ln(Q2/Q2) Q2 ≥ Q2 eff 1 1 1 ln(Q2/Q2) ǫ = ǫ +(ǫ −ǫ ) 0 Q2 < Q2 < Q2 eff 0 1 0 ln(Q2/Q2) 0 1 1 0 Thistermhasfivefreeparameters(ǫ ,ǫ ,ǫ′,Q2,Q2). Thecoherence lengthismeasuredinfm. 0 1 0 1 3.2.3 PredictionforF 2 Thephoton-proton crosssectionistherefore fullyparametrized withtheeightparameters as M2 l ǫeff(ǫ0,ǫ1,ǫ′,Q20,Q21) σγp = σ , (6) 0Q2+M2 (cid:18)l (cid:19) 0 Theprediction for F (and subsequently the reduced cross section) is then achieved using Eq. (1), with 2 the addition of one parameter for each data set to account for relative normalization uncertainties. The bin-centereddataareshowninFig.2. Asisseeninthefigure,thedatacoverseveralordersofmagnitude incoherence length atthesmallervaluesofQ2,reducing tooneorderofmagnitude atQ2 = 120GeV2. Itisalsoclearthattheslopeofthecrosssectionincreases withincreasing Q2. Photon-Proton Cross Section -1 10 -2 10 -3 10 2 3 4 5 10 10 10 10 Figure 2: Bin-centered photon-proton cross section data, shown in alternating colors for clarity. The valuesofQ2 inGeV2 foragivensetofpointsaregivenattheleftedgeoftheplot. 4 4 Global Fit Results TheBayesian Analysis Toolkit (BAT)[12]was used to extract the 15-dimensional posterior probability distribution of the parameters (the 8 parameters for the photon-proton cross section and additionally 7 normalizations) assuming Gaussian probability distributions for all data points. The prior probability distributions chosen are given in Table 1. For the parameters of the photon-proton cross section, the priors reflect the knowledge that exists on the parameters. For the normalization parameters, the priors reflecttheexperimentally quoteduncertainties. Table 1: Prior probability definitions for the parameters in the fit function and results of the global fit to the data. The notation x ∼ G(◦|µ,σ) means that x is assumed to follow a Gaussian probability distribution centered on µ with variance σ2. The parameter boundaries are also given. The parameter valuesattheglobalmodeaswellasthe68%smallestmarginalizedintervalsfromtheglobalfitaregiven inthelasttwocolumns. Parameter priorfunction globalmode 68%interval σ σ ∼ G(◦|0.07,0.02) 0.01 < σ < 0.2 0.062 0.059−0.067 0 0 0 M2 M2 ∼ G(◦|0.75,0.5) 0.1 < M2 < 2.0 0.63 0.59−0.67 0 l l ∼ G(◦|1.0,0.5) 0.1 < l < 2.0 1.6 1.54−1.77 0 0 0 ǫ ǫ ∼ G(◦|0.09,0.01) 0.05 < ǫ < 0.2 0.106 0.102−0.110 0 0 0 ǫ ǫ ∼ G(◦|0.2,0.2) 0.1 < ǫ < 0.3 0.156 0.152−0.160 1 1 1 ′ ′ ′ ǫ ǫ ∼ G(◦|0.05,0.02) 0.0 < ǫ < 0.1 0.052 0.051−0.054 Q2 Q2 ∼ G(◦|1.0,1.0) 0.01 < Q2 < 2.0 0.37 0.33−0.41 0 0 0 Q2 Q2 ∼ G(◦|3.0,3.0) 2.0 < Q2 < 10.0 3.13 2.96−3.36 1 1 1 S S ∼ G(◦|1.0,0.018) 0.9 < S < 1.1 0.97 0.958−0.982 E665 E665 S S ∼ G(◦|1.0,0.025) 0.9 < S < 1.1 0.94 0.093−0.096 NMC NMC Se−p S ∼ G(◦|1.0,0.015) 0.9 < Se−p < 1.1 0.997 0.988−1.006 S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.020 1.014−1.030 e+p460 e+p460 S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.014 1.008−1.024 e+p575 e+p575 S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.009 1.002−1.020 e+p820 e+p820 S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 0.998 0.992−1.008 e+p920 e+p920 ThevaluesoftheparametersattheglobalmodearegiveninTable1togetherwiththe68%smallest intervalsforthe1Dmarginalizeddistributions. Thefitwassatisfactoryandwasfurtherusedtobin-center thedata points. Theposterior probability at theglobal modecorresponded toa χ2 value of 761 for 707 fitteddatapoints. Thebin-centered datawerethenfittothesimpleform l λeff(Q2) σ(l,Q2) = σ (Q2) (7) 1 (cid:18)1 fm(cid:19) forfixedvalues ofQ2. Somesample fitsareshown inFig3. These two-parameter fitswere universally good, indicating that at fixed Q2 values, the data are completely consistent with a simple power law behavior. Theparameters resulting from these fits are shown inFig. 4, together with the expectation from the formula used in the global fit of the data. It is seen that the global fit does a decent job or representing the data, in particular the Q2 dependence. The dependence on the coherence length has the expected 5 Photon-Proton Cross Section Photon-Proton Cross Section 0.1837 0.0629 0.1506 0.0508 0.1175 0.0386 0.0845 0.0264 0.0514 0.0143 2 3 4 2 3 10 10 10 10 10 -2 x 10 Photon-Proton Cross Section Photon-Proton Cross Section 0.0139 0.38 0.0112 0.312 0.0086 0.243 0.006 0.174 0.0033 0.106 2 3 2 10 10 10 Figure 3: Fits of the data using the function given in (7) for selected values of Q2. The values of Q2 are given in the panels, as well as the number of data points fit and the value of χ2 using the best-fit parameters. The band indicates the 68 % credible interval from the fit function. The error bars on the pointsarefromadding thestatistical andsystematicuncertainties inquadrature. 