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Diffractive cross sections at HERA and diffractive PDFs
Laurent Schoeffela (on behalf of the H1 and ZEUS collaborations)
a CEA Saclay, Irfu/SPP, 91191 Gif-sur-Yvette Cedex, France
A large collection of results for the diffractive dissociation of virtual photons, γ�p → Xp, have been obtainedwith the H1 and ZEUS detectors at HERA. Different experimental techniques have been used, by requiring alarge rapidity gap between X and the outgoing proton, by analyzing the mass distribution, MX , of the hadronicfinal state, as well as by directly tagging the proton. A reasonable compatibility between those techniques andbetween H1 and ZEUS results have been observed. Some common fundamental features in the measurements arealso present in all data sets. They are detailed in this document. Diffractive PDFs can give a good account ofthose features. Ideas and results are discussed in the following.
1. Experimental diffraction at HERA
One of the most important experimental re-sults from the DESY ep collider HERA is theobservation of a significant fraction of events inDeep Inelastic Scattering (DIS) with a large ra-pidity gap (LRG) between the scattered proton,which remains intact, and the rest of the finalsystem. This fraction corresponds to about 10%of the DIS data at Q2 = 10 GeV2. In DIS, suchevents are not expected in such abundance, sincelarge gaps are exponentially suppressed due tocolor string formation between the proton rem-nant and the scattered partons. Events are ofthe type ep → eXp, where the final state protoncarries more than 95 % of the proton beam en-ergy. A photon of virtuality Q2, coupled to theelectron (or positron), undergoes a strong inter-action with the proton (or one of its low-massexcited states Y ) to form a hadronic final statesystem X of mass MX separated by a LRG fromthe leading proton (see Fig. 1). These events arecalled diffractive. In such a reaction, ep → eXp,no net quantum number is exchanged, and thelongitudinal momentum fraction 1−xIP is lost bythe proton. Thus, the longitudinal momentumxIP P is transferred to the system X . In additionto the standard DIS kinematic variables and xIP ,a diffractive event is also often characterized bythe variable β = xBj/xIP , which takes a simpleinterpretation in the parton model discussed in
the following.Experimentally, a diffractive DIS event, ep →
eXp, is presented in Fig. 2 (bottom). The disso-ciating particle is the virtual photon emitted bythe electron. The final state consists of the scat-tered electron and hadrons which populate thephoton fragmentation region. The proton is scat-tered in the direction of the initial beam protonwith little change in momentum and angle. Inparticular, we detect no hadronic activity in thedirection of the proton flight, as the proton re-mains intact in the diffractive process. On the
2X
Y
Largest rapiditygap in eventP
e
e’
q
t
Q2
W
Figure 1. Illustration of the process ep → eXY .The hadronic final state is composed of two dis-tinct systems X and Y , which are separated bythe largest interval in rapidity between final statehadrons.
Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213
0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
www.elsevierphysics.com
doi:10.1016/j.nuclphysbps.2009.03.127
Figure 2. Usual (top) and diffractive (bottom)events in the H1 experiment at HERA. For adiffractive event, no hadronic activity is visible inthe the proton fragmentation region, as the pro-ton remains intact in the diffractive process. Onthe contrary, for a standard DIS event, the pro-ton is destroyed in the reaction, and the flow ofhadronic clusters is clearly visible in the protonfragmentation region (+z direction, i.e. forwardpart of the detector).
contrary, for a standard DIS event (Fig. 2 top),the proton is destroyed in the reaction and theflow of hadronic clusters is clearly visible in theproton fragmentation region (forward part of thedetector).
The experimental selection of diffractive eventsin DIS proceeds in two steps. Events are firstselected based on the presence of the scatteredelectron in the detector. Then, for the diffractiveselection itself, three different methods have beenused at HERA:
1. A reconstructed proton track is required inthe leading (or forward) proton spectrom-eter (LPS for ZEUS or FPS for H1) witha fraction of the initial proton momentumxL > 0.97. Indeed, the cleanest selection ofdiffractive events with photon dissociationis based on the presence of a leading pro-ton in the final state. By leading proton wemean a proton which carries a large frac-tion of the initial beam proton momentum.This is the cleanest way to select diffractiveevents, but the disadvantage is a reducedkinematic coverage.
2. The hadronic system X measured in thecentral detector is required to be separatedby a large rapidity gap from the rest of thehadronic final state. This is a very effi-cient way to select diffractive events in alarge kinematic domain, close to the stan-dard DIS one. The prejudice is a large back-ground as discussed in the following.
