Hard Diffraction with Dynamic Gap Survival
PPrepared for submission to JHEP
Hard Diffractionwith Dynamic Gap Survival
Christine O. Rasmussen and Torbjörn Sjöstrand
Theoretical Particle PhysicsDepartment of Astronomy and Theoretical PhysicsLund Unicersity,Sölvegatan 14ASE-223 62 Lund, Sweden
E-mail: [email protected] , [email protected] Abstract:
We present a new framework for the modelling of hard diffraction in pp and pp collisions. It starts from the the approach pioneered by Ingelman and Schlein,wherein the single diffractive cross section is factorized into a Pomeron flux and a PomeronPDF. To this it adds a dynamically calculated rapidity gap survival factor, derived fromthe modelling of multiparton interactions. This factor is not relevant for diffraction in ep collisions, giving non-universality between HERA and Tevatron diffractive event rates.The model has been implemented in Pythia pp and pp data reveal improvement in the description of single diffractive events. a r X i v : . [ h e p - ph ] J a n ontents W / Z production at the Tevatron 254.2 Diffractive dijets at the Tevatron 274.3 CMS diffractive contribution to dijet production 294.4 ATLAS dijets with large rapidity gaps 31 The nature of diffractive excitation in hadron-hadron collisions remains a bit of a mystery.We may motivate why it happens, e.g. based on the optical analogy that lies behind itsname, or in the related Good-Walker formalism [1]. But to explain how diffractive eventsare produced, and with what properties, is a longer story. In a first step the single diffractivecross section should be describable as a function of the diffractive mass M and the squaredmomentum transfer t . In a second step the generic properties of a diffractive system of agiven mass should be explained: multiplicity distributions, rapidity and transverse momen-tum spectra and other event characteristics. In a third step the existence and characterof exclusive diffractive processes and the underlying events associated therewith should beunderstood.Over the years much data has accumulated, and many models have been presented, butso far without any model that explains all aspects of the data, and without any consensuswhich models are the most relevant ones. It is beyond the scope of the current article toreview all the data and models; for a selection of relevant textbooks and reviews see [2–8].For the path we will follow in this article, Regge theory provides the basic mathematicalframework. In it, poles in the plane of complex spin α may be viewed as the manifestations– 1 –f hadronic resonances in the crossed channels. A linear trajectory of poles α ( t ) = α (0)+ α (cid:48) t corresponds to a σ tot ∼ s α (0) − . Several trajectories appear to exist, but for high-energyapplications the most important is the Pomeron ( P ) one, which with its α (0) > is deemedresponsible for the observed rise of the total cross section, and in modern terminologywould correspond to a set of glueball states. With single-Pomeron exchange as the startingpoint, higher orders involve multiple Pomeron exchanges, and also interactions between thePomerons being exchanged, driven by a triple-Pomeron vertex. Out of this framework thecross section for various diffractive topologies can be derived, differentially in mass and t ,given a set of numbers that must be extracted from data.Such models do not address the structure of the diffractive system. The fireball mod-els of older times implied isotropic decay in the rest frame of the diffractive system, orpossibly elongated along the collision axis, but without internal structure. The Ingelman-Schlein (IS) model [9] made the bold assumption that the exchanged Pomeron could beviewed as a hadronic state, and that therefore a diffractive system could be described as ahadron-hadron collision at a reduced energy. This implied the existence of Parton Distri-bution Functions (PDFs) for the Pomeron. Thereby also hard processes became available,confirmed by the observation of jet production in diffractive systems [10]. The PomPytprogram [11] combined Pomeron fluxes and PDFs, largely determined from HERA data,with the Pythia event generator of the time [12] to produce complete hadronic final states,and PomWig [13] did similarly for Herwig [14].One limitation of these models is that they are restricted to the exchange of onePomeron per hadron-hadron collision, not the multiple ones expected in Regge theory.Translated into a QCD-based, more modern view of such collisions, Multiple Partonic In-teractions (MPIs) occur between the incoming hadrons [15]. That is, since hadrons arecomposite objects, there is the possibility for several partons from a hadron to collide,predominantly by semisoft → QCD interactions. These together create colour flows(strings [16]) criss-crossing the event, typically filling up the whole rapidity range betweenthe two beam particles with hadron production. Thereby a “basic” process containing arapidity gap can lose that by MPIs. (MPIs and soft colour exchanges could also be sourcesof gaps [17, 18], a possibility we will not study further in this article, so as to keep thediscussion focussed.)A spectacular example is Higgs production by gauge-boson fusion, W + W − → H and Z Z → H , where the naive process should result in a large central gap only populatedby the Higgs decay products, since no colour exchange is involved. Including MPIs, thisgap largely fills up [19], although a fraction of the events should contain no further MPIs[20], a fraction denoted as the Rapidity Gap Survival Probability (RGSP). Such a picturehas been given credence by the observation of “factorization breaking” between HERA andthe Tevatron: the Pomeron flux and PDFs determined at HERA predicts about an orderof magnitude more QCD jet production than observed at the Tevatron, e.g. [21].In this article the intention is to provide a dynamical description of such factorizationbreaking, as a function of the hard process studied and its kinematics, and to predict theresulting event structure for hard diffraction in hadronic collisions. This is done in threesteps. Firstly, given a hard process selected based on the inclusive PDFs, the fraction of– 2 – PDF that should be associated with diffraction is calculated, as a convolution of thePomeron flux and its PDFs. Secondly, the full MPI framework of Pythia , including alsothe effects of initial- and final-state radiation, is applied to find the fraction of eventswithout any further MPIs. Those events that survive these two steps define the diffractiveevent fraction, while the rest remain as regular nondiffractive events. Thirdly, diffractiveevents may still have MPIs within the P p subsystem, and therefore the full hadron-hadronunderlying-event generation machinery is repeated for this subsystem. The nondiffractiveevents are kept as they are in this step.One should not expect perfect agreement with data in this approach; there are too manyuncertainties that enter in the description. Neveretheless a qualitative description can behelpful, not only to understand the trend of existing data, but also to pave the way forfuture studies. The new framework we present here has been implemented as an integratedpart of the Pythia