6 features: atsmall Q2, the exponent λ ≈ 0.1, avalue compatible with the results from hadron-hadron eff scattering data. For values above a few GeV2, the exponent λ ∝ ln(Q2), with a transition region in eff between 0.35 0.275 0.2 0.125 0.05 2 1 10 10 -2 10 -3 10 2 1 10 10 Figure 4: Values of the parameters λ and σγp(l = 1 fm) = σ (see Eq. 7) as a function of Q2. The eff 1 values expected from theglobal fittothedata (see Eq.6)isshownasthegreen lines. Theuncertainties onthefitparameters aretypicallysmallerthanthensizeofthesymbols. From Fig. 4, we note that while the cross section at l = 1 fm decreases approximately as 1/Q2 for the larger values of Q2, the dependence on the coherence length becomes steeper with increasing Q2. An extrapolation of the cross sections will result in the cross section for high Q2 photons eventually becoming larger than the cross section for lower Q2 photons. We study this for Q2 > 3 GeV2, where the data in Fig. 4 indicate that we have reached the simple scaling behavior of the cross section. An extrapolation of thecross sections isshowninFig. 5andreveals that forl ≈ 108 fm, theextrapolations merge. 5 Discussion At small values of the coherence length, the photon-proton scattering cross section scales roughly as 1/Q2, as expected from color transparency. However, as the coherence length increases, the scattering cross sections increase at different rates, and the scattering cross section for an initially small photon configuration willgrowtobecomparable tothatforasmallerQ2,orlarger, photonconfiguration unless thegrowthofthecross section ismodified. Ifthephoton statesbecome comparable insize, theyshould evolveinthesamewaysothatthecrosssectionsevolveuniformlywithcoherencelength,independently of their initial size, and with the same energy dependence as typical for hadronic cross sections. This 7 Photon-Proton Cross Section -1 10 -2 10 -3 10 3 5 7 8 10 10 10 10 Figure5: ExtrapolationofthefitfunctionstolargerlforvaluesofQ2intherange3.5 ≤ Q2 ≤ 90GeV2. Thewidthofthebandsrepresentthe68%credibleintervalsforthefunctionsfromthedatafitsdescribed in the text. The values of Q2 range from 3.5 GeV2 (top curve) to 90 GeV2 (bottom curve) with the intermediate curvescorresponding tointermediate Q2 values. 8 hadron-like energy dependence is observed for the smallest Q2 values investigated (the energy depen- denceisthesameasforhadron-hadronscattering). Wethereforeexpectthattheslopeofthecrosssection withcoherencelengthshouldflattenasafunctionofcoherencelength;thiscouldbeanindicationforsat- urationofthepartondensities. Itisalsopossiblethatthepowerlawbehaviorofthecrosssectionchanges withcoherence length withoutsaturation [16]butthiswouldagainimplyaninteresting change ingluon dynamics. Theneweffectsshouldsetinbelowl = 108 fmtoavoidthecrosssectioncrossing. Giventhe relation giveninEq.4,thismeansthatachange shouldsetinforx > 10−9. Firstsignsofthechangein slopewouldpresumablysetinconsiderablyearlier. Theapproachtosaturationandafundamentallynew state of matter is therefore perhaps within reach of next generation lepton-hadron colliders [13–15], an excitingprospect. 6 Acknowledgments I would like to thank Matthew Wing and Henri Kowalski for many interesting discussions concerning thisanalysis. References [1] A.Donnachie, P.V.Landshoff, Phys.Lett.B296,227(1992). [2] J.Breitwegetal.[ZEUSCollaboration], Eur.Phys.J. C7,609(1999). [3] A.Caldwell,‘Behaviorofsigma(gammap)atLargeCoherenceLengths’, arXiv:0802.0769v1. [4] V.N.Gribov,‘Space-timedescription ofthehadroninteraction athighenergies’, arXiv:0006158v1. [5] L.N.Hand,Phys.Rev.129,1834(1963). [6] L.Stodolsky, Phys.Lett.B325,505(1994). [7] J.BartelsandH.Kowalski,‘DiffractionatHERAandtheConfinementProblem’,Eur.Phys.J. C19, 693(2001). [8] H.Abramowiczetal.[H1andZEUSCollaborations], Eur.Phys.J.C751(2015). [9] M.Adamsetal.[E665Collaboration], Phys.Rev.D54,3006(1996). [10] M.Benvenutietal.[NMCCollaboration], Nucl.Phys.B483,3(1997). 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