3. The diffractive contribution is identified asthe excess of events at small MX abovethe exponential fall-off of the non-diffractivecontribution with decreasing ln M 2
X . Theexponential fall-off, expected in QCD, per-mits the subtraction of the non-diffractivecontribution and therefore the extraction ofthe diffractive contribution without assum-ing the precise MX dependence of the lat-ter. This is also a very efficient way to se-lect diffractive events in a large kinematicdomain.
Extensive measurements of diffractive DIS crosssections have been made by both the ZEUS and
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213206
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H1 collaborations at HERA, using different ex-perimental techniques [1–5]. Of course, the com-parison of these techniques provides a rich sourceof information to get a better understanding ofthe experimental gains and prejudices of thosetechniques. In Fig. 3 and 4, the basis of the lastZEUS experimental analysis is summarized [5].Data are compared to Monte-Carlo (MC) expec-tations for typical variables. The MC is based onspecific models for signal and backgrounds, andthe good agreement with data is proof that themain ingredients of the experimental analysis areunder control: resolutions, calibrations, efficien-cies... These last sets of data (Fig. 3 and 4) [5]contain five to seven times more statistics thanin preceding publications of diffractive cross sec-tions, and thus opens the way to new develop-ments in data/models comparisons. A first rela-tive control of the data samples is shown in Fig. 5,where the ratio of the diffractive cross sectionsis displayed, as obtained with the LPS and theLRG experimental techniques. The mean value
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of the ratio of 0.86 indicates that the LRG samplecontains about 24% of proton-dissociation back-ground, which is not present in the LPS sam-ple. This background corresponds to events likeep → eXY , where Y is a low-mass excited stateof the proton (with MY < 2.3 GeV). It is ob-viously not present in the LPS analysis whichcan select specifically a proton in the final state.This is the main background in the LRG analysis.Due to a lack of knowledge of this background,it causes a large normalisation uncertainty of 10to 15% for the cross sections extracted from theLRG analysis. We can then compare the resultsobtained by the H1 and ZEUS experiments fordiffractive cross sections in Fig. 6, using the LRGmethod. A good compatibility of both data setsis observed, after rescaling the ZEUS points bya global factor of 13%. This factor is compati-ble with the normalisation uncertainty describedabove. We can also compare the results obtainedby the H1 and ZEUS experiments in Fig. 7, usingthe tagged proton method (LPS for ZEUS and
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213 207
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Figure 5. Ratio of the diffractive cross sections,as obtained with the LPS and the LRG experi-mental techniques. The lines indicate the aver-age value of the ratio, which is about 0.86. It im-plies that the LRG sample contains about 24% ofproton dissociation events, corresponding to pro-cesses like ep → eXY , where MY < 2.3 GeV.This fraction is approximately the same for H1data (of course in the same MY range).
FPS for H1). In this case, there is no proton dis-sociation background and the diffractive sampleis expected to be clean. It gives a good referenceto compare both experiments. A global normal-isation difference of about 10% can be observedin Fig. 7, which can be studied with more data.It remains compatible with the normalisation un-certainty for this tagged proton sample. It is in-teresting to note that the ZEUS measurementsare globally above the H1 data by about 10% forboth techniques, tagged proton or LRG. In Fig. 8,we compare the results using the LRG and theMX methods, for ZEUS data alone. Both setsare in good agreement, which shows that there isno strong bias between these experimental tech-
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niques. The important message at this level is notonly the observation of differences as illustratedin Fig. 6 and 7, but the opportunity opened withthe large statistics provided by the ZEUS mea-surements. Understanding discrepancies betweendata sets is part of the experimental challengeof the next months. It certainly needs analysisof new data sets from the H1 experiment. How-ever, already at the present level, much can bedone with existing data for the understanding ofdiffraction at HERA.
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213208
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2. Diffractive PDFs at HERA
In order to compare diffractive data with per-turbative QCD models, or parton-driven models,the first step is to show that the diffractive crosssection shows a hard dependence in the centre-of-mass energy W of the γ∗p system. In Fig. 9,we observe a behavior of the form ∼ W 0.6 , com-patible with the dependence expected for a hardprocess. This observation is obviously the keyto allow further studies of the diffractive processin the context of perturbative QCD. Events withdiffractive topology can be studied in terms of thePomeron trajectory exchanged between the pro-ton and the virtual photon. In this view, theseevents result from a colour-singlet exchange be-tween the diffractively dissociated virtual photonand the proton (see Fig. 10).