In this article we study hard diffraction, so this means we assume the presence of some hardprocess in the events of interest. Standard examples would be jet, Z and W ± production.By factorization a cross section involving partons i, j from incoming beams A, B can bewritten as σ = (cid:88) i,j (cid:90) (cid:90) d x d x f i/A ( x , Q ) f j/B ( x , Q ) ˆ σ ij (ˆ s = x x s, Q ) , (2.1)where ˆ σ is the parton-level cross section, integrated over relevant further degrees of freedom,like a p ⊥ range for jets.Assuming Pomerons to have some kind of existence inside the proton, in the Ingelman-Schlein spirit, we introduce a Pomeron flux f P / p ( x P , t ) , where x P is the P momentum fractionand t its (spacelike) virtuality. The P has a partonic substructure, just like a hadron, andthus we can define PDFs f i/ P ( x, Q ) . The PDF could also depend on the t scale, just likethe photon has a PDF strongly dependent on its virtuality. For lack of a model for sucha dependence we assume the P PDF is a suitable average over the t range probed. As aconsequence we will not need t for most of the studies, and so it can be integrated out ofthe flux, f P / p ( x P ) = (cid:82) f P / p ( x P , t ) d t . – 3 –iven the ansatz with Pomeron flux and PDF, the PDF of a proton can be split intoone regular nondiffractive (ND) and one P -induced diffractive (D) part, f i/ p ( x, Q ) = f ND i/ p ( x, Q ) + f D i/ p ( x, Q ) , (2.2)where f D i/ p ( x, Q ) = (cid:90) d x P f P / p ( x P ) (cid:90) d x (cid:48) f i/ P ( x (cid:48) , Q ) δ ( x − x P x (cid:48) )= (cid:90) x d x P x P f P / p ( x P ) f i/ P (cid:18) xx P , Q (cid:19) . (2.3)The assumption that the diffractive part f D i/ p ( x, Q ; x P , t ) of the full PDF can be decomposedin this way is in approximate agreement with the HERA data [23].For two incoming protons (or antiprotons, or other hadrons) A and B , an initial prob-ability for diffraction P D ≈ P D A + P D B is obtained from the ratio of diffractive to inclusivePDFs, P D A = f D i/B ( x B , Q ) f i/B ( x B , Q ) for AB → XB , P D B = f D i/A ( x A , Q ) f i/A ( x A , Q ) for AB → AX , (2.4)where P D A/B is the probability for side
A/B to be the diffractive system, thus being depen-dent on the variables of the opposite side.This probability is used to determine, on an event-by-event basis, the nature of theselected hard scattering, whether diffractive or not. Currently we concentrate on singlediffraction. A natural extension would be to associate the product P D A P D B with centraldiffraction (CD), where two Pomerons collide and one parton is extracted from each P . Itwould also be possible to extend the formalism such that part of the SD rate is reassigned asdouble diffraction (DD), where the hard collision happens inside one of the two diffractivesystems. Neither CD nor DD are considered in this first study, however. Instead, for thefraction P D A P D B of events, which normally is small anyway, a random choice is made between AB → AX and AB → XB .The key aspect of the model is now that it contains a dynamical gap survival. Thismeans that we do not allow any further MPIs to occur between the two incoming hadrons,so as to ensure that the gap survives. In practise the tentative classification as diffractive,based on eq. (2.4), initially has no consequences: all events are handled as if they werenondiffractive hadron-hadron collisions.Only if no additional MPIs occur does a diffractive classification survive and only thenis the P p subsystem set up. Specifically the x P value is selected according to the distributionimplied by eq. (2.4), and also a t value is selected for the outgoing proton. Technically, it isonly at this stage that “pure” samples of diffractive events can be selected, should one wishto single out such events. – 4 –nce the P p system has been set up, it is allowed to develop a partonic structure justlike any hadron-hadron collision. Both initial-state radiation (ISR) and final-state radiation(FSR) thereby dress the original hard process by the emission of further softer partons. Alsofurther MPIs inside this system are allowed, based on the f i/ P ( x, Q ) PDFs, successivelymodified to take into account the momentum and flavours already carried away by theMPI, ISR and FSR activity at p ⊥ scales above the currently considerd one, just like fornondiffractive systems.The ISR/FSR/MPI description is based on the perturbative parton picture. Nonper-turbative aspects have to be added to this. Beam remnants carry the momentum not kickedout of the incoming P and p . For the former a fictitious “valence quark” content of either dd or uu is chosen at random for each new event. It is essentially equivalent to having a gluonas remnant, but is slightly more convenient. All outgoing partons are colour-connected bycolour flux lines - strings - that fragment to produce the primary hadrons of the final state.The colour flow in an event is not unambiguously determined, however, and data suggestthat colours tend to be more correlated than naively comes out of the perturbative picture,a phenomenon known as Colour Reconnection (CR).We can by combining these two simple ideas give an explanation of the discrepan-cies between Tevatron and HERA. The dynamical gap survival introduces an additionalsuppression factor, reducing the number of diffractive events without any additional pa-rameters. For numerical studies it is necessary to specify Pomeron flux and PDF parametrizations.There are currently seven parametrizations/models for the former and five for the latteravailable in
Pythia .The parametrizations for the Pomeron flux f P / p ( x P , t ) are • Schuler-Sjöstrand model (SaS) [24], • the Bruni-Ingelman model [25], • the Streng-Berger model [26], • the Donnachie-Landshoff model [27], • the Minimum Bias Rockefeller model (MBR) [28] with an option to renormalize theflux, and • the H1 models Fit A and B [29, 30].All have to obey an approximate form f P / p ( x P ) ∼ /x P in order to obtain an approximatediffractive mass spectrum ∼ d M X /M X , as required by Regge theory and by data. Justlike the rise of the total cross section requires a supercritical Pomeron α (0) = 1 + (cid:15) > ,with (cid:15) ≈ . , several of the fluxes have adapted this steeper slope f P / p ( x P ) ∼ /x (cid:15) P (where the factor of 2 in front of (cid:15) comes from the optical theorem). There are also someattempts to account for an excess in the low-mass resonance region. The t dependence is– 5 –ypically parametrized as a single exponential f P / p ( x P , t ) ∼ exp( B SD t ) , but also as a sum oftwo exponentials, or as a (power-like) dipole form factor. The MBR model differs from theothers, since the model renormalizes the flux to unity. This renormalization suppresses theflux, thus making the dynamical gap survival obsolete. In order to make direct comparisonsto the other available flux-models, we have implemented the renormalization as an optionwith the default being the non-renormalized flux.The parametrizations for the Pomeron PDFs f i/ P ( x, Q ) are • PomFix, a simple (toy) Q -independent parametrization, • the H1 Fit A and B NLO PDFs [29], • the H1 Jets NLO PDF [31], and • the H1 Fit B LO PDF [29], • the ACTW B PDF with (cid:15) = 0 . [32], • the ACTW D PDF with (cid:15) = 0 . [32], • the ACTW SG PDF with (cid:15) = 0 . [32], • the ACTW D PDF with (cid:15) = 0 . [32].The first of these has a momentum sum of unity, whereas the latter are not normalized toany specific value, the argument used being that the Pomeron is not a real particle and sodoes not obey that kind of constraints [33, 34]. (Technically H1 chose to normalize the P flux to unity at x P = 0 . , and then let the PDF normalization float. No normalisationconstraints are included in the ACTW PDFs, as this is primarily set by the normalisationof the DL flux. Thus the momentum sum of these PDFs range from 0.5 to 2, dependingon fit.) Pragmatically it could be argued that what is measured is the convolution of the P flux and the P PDF, so that is is feasible to shuffle any constant number between the two.Unfortunately this makes it less trivial to mix freely, and makes it almost a necessity tocombine H1 PDFs with H1 fluxes. In
Pythia
8, it is only allowed to combine the ACTWPDFs with the DL flux, as these have been fitted together, and each of the fits uses different (cid:15) values.No attempts have been made to exclude or validate different flux–PDF combinationsin the light of more recent HERA data than available at the time of the fits; this woud bea separate project. We do note, however, that a more recent ZEUS article [35] compares anew ZEUS DPDF SJ fit with the H1 Fit B, showing disagreements on the 10–20% level. Forour purposes this is an acceptable uncertainty, and we will often use Fit B as a reference,but keep an open mind to variations.This is not the end of the story from an event-generator point of view, however. Inmost of the available PDF parametrizations the momentum sums to approximately 0.5, butthis does not mean that half of the P momentum in the P p collision can just be thoughtaway. At the very least this other half has to be considered as an inert component that sails– 6 –hrough without interacting, but is present in the beam remnant. A further complicationarises when MPIs are introduced. Normally these are generated in a sequence of decreasing p ⊥ , with the PDFs for an MPI adjusted to take into account the momentum and flavourscarried away by the preceding MPIs. So if 0.4 of the P momentum has already been taken,does it mean that 0.1 or 0.6 of it remains? This is an issue that did not exist at HERA, whereMPIs are negligible outside of the photoproduction region. The choice made in Pythia is to assume that the full P momentum is available for MPIs. Furthermore we allow theoption to rescale the PDFs by a constant factor so as to change the momentum, notablyby a factor of two to restore (approximately) the momentum sum rule. This should thenbe compensated by a corresponding rescaling of the P flux in the opposite direction. Thatway the P can be brought closer to an ordinary hadron, and more P flux-PDF combinationscan be used.Another problem is that most PDF fits are NLO ones. Given the sparsity of data,it should be clear that “NLO accuracy” does not mean the same thing as it does for theinclusive proton PDF. Since Pythia only contains LO matrix elements (MEs) for QCDprocesses there is no extra bonus for using NLO PDFs. Worse, it is well known that thegluon PDF (of the proton) is much smaller in NLO than in LO for small x and Q ; inprinciple it can even become negative. This behaviour compensates for the NLO MEsbeing larger than the LO ones in this region, but the compensation is nontrivial. Thereforean all-LO description, for all its weaknesses, is more robust in the small- p ⊥ region, whichis where the MPI machinery largely operates. The default choice thus is H1 Fit B LO.Finally also the inclusive proton PDF f i/ p ( x, Q ) should be chosen. Here several optionscome with Pythia , and many more can be obtained via the interfaces to LHAPDF5 andLHAPDF6 [36, 37]. The current default set is the NNPDF 2.3 QCD+QED LO one with α s ( M Z ) = 0 . [38]. The argument for using LO has already been outlined above. Sincethe proton PDF is much better constrained than that of the P , there is less of a point invarying it between different options consistent with current p data. Note that, for diffractiveevents, the dependence on the original choice of proton PDF is largely removed on the P sideby applying eq. (2.4). It does remain on the proton side, and in the dynamical calculationof rapidity gap survival, however. The QCD → processes are dominated by t -channel gluon exchange, which gives aperturbative cross section dˆ σ/ d p ⊥ ∼ α ( p ⊥ ) /p ⊥ that diverges in the p ⊥ → limit. Twomodifications are needed to make sense out of this divergence.Firstly a divergent integrated QCD cross section should not necessarily be construedas a divergent total pp cross section. Rather a µ = σ tot2 → /σ totpp > for p ⊥ > p ⊥ min shouldbe interpreted as implying an average of µ such partonic interactions per pp collision.Overall energy-momentum conservation will reduce the naively calculated rate, but wouldstill kick out essentially all beam momentum if we allow p ⊥ min → , in contradiction withthe presence of well-defined beam jets wherein a single particle can carry an appreciablefraction of the incoming beam momentum.– 7 –econdly, therefore, it is important to note the presence of a screening mechanism:whereas standard perturbation theory is based on asymptotically free incoming states,reality is that partons are confined inside colour singlet states. This introduces a nonper-turbative scale of the size of a hadron, or rather of the average distance d between twoopposite-colour charges. In this spirit we introduce a free parameter p ⊥ ∼ /d that is usedto dampen the cross section d σ d p ⊥ ∝ α ( p ⊥ ) p ⊥ −→ α ( p ⊥ + p ⊥ )( p ⊥ + p ⊥ ) . (2.5)Technically the dampening is implemented as an extra factor multiplying the standard QCD → cross sections, but could equally well have been associated with a dampening of thePDFs; it is only the product of these that enters in measurable quantities.Empirically, a p ⊥ scale of 2 – 3 GeV is required to describe data. This scale is largerthan expected from the proton size alone, and is also in a regime where normally one wouldexpect perturbation theory to be valid. The p ⊥ scale appears to increase slowly withenergy, which is consistent with the growth of the number of gluons at smaller x values,leading to a closer-packing of partons and thereby a reduced screening distance d . A similarparametrization is chosen as for the rise of total cross section p ⊥ ( E CM ) = p ref ⊥ × (cid:18) E CM E refCM (cid:19) E powCM , (2.6)with E powCM and p ref ⊥ being tunable parameters and E refCM a reference energy scale.With the protons being extended objects, the amount of overlap between two incomingones strongly depends on the impact parameter b . A small b will allow for many parton-parton collisions, i.e. a high level of MPI activity, and a close-to-unity probability for theincoming protons to interact. A large b , on the other hand, gives less average activityand a higher likelihood that the protons pass by each other unaffected. Diffractive eventspredominantly occur in peripheral collisions, a concept well-known already from the opticalpoint of view. In our approach it comes out naturally since we only allow one interaction tooccur, namely the hard process of interest; if there is a second one this will fill the rapiditygap and kill the diffractive nature.The shape of the proton and the resulting overlap – the convolution of the two incomingproton distributions – is not known in any detail. The proton electric charge distributionmay give some hints, but measures quarks only and not gluons, and is in the static limit.Instead a few different simple parametrizations can be chosen: • a simple Gaussian, offering no free parameters, • a double Gaussian, i.e. a sum of two Gaussians with different radii and proton mo-mentum fractions, and • an overlap of the form exp( − b p ) (which does not correspond to a simple shape for theindividual proton), with p a free parameter.