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213 209
A diffractive structure function FD(3)2 can then
be defined as a sum of two factorized contribu-tions, corresponding to a Pomeron and secondaryReggeon trajectories:
FD(3)2 (Q2, β, xIP ) = fIP/p(xIP )FD(IP )
2 (Q2, β) +
fIR/p(xIP )FD(IR)2 (Q2, β) ,
where fIP/p(xIP ) is the Pomeron flux. It dependsonly on xIP , once integrated over t, and F
D(IP )2
can be interpreted as the Pomeron structure func-tion, depending on β and Q2. The other func-tion, F
D(IR)2 , is an effective Reggeon structure
function taking into account various secondaryRegge contributions which cannot be separated.The Pomeron and Reggeon fluxes are assumed tofollow a Regge behavior with linear trajectoriesαIP,IR(t) = αIP,IR(0) + α
′IP,IRt, such that
fIP /p,IR/p(xIP ) =∫ tmin
tcut
eBIP,IRt
x2αIP ,IR(t)−1IP
dt ,
where |tmin| is the minimum kinematically al-lowed value of |t|, and tcut = −1 GeV2 is thelimit of the measurement. We take α
′IP = 0.06
GeV−2, α′IR = 0.30 GeV−2, BIP = 5.5 GeV−2
and BIR = 1.6 GeV−2. The Pomeron interceptαIP (0) is left as a free parameter in the QCD fitand αIR(0) is fixed to 0.50.
The next step is then to model the Pomeronstructure function F
D(IP )2 [1,7–9]. Among the
most popular models, the one based on a point-like structure of the Pomeron has been studied ex-tensively, using a non-perturbative input supple-mented by a perturbative QCD evolution equa-tions [7–9]. In this formulation, it is assumed thatthe exchanged object, the Pomeron, is a colour-singlet quasi-particle whose structure is probedin the DIS process. As for standard DIS, diffrac-tive parton distributions related to the Pomeroncan be derived from QCD fits to diffractive crosssections. The procedure is standard: we as-sign parton distribution functions to the Pomeronparametrised in terms of non-perturbative inputdistributions at some low scale Q2
0. The quark fla-vor singlet distribution (zS(z, Q2) = u + u + d +d + s + s) and the gluon distribution (zG(z, Q2))are parametrised at this initial scale Q2
0, where
X
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e
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Figure 10. Schematic diagram of a diffractive pro-cess. Events with a diffractive topology can bestudied in terms of the Pomeron trajectory ex-changed between the proton and the virtual pho-ton.
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Figure 11. Singlet and gluon distributions of thePomeron (DPDFs) as a function of z ≡ β, thefractional momentum of the Pomeron carried bythe struck parton (see text), obtained by a QCDfit to the H1 diffractive cross sections.
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213210
H1 data (H1RAP)
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Figure 12. Singlet and gluon DPDFs as a functionof z ≡ β, where the results of fitting H1 or ZEUSdata are compared. The ZEUS data consideredhere [3] are derived using the MX method. Aglobal fit of all published data is also presented.Note that the last ZEUS data set [5] is not usedfor this plot.
z = xi/IP is the fractional momentum of thePomeron carried by the struck parton. Func-tions zS and zG are evolved to higher Q2 us-ing the next-to-leading order DGLAP evolutionequations. For the structure of the sub-leadingReggeon trajectory, the pion structure function[6] is assumed with a free global normalizationto be determined by the data. Diffractive PDFs(DPDFs) extracted from H1 and ZEUS data areshown in Fig. 11 and 12 [1,7–9]. We observe thatsome differences in the data are reflected in theDPDFs, but some basic features are common forall data sets and the resulting DPDFs. Firstly,the gluon density is larger than the sea quark den-sity, which means that the major fraction of themomentum (about 70%) is carried by the gluonfor a typical value of Q2 = 10 GeV2. Secondly,we observe that the gluon density is quite large
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H1 DataH1 2006 DPDF Fit A(extrapol. fit)
Figure 13. Scaling violations for H1 diffractivecross sections for one value of xIP (xIP = 0.01)and a large range of β values, from low (< 0.01)to large values (> 0.5).
at large β, with a large uncertainty, which meansthat we expect positive scaling violations still atlarge values of β. This is shown in Fig. 13. Wenote that even at large values of β ∼ 0.5, thescaling violations are still positive, as discussedabove. The strength of the DPDFs approach isto give a natural interpretation of this basic obser-vation and to describe properly the Q2 evolutionof the cross sections. Other approaches are alsowell designed to describe all features of the data[12], but this is another story. The near futureof the study of DPDFs is to combine all existingdata and check their compatibility with respect to
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213 211
the QCD fit technique. If this is verified, a newglobal analysis can be followed to get the mostcomplete understanding of DPDFs [7].