– 8 –A further option is a Gaussian with an x -dependent width, but this has not been im-plemented in a diffractive context.) All are normalized to unit momentum sum for theincoming partons, and an overall radius normalization factor is fixed by the total crosssection.The more uneven the matter distribution, the broader will the charged multiplicitydistribution be. Notably the higher the overlap for central collisions, the higher the tail tovery large multiplicities. Also other measures, like forward-backward correlations, probethe distribution. Unfortunately it is always indirectly, and closely correlated with othermodel details. As an example we can mention that the earliest tunes worked with a muchlower p ⊥ than today and with double Gaussians rather far away from the single-Gaussianbehaviour. This changed when more modern PDFs started to assume a steeper rise of thegluon PDF at small x , and when the Pythia parton showers were extended to apply to allMPIs rather than only the hardest one, and for some other improvements over the years.Currently best fits are not very far away from a simple Gaussian, e.g. with an overlap like exp( − b . ) , but still on the side of more peaked than a Gaussian.An event that contains a high- p ⊥ interaction is likely to be more central than one thatdoes not, since the former has more MPIs and therefore more chances that the hardest ofthese reaches a high p ⊥ . This bias effect is included in the choice of a b for an event wherethe hardest interaction has been given, and is used in the subsequent generation of MPIs.For the current study of hard diffraction this means that the hard process is initially pickedbiased towards smaller b values, but afterwards the central b region is strongly suppressedbecause the likelihood of several MPIs is so big there.Starting from a hard interaction scale, and a selected b , the probability for an MPI ata lower scale has the characteristic form d P d p ⊥ = O ( b ) 1 σ ref d σ QCD d p ⊥ . (2.7)Here O ( b ) is the overlap enhancement/depletion factor, d σ QCD the differential cross sectionfor all → QCD processes, and σ ref the total cross section for the event classes affected bythe QCD processes. Historically σ ref has been equated with the nondiffractive cross sectionin Pythia , on the assumption that diffraction only corresponds to a negligible fraction of d σ QCD . Within the current framework a reformulation to use the full inelastic cross sectionwould make sense, but would require further work and retuning, and is therefore left asidefor now.Given eq. (2.7) as a starting point, MPIs can be generated in a falling p ⊥ sequence,using a Sudakov-style formalism akin to what is used in parton showers. Actually, in thecomplete generation the MPI, ISR and FSR activity is interleaved into one common p ⊥ -ordered chain of interactions and branchings, with one common “Sudakov form factor”,down to the respective cutoff scales.In the current case, the MPI formalism is used twice. Firstly, to determine whetheran event is diffractive, and if not to generate the complete nondiffractive event. Secondly,for diffractive events, to determine the amount of MPI activity within the P p system. Hereeq. (2.7) can be reused, but with new meaning for the components of the equation.– 9 – The d σ QCD / d p ⊥ is now evaluated using the P PDF on one side, but with the samedamping as in eq. (2.5), where E CM in eq. (2.6) is now the P p invariant mass. If the P is supposed to have a smaller size than the proton then this could be an argumentfor a higher p ⊥ in this situation, but we have not here pursued this. • The σ ref now represents the P p total cross section, an unknown quantity that relatesto the normalizations of the P flux and P PDF. By default is is chosen to have afixed value of 10 mb, higher than is normally quoted in literature. This way, withother quantities at their default settings, the charged multiplicity of a P p collisionagrees reasonably well with that of a nondiffractive pp one at the same invariantmass. This may not be the best of arguments, but is a reasonable first choice that isexperimentally testable, at least in principle. • The O ( b ) factor may be changed, see next.The impact parameter b P p of the P p subcollision does not have to agree with the b pp ofthe whole pp collision. It introduces the transverse matter profile of the Pomeron, even lessknown than that of the proton. Generally a Pomeron is supposed to be a smaller object ina localized part of the proton, but one should keep an open mind. For lack of better, threepossibilities have been implemented, which can be compared to gauge the impact of thisuncertainty. • b P p = b pp . This implicitly assumes that a Pomeron is as big as a proton and centeredin the same place. Since small b pp values already have been suppressed, by the MPIselection step, it implies that few events will have high enhancement factors. • b P p = (cid:112) b pp (where normalization is such that (cid:104) b (cid:105) = 1 for minimum-bias events). Thiscan crudely be motivated as follows. In the limit that the P is very tiny, such that theproton matter profile varies slowly over the width of the P , then what matters is wherethe Pomeron strikes the other proton. Thus the variation of O ( b ) with b is that ofone proton, not two, and so the square root of the normal variation, loosely speaking.Technically this is messy to implement, but the current simple recipe provides themain effect of reducing the variation, bringing all b values closer to the average. • Pick a completely new b P p , as was done with b pp in the first place. This allows abroad spread from central to peripheral values, and thereby also a larger and morevarying MPI activity inside the diffractive system than the other two options, andthereby offers a useful contrast. In this section we summarise some of the tests and sanity checks we have performed on themodel implementation. This provide insight into how the model operates and with whatgeneral results, but also highlights the uncertain nature of many of the components of themodel. – 10 – . . . . . . x P − − − − x P f P / p ( x P ) Schuler-SjostrandBruni-IngelmanStreng-BergerDonnachie-LandshoffMBRH1 Fit AH1 Fit B (a) − − − − − x P − − − − x P f P / p ( x P ) Schuler-SjostrandBruni-IngelmanStreng-BergerDonnachie-LandshoffMBRH1 Fit AH1 Fit B (b)
Figure 1 . The seven different Pomeron fluxes included in
Pythia on linear (a) and logarithmicscale (b). Note that the MBR flux has not been renormalized (see [28]).
In the model we have two options for when an event is classified as diffractive: eitherright after the event has passed the PDF selection criterion, eq. (2.4), or after passing thefurther MPI criterion. Results using only the former will from now on be denoted “PDFselected”, and with the latter in addition “MPI selected”. Our full model for hard diffractioncorresponds to the latter, but the intermediate level is helpful in separating the effects ofthese two rather different physics components.Notably, many distributions tend to be mainly determined by one of the two criteria.Those mainly sensitive to the PDF selection include the x P and thereby the mass of thediffractive system, and the squared momentum transfer t of the process and thereby thescattering angle θ of the outgoing proton. In particular we will explore the dependence onPomeron fluxes and PDFs. Aspects that depend on the details of the MPI model includeseveral particle distributions, such as multiplicities, and that will also be highlighted.The key number where both components are comparably important is the overalldiffractive rate, where each of them gives an order-of-magnitude suppression, resulting in a ∼
1% fraction of hard events being of a diffractive nature. This number thereby receives aconsiderable overall uncertainty.
We begin by studying the effects of variations of the P parametrizations. In figures 1a and1b the seven different Pomeron fluxes are compared. As can be seen there is a considerablespread. Even in the region of medium x P values, x P ∼ . , this corresponds to more thana factor of two between the extremes. The dramatic differences at large x P are not readilyvisible, since a large- x P event usually corresponds to a small rapidity gap and thereforeis difficult to discern from non-diffractive events. The limit of small x P generally is moreinteresting, tying in with the intercept of the Pomeron trajectory, but plays a lesser role forthe current study of hard diffraction. – 11 – .0 0.2 0.4 0.6 0.8 1.0 x − − − − − F i / P ( x , Q ) Q = 100 NNPDF 2.3H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LOPomFix (a) − − − − − x − − − − F i / P ( x , Q ) Q = 100 NNPDF 2.3H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LOPomFix (b) x − − − − − F i / P ( x , Q ) Q = 100 NNPDF 2.3ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . (c) − − − − − x − − − − F i / P ( x , Q ) Q = 100 NNPDF 2.3ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . (d) Figure 2 . The QCD charge-weighted sum, eq. (3.1), of the H1 PDFs and the toy PDF PomFixcompared to the NNPDF 2.3 proton PDF on linear (a) and logarithmic scale (b). The QCD charge-weighted sum, eq. (3.1), of the ACTW PDFs compared to the NNPDF 2.3 proton PDF on linear(c) and logarithmic scale (d).
Turning to the Pomeron PDFs, a detailed comparison would entail the separate quarkand gluon distributions at varying Q scales. To simplify we show the QCD-charge-weightedsum F P ( x, Q ) = 49 (cid:88) i =q , q xf i/ P ( x, Q ) + xg P ( x, Q ) (3.1)at a single value Q = 100 GeV , figures 2a to 2d. We notice that they all tend to besignificantly harder than the corresponding proton PDF, here represented by the NNPDF2.3 QCD+QED LO one. (The PomFix option is just a toy one, shown for completeness,but not used in the following.) For the gluon on its own, the P is significantly harder thanthe p , consistent with the idealized picture of a P as a glueball state with two “valencegluons”, figures 3a, b and 4a, b. Surprisingly, also the quark PDFs of the P (figures 3c, dand 4c, d) are harder than proton ones, suggesting the presence of “valence quarks” in the P , although an order of magnitude below the gluons. Another observation is that the P – 12 – .0 0.2 0.4 0.6 0.8 1.0 x − − − x g P ( x , Q ) H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LONNPDF 2.3 g (a) − − − − − x − − − x g P ( x , Q ) H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LONNPDF 2.3 g (b) . . . . . . x − − − x q P ( x , Q ) H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LO NNPDF 2.3 dNNPDF 2.3 uNNPDF 2.3 ¯ d NNPDF 2.3 ¯ u (c) − − − − − x − − − x q P ( x , Q ) H1 Fit A NLOH1 Fit B NLOH1 JetsH1 Fit B LO NNPDF 2.3 dNNPDF 2.3 uNNPDF 2.3 ¯ d NNPDF 2.3 ¯ u (d) Figure 3 . The H1 P gluon distribution on linear (a) and logarithmic (b) scale. The H1 P quarkand antiquark distributions on linear (c) and logarithmic (d) scale. Both compared to the NNPDF2.3 proton PDF distributions. Note that for the P we have d = u = s = d = u = s(= c = c) , wherethe c , c are only included in H1Jets. PDF sets we compare are all primarily based on H1 analyses, with largely the same dataand with correlated assumptions for the definition of diffractive events. This is especiallynotable in the quark distributions, which are close to identical. Also the close affinity ofgluons at lower x values should not be overstressed. The slightly larger variations in theACTW PDFs are due to both the different values of the flux-parameter (cid:15) , as well as differentparametrisations of the PDFs. Finally, note that the H1 parametrizations only apply downto x = 10 − , and are frozen below that. This is likely to underestimate the low- x rise ofPDFs, which as well could have been of the same shape as in the proton. A small kink in theACTW PDFs around x = 10 − is due to regions of phase space where the parametrizationof the initial quark distribution would become negative and has been reset to vanish.In the end, what matters is the convolution of the P flux with its PDFs, and that isshown in figure 5. There would be too many combinations possible to show individually, so– 13 – .0 0.2 0.4 0.6 0.8 1.0 x − − − x g P ( x , Q ) ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . NNPDF 2.3 g (a) − − − − − x − − − x g P ( x , Q ) ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . NNPDF 2.3 g (b) . . . . . . x − − − x q P ( x , Q ) ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . NNPDF 2.3 dNNPDF 2.3 uNNPDF 2.3 ¯ d NNPDF 2.3 ¯ u (c) − − − − − x − − − x q P ( x , Q ) ACTW B, (cid:15) = 0 . ACTW D, (cid:15) = 0 . ACTW SG, (cid:15) = 0 . ACTW D, (cid:15) = 0 . NNPDF 2.3 dNNPDF 2.3 uNNPDF 2.3 ¯ d NNPDF 2.3 ¯ u (d) Figure 4 . The ACTW P gluon distribution on linear (a) and logarithmic (b) scale. The ACTW P quark and antiquark distributions on linear (c) and logarithmic (d) scale. Both compared to theNNPDF 2.3 proton PDF distributions. Note that for the P we have d = u = s = d = u = s(= c = c) ,where the c , c are only included in H1Jets. we only indicate the range of possibilities and a few specific combinations. This may be onthe extreme side, since some fluxes and PDFs come as fixed pairs, not really intended to bemixed freely. The key feature to note is that in this convolution the Pomeron part is nowfalling steeper at large x than the proton as a whole. This has the immediate consequencethat diffractive hard subcollisions are not necessarily going to be produced more in theforwards direction than the bulk of corresponding nondiffractive ones, but on the contrarymay be more central. The difference is not all that dramatic, however. It is also partlycompensated by a somewhat slower increase of the P towards lower x values, a feature thatfor the H1 P PDF derives from the artificial freezing of below x = 10 − . Note that the fourdifferent ACTW PDFs differ by up to an order of magnitude. The two D fits are similarin shape and size as expected, but especially the SG fit stands out being up to a factor 10smaller than the D fits. Most of this discrepancy is also seen in figure 2c,d but also arise– 14 – . . . . . . x − − − − − F D ( x , Q ) NNPDF 2.3SaS + H1 Fit B LOH1 Fit A + H1 Fit A NLO F D ( x, Q ) = R x d x P f P / p ( x P ) [ xx P g ( xx P , Q ) + P xx P q ( xx P , Q )] (a) − − − − − x − − − F D ( x , Q ) NNPDF 2.3SaS + H1 Fit B LOH1 Fit A + H1 Fit A NLO F D ( x, Q ) = R x d x P f P / p ( x P ) [ xx P g ( xx P , Q ) + P xx P q ( xx P , Q )] (b) . . . . . . x − − − − − − F D ( x , Q ) NNPDF 2.3DL + ACTW B, (cid:15) = 0 . DL + ACTW D, (cid:15) = 0 . DL + ACTW SG, (cid:15) = 0 . DL + ACTW D, (cid:15) = 0 . F D ( x, Q ) = R x d x P f P / p ( x P ) [ xx P g ( xx P , Q ) + P xx P q ( xx P , Q )] (c) − − − − − x − − − F D ( x , Q ) NNPDF 2.3DL + ACTW B, (cid:15) = 0 . DL + ACTW D, (cid:15) = 0 . DL + ACTW SG, (cid:15) = 0 . DL + ACTW D, (cid:15) = 0 . F D ( x, Q ) = R x d x P f P / p ( x P ) [ xx P g ( xx P , Q ) + P xx P q ( xx P , Q )] (d) Figure 5 . The convolution of Pomeron fluxes and H1 PDFs for a few cases, with the rangebetween the extremes marked in yellow; (a) linear and (b) logarithmic x scale. The convolution ofDL flux and ACTW PDFs on (c) linear and (d) logarithmic x scale. from the difference in normalisation, the D and SG fits having momentum sums of ∼ . and ∼ . , respectively. The lack of major shape differences between the P part and the restwill be visible in the more detailed studies later on. Because of the close similarity of mostof the different (but related) P PDFs at low-to-medium x , the bulk of the differences comefrom the P fluxes. We have chosen to exemplify this for → QCD processes with p ⊥ > GeV in √ s = 8 TeV pp collisions, with the diffractive fractions for a few combinationsshown in table 1.Note that changing the Pomeron parametrizations changes the fraction of events passingthe PDF selection, but that the suppression factor introduced by the dynamical gap survivalis about ∼ . for all combinations in table 1. This reflects the fact that neither the MPImodel nor the proton PDF are influenced by the Pomeron parametrization, hence theprobability for obtaining no additional MPIs in the pp system should not change. (This– 15 –iffractive fractions pp collisions at √ s = 8 TeV P PDF P flux PDF selection MPI selectionH1 Fit B LOSaS (14.33 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 1 . Diffractive fractions for the → QCD processes with p ⊥ > GeV obtained with
Pythia
8. The samples have been produced without any phase-space cuts. does not have to hold in general, but here we compare very similar distributions of x and p ⊥ values of the hard interaction, and then also the MPI effects are closely the same.) Notealso that some of the ACTW PDFs gives substantially larger fractions than the HERAPDFs. This is related to the fact that the intercept of the P trajectory is larger in ACTWfits than in the H1 ones, (cid:15) = 0 . − . vs. 0.085. This gives a larger flux at high-energyhadron colliders. A similar flux increase can of course be obtained for the H1 PDFs, withthe caveat that the flux might not be able to describe the total cross section and otherassociated quantities. Additionally the gluon is only probed indirectly in DIS, and so ispoorly constrained, while it dominates for QCD jet rates.Differential distributions of the diffractive events are also affected, since the kinematicsof the P p system is set up using the Pomeron flux parametrizations. A subset of thesedistributions is shown in figure 6, for some of the same combinations as in table 1. Asexpected, P PDF variations do not have a large impact on the shapes (cf. figure 6), whilethe P flux gives rise to large effects in x P , hence on the broadening of the mass spectrum andon the tails of the t and θ distributions. In view of these observations, we do not expect tobe able to discrimate between the available Pomeron PDFs when comparing to data. Thuswe will leave out this variation from now on, and focus on variations in the Pomeron flux.The diffractive event fraction is not process-independent. One reason is that processesmay be dominated by different initial states, another that different x and Q scales are– 16 – − − − − − x P − − / N d N / d x P SaS + H1 Fit B LOMBR + H1 Fit B LOSaS + H1 Jets (a) M X . . . . . / N d N / d M X SaS + H1 Fit B LOMBR + H1 Fit B LOSaS + H1 Jets (b) − . − . − . − . − . . t . . . . . / N d N / d t SaS + H1 Fit B LOMBR + H1 Fit B LOSaS + H1 Jets (c) . . . . . . θ / N d N / d θ SaS + H1 Fit B LOMBR + H1 Fit B LOSaS + H1 Jets (d)
Figure 6 . Some kinematics distributions obtained with variations of the Pomeron parametriza-tions: (a) x P , (b) M X , (c) t and (d) θ . probed. In table 2 we show the fraction of events passing either selection for various hardprocesses available in Pythia x needed to produce these particles, cf. figures 1,2. This is why top, being the heaviest,has the smallest diffractive fraction. In addition there is a notable difference between thegluon-dominated Higgs production and the quark-induced production of W ± /γ ∗ / Z , owingto the hard gluon PDF in the P . If top production is considered separately for qq → tt and gg → tt , the PDF survival rate is (9.74 ± ± W -boson produced in the process qq → W ± at an TeV pp collision, comparing three samples; nondiffractive, PDF selected and MPI selected.It is observed that the diffractive W ’s are slighly more central than the nondiffractive inthe CM frame, as expected from figure 5. The differences are small, however, being on theorder of (5-10)%, and might reduce when phase-space cuts are introduced. We will study– 17 –iffractive fractions pp collisions at √ s = 8 TeVPDF selection MPI selection qq → W ± (11.16 ± ± qq → γ ∗ / Z (10.69 ± ± ± ± ± ± Table 2 . Diffractive fractions obtained with
Pythia without any phasespace cuts at √ s = 8 TeVfor various hard processes.
Pythia is run with the SaS flux and the H1 Fit B LO PDF. -10 -5 0 5 10 η / N d N / d η NDPDF selectionMPI selection
Figure 7 . The rapidity of the W -boson produced in qq → W ± at √ s = 8 TeV. this process further in section 4.1.
In the above section we studied how the parametrization of the Pomeron flux and PDFaffected the diffractive fractions and distributions, and notably by the choice of P flux. Bycontrast, we saw that the survival fraction in the MPI selection step was not significantlyaffected by these choices. A dependence does enter both via the x and the p ⊥ distributions ofa process: larger x scales leaves less energy for MPIs and thereby gives a higher MPI survivalprobability, whereas larger p ⊥ values gives a longer MPI evolution range and thereby a lowerMPI survival. Such effects are not too prominent, however, and tend to be overshadowedby the sensitivity to the parameters of the MPI model. These enter twice. Firstly, for theMPI selection, since the dynamical gap survival is tied to the number of MPIs in the pp system. Secondly, for the properties of the diffractive system, where the number of MPIsaffects e.g. charged multiplicities.The probability for obtaining MPIs is given by eq. (2.7), and hence depends on boththe overlap function and the regulator p ref ⊥ . The related parameters are primarily tunedto minimum bias and underlying event data, e.g. charged particle pseudorapidity d n/ d η ,– 18 – .0 0.5 1.0 1.5 2.0 2.5 b / N d N / d b NDPDF selectionMPI selection (a) . . . . . . b . . . . . / N d N / d b p ref ⊥ = 2 . p ref ⊥ = 1 . p ref ⊥ = 2 . (b) . . . . . . b . . . . . . . . / N d N / d b b P p = b pp b P p = p b pp b P p new (c) . . . . . . b . . . . . . . . . / N d N / d b Default overlapSG profileDG profileExp. overlap (d)
Figure 8 . Impact-parameter distribution of → QCD processes with p ⊥ > GeV in √ s = 8 TeV pp collisions. (a) The change during the selection steps. (b) The dependence on p ref ⊥ . (c) Thedistribution in the P p subcollision. (d) The dependence on impact-parameter profile. multiplicity P ( n ) and transverse momenta d n/ d p ⊥ and (cid:104) p ⊥ (cid:105) ( n ch ) spectra of charged par-ticles. This means that a change of MPI parameters for the diffractive studies would spoilagreement with nondiffractive data. Nevertheless, it is interesting to study how the survivalrate changes with these parameters for the pp collision itself.The MPI modelling of the P p collision can be decoupled from that of the pp one. Thenthe MPI survival rate would not be affected by changes, but only the particle distributions inthe diffractive system. One inevitable free parameter is the effective P p total cross section.It is currently set always to be 10 mb, but could be made to depend on the mass of thediffractive system. Also the relative normalization of P flux and PDFs can influence theevent activity. We will study the normalization dependence in the last part of this section.To begin with, consider the impact-parameter picture associated with hard collisionsin our model, figure 8a. The b scale is normalized such that (cid:104) b (cid:105) = 1 for inclusive minimum-bias events. Events with a hard interaction tend to be more central than that, since central– 19 –iffractive fractions pp collisions at √ s = 8 TeVPDF selection MPI selection p ⊥ = 1 . (14.50 ± ± p ⊥ = 2 . (14.33 ± ± p ⊥ = 2 . (14.19 ± ± Table 3 . Diffractive fractions for the → QCD processes with p ⊥ > GeV in √ s = 8 TeV pp collisions. Pythia is run with the SaS flux and the H1 Fit B LO PDF. events have more MPIs in general and thereby a bigger likelihood that at least one of themis at large p ⊥ . The PDF selection step does not have a significant impact, but the MPI onekills most low- b events and pushes (cid:104) b (cid:105) above unity. The reason is obvious: for central eventsthe average number of MPIs is high, and thus the likelihood of only having the trigger hardprocess and no further MPIs is small, while more peripheral collisions give fewer MPIs andthereby a higher surviving fraction. Ultimately, when (cid:104) n MPI ( b ) (cid:105) (cid:28) , most protons passby each other without colliding at all. Thus the interesting region for diffraction is where (cid:104) n MPI ( b ) (cid:105) ∼ .The p ref ⊥ regulator is by default 2.28 GeV. Since an increase in this parameter givesless MPI in the pp system, we expect an increase in the diffractive fractions, and viceversa. table 3 confirms that this is indeed the case: variations of ± . GeV around thedefault p ref ⊥ value gives about a factor of two in the MPI selection rate. This major p ref ⊥ dependence holds also for many other nondiffractive event properties, however; keepingeverything else fixed even a variation of ± . GeV would be unacceptable. In figure 9we show the charged multiplicity distribution, when we change the regulator p ref ⊥ for bothdiffractive and nondiffractive events, with minor/major effects for the former/latter. Thestability in the diffractive case is because a change in the regulator also affects the impactparameter picture. Specifically, in this case b P p = b pp has been assumed. A lower valueof the regulator, giving rise to a larger number of MPIs in the pp system, pushes (cid:104) b pp (cid:105) tolarger values for those events that survive the diffractive MPI criterion, figure 8b. Moreprecisely, the change is to b values where the average pp MPI activity is restored to itsoriginal level. With b P p = b pp the same then holds when MPI activity is generated in thediffractive system, such that the effects of a smaller regulator and a larger impact parameteralmost completely cancel.As we have already discussed, the modelling of the P size could also affect the MPImachinery for the P p subcollision via the impact parameter b P p . The currently implementedthree alternatives are compared in figure 8c. The maybe less realistic last option of pickinga new b P p value at random implies a significant fraction of events with small b P p and therebythe possibility of many MPIs. The average (cid:104) n MPI (cid:105) for the three options is 1.66, 2.04 and4.09, respectively, thus giving rise to 0.66, 1.04 and 3.09 additional MPIs besides the hardestprocess. This is reflected notably in the charged multiplicity distribution, figure 10a.The MPI survival rate is highly dependent on the proton matter profile, table 4 and– 20 –
50 100 150 200 n ch . . . . . . / N d N / d n c h p ref ⊥ = 2 . p ref ⊥ = 1 . p ref ⊥ = 2 . (a) n ch . . . . . . . . . / N d N / d n c h p ref ⊥ = 2 . p ref ⊥ = 1 . p ref ⊥ = 2 . (b) Figure 9 . Charged multiplicity distributions in the (a) P p subsystem for diffractive events, (b) pp system for nondiffractive events, in → QCD processes with p ⊥ > GeV as before. n ch . . . . . . / N d N / d n c h b P p = b pp b P p = p b pp b P p new (a) n ch . . . . . . . . / N d N / d n c h H1 Fit A + H1 Fit A NLOH1 Fit A + H1 Fit A NLO rescaledH1 Fit A + H1 Fit A NLO rescaled, σ ref P p = 20 mb (b) Figure 10 . Charged multiplicity distribution distributions for the P p diffractive subsystem,for events with → QCD processes with p ⊥ > GeV as before. (a) For three different b P p impact-parameter profiles. (b) With or without rescaled P flux and PDFs, see text. figure 8d. Diffraction thrives when (cid:104) n MPI ( b ) (cid:105) ∼ , so this b region should be as broad aspossible for a large diffractive rate. Conversely, a sharp proton edge implies less diffraction.The default overlap function exp( − b . ) is close to a Gaussian, and the two have about thesame MPI selection rate. The double Gaussian and the exponential overlap are examples ofbroader distributions, thus with more diffraction, whereas the option without any b depen-dence represents the other extreme (not shown in figure 8d), with less diffraction. Overallthe variation is not so dramatic, however, if only experimentally acceptable variations areconsidered.Finally we turn to the relative normalization of the P PDF and flux. From eq. (2.3)we know that the PDF selection step depends on the convolution of the P flux and PDFs.– 21 –iffractive fractions pp collisions at √ s = 8 TeVPDF selection MPI selectionNo impact parameter dependence (14.36 ± ± ± ± ± ± ± ± ± ± Table 4 . Diffractive fractions for the → QCD processes with p ⊥ > GeV in √ s = 8 TeV pp collisions. Pythia is run with the SaS flux and the H1 Fit B LO PDF.
Thus it has no net effect if the flux is scaled down by a factor of two and the PDFs arescaled up by the same amount, so as to bring the H1 PDFs to be approximately normalizedto unit momentum sum. It does have consequences for the MPI selection step, however,since the average MPI rate comes up in the P p system.Compared with the (1.35 ± ± d σ MPI in eq. (2.7). This could be compensated by acorresponding increase of σ ref from the default 10 mb to 20 mb, thereby restoring both theMPI selection rate and the multiplicity distribution, cf. the blue line in figure 10b. Here we study the model dependence on the scales in the hard process and the energy ofthe collision.In figure 11 the diffractive fractions are compared at different collision energies, √ s , for → QCD processes with p ⊥ > GeV, and for W ± production. In the PDF selectionstep the diffractive rate increases with energy. The difference between the two processesindicates that this rise can depend on the incoming flavours and the relevant ranges of x values. Depending on the P flux and PDF, such as a freezing of the latter at small x , thefraction might even decrease with energy.A larger collision energies also implies a higher average number of MPIs, in additionto the hardest collision, thus implying a reduced fraction of events passing the MPI crite-rion, see figure 11. There is a compensatory effect of diffraction shifting to larger impactparameters, as already discussed for the p ref ⊥ variations. For the close-to-Gaussian defaultoverlap the relative size of the (cid:104) n MPI (cid:105) ≈ region decreases with energy, however, result-ing in the trend shown. By comparison an exponential overlap decreases slower than theclose-to-Gaussian, hence resulting in less suppression with increasing energy.Finally, table 5 shows the number of events passing the PDF and MPI selections whenthe mass of the produced particle is changed. In the PDF selection step heavier particles areless likely to be produced diffractively, as they require larger x -values, where the probability– 22 – √ s D i ff r a c t i v e f r a c t i o n ( % ) QCD, PDFQCD, MPI W, PDFW, MPI (a) √ s D i ff r a c t i v e f r a c t i o n ( % ) Default, PDFDefault, MPI Exponential, PDFExponential, MPI (b)
Figure 11 . (a) The diffractive fractions obtained in → QCD processes with p ⊥ > GeV(circles and solid lines) and qq → W ± (squares and dashed lines) in pp collisions at differentenergies. (b) The diffractive fractions obtained in qq → W ± with the default overlap function(squares and dashed lines) and the exponential overlap function (crosses and dashed-dotted lines). Pythia is run with the SaS flux and the H1 Fit B LO PDF.
Diffractive fractions pp collisions at √ s = 8 TeVPDF selection MPI selection M W = 50 GeV (11.52 ± ± M W = 80 . GeV (10.69 ± ± M W = 150 GeV (10.46 ± ± M W = 500 GeV ( 9.47 ± ± Table 5 . Diffractive fractions for the process qq → Z in √ s = 8 TeV pp collisions. Pythia isrun with the SaS flux and the H1 Fit B LO PDF. for diffraction is lower (cf. figure 5). The same trend was observed in table 2, but was theremixed up by the use of different production channels. After the MPI selection step themass dependence is not as clearly visible. A partial compensation can indeed occur, sincea higher subcollision mass implies more momentum taken out of the incoming protons andthereby less left for subsequent collisions.
The new model for hard diffraction complements the existing one for soft (or rather inclu-sive) diffraction. The latter already has a hard component arising from the MPI model,which is used to pick the hardest process and all subsequent scatterings in the P p system,except for low-mass diffractive systems where no perturbative framework can be applied.The soft diffractive model only allows for → QCD processes, unlike the new hard one,but for QCD processes a comparison between the two is meaningful. To this end, the p ⊥ of the hardest process in an event will be used. This is not a physically measurable observ-– 23 –
20 40 60 80 100 p ⊥ − − − − − − − d σ / d p ⊥ ( m b / G e V ) NDSDDD (a) p ⊥ − − − − − − − d σ / d p ⊥ ( m b / G e V ) ND softSD soft ND hardSD hard (b)
Figure 12 . The p ⊥ of the hardest process obtained with (a) the soft (or inclusive) diffractionframework, and (b) both the soft and hard diffraction frameworks for events with p ⊥ > GeV.
Cross sections pp collisions at √ s = 8 TeVSoft diffraction Hard diffractionND sample, p ⊥ > GeV (mb) 3.730 4.239ND sample, p ⊥ > GeV (mb) 0.348 0.353SD sample, p ⊥ > GeV (mb) 0.084 0.048SD sample, p ⊥ > GeV (mb) 0.0066 0.0035
Table 6 . Cross sections obtained with the two diffractive frameworks. Extracted from figure 12by integration. able, unlike e.g. the closely related p ⊥ of the hardest jet in an event, but for the relativecomparison of hard and soft diffraction it is cleaner.The MPI framework predominantly gives low- p ⊥ interactions, be it for diffractive ornondiffractive events. Thus only a small fraction of the events will have p ⊥ values at ∼ GeV or more, see figure 12a. Note that the p ⊥ spectrum falls faster for diffractive thannondiffractive events, mainly as a consequence of the former having a P p invariant massspectrum peaked towards lower values.In figure 12b the p ⊥ of the hardest process for the two samples is compared. One isobtained by generating inclusive (soft) events and keeping only those with large enough p ⊥ ,the other by generating only hard events above 10 GeV. Nondiffractive events are shownas a sanity check, as for them the two approaches should give the same results. A closerlook at integrated cross sections, table 6, shows a small discrepancy for the p ⊥ > GeVcase, while the p ⊥ > GeV agree much better. This discrepancy is caused by not havinga “Sudakov factor” in the hard model. That is, in the soft model the rate at lower p ⊥ scalesis reduced by the requirement of not having an interaction at a higher p ⊥ scale, whereasno such reduction is implemented in the hard framework, which only uses pure matrix– 24 –lements.The single diffractive events show differences in the normalisation, while the shape ofthe p ⊥ distributions agree between the two frameworks. The normalisation differences arisefrom the two different ways of handling the survival rate. The soft diffractive frameworkassumes an effective flux of P ’s inside the proton, rescaled to get the desired total diffractivecross section, and thereby implicitly includes an average rapidity gap survival factor. Thehard diffractive framework has a higher initial P flux but then explicitly implements adynamical event-by-event survival factor. As it works out, single diffractive high- p ⊥ eventsare