3. Diffractive PDFs and the LHC
Note that diffractive distributions are process-independent functions. They appear not onlyin inclusive diffraction but also in other pro-cesses where diffractive hard-scattering factoriza-tion holds. The cross section of such a processcan be evaluated as the convolution of the rele-vant parton-level cross section with the DPDFs.For instance, the cross section for charm produc-tion in diffractive DIS can be calculated at lead-ing order in αs from the γ∗g → cc cross sectionand the diffractive gluon distribution. An analo-gous statement holds for jet production in diffrac-tive DIS. Both processes have been analyzed atnext-to-leading order in αs and are found to beconsistent with the factorization theorem [10]. Anatural question to ask is whether one can usethe DPDFs extracted at HERA to describe harddiffractive processes such as the production ofjets, heavy quarks or weak gauge bosons in ppcollisions at the Tevatron. Fig. 14 shows resultson diffractive dijet production from the CDF col-laboration compared to the expectations basedon the DPDFs from HERA [11]. The discrep-ancy is spectacular: the fraction of diffractive di-jet events at CDF is a factor 3 to 10 smaller thanwould be expected on the basis of the HERAdata. The same type of discrepancy is consis-tently observed in all hard diffractive processesin pp events. In general, while at HERA harddiffraction contributes a fraction of order 10% tothe total cross section, it contributes only about1% at the Tevatron. This observation of QCD-factorization breaking in hadron-hadron scatter-ing can be interpreted as a survival gap proba-bility or a soft color interaction which needs tobe considered in such reactions. In fact, froma fundamental point of view, diffractive hard-scattering factorization does not apply to hadron-hadron collisions. Attempts to establish corre-sponding factorization theorems fail, because ofinteractions between spectator partons of the col-liding hadrons. The contribution of these interac-
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tions to the cross section does not decrease withthe hard scale. Since they are not associatedwith the hard-scattering subprocess, we no longerhave factorization into a parton-level cross sectionand the parton densities of one of the collidinghadrons. These interactions are generally soft,and we have at present to rely on phenomenologi-cal models to quantify their effects [11]. The yieldof diffractive events in hadron-hadron collisions isthen lowered precisely because of these soft inter-actions between spectator partons (often referredto as reinteractions or multiple scatterings). Theycan produce additional final-state particles whichfill the would-be rapidity gap (hence the often-used term rapidity gap survival). When such ad-ditional particles are produced, a very fast pro-ton can no longer appear in the final state be-cause of energy conservation. Diffractive factor-ization breaking is thus intimately related to mul-tiple scattering in hadron-hadron collisions. Un-derstanding and describing this phenomenon is achallenge in the high-energy regime that will bereached at the LHC [13]. We can also remark sim-
L. Schoeffel / Nuclear Physics B (Proc. Suppl.) 191 (2009) 205–213212
ply that the collision partners, in pp or pp reac-tions, are both composite systems of large trans-verse size, and it is not too surprising that multi-ple interactions between their constituents can besubstantial. In contrast, the virtual photon in γ∗pcollisions has small transverse size, which disfa-vors multiple interactions and enables diffractivefactorization to hold. According to our discus-sion, we may expect that for decreasing virtual-ity Q2 the photon behaves more and more likea hadron, and diffractive factorization may againbe broken.
4. Conclusions
We have presented and discussed the mostrecent results on inclusive diffraction from theHERA experiments. A large collection of datasets and diffractive cross sections are published,which present common fundamental features inall cases. The different experimental techniques,for both H1 and ZEUS experiments, provide com-patible results, with still some global normaliza-tion differences of about 10%. DPDFs give a goodaccount of the main features of the diffractivedata. There is still much to do on the experi-mental side with large statistics analyses, in or-der to obtain a better understanding of data andbackgrounds. This is an essential task for thenext months with the purpose to understand andreduce the normalisation uncertainties of diffrac-tive measurements at HERA. This will make thecombination of cross sections between the two ex-periments much easier, with a common messagefrom HERA on inclusive diffraction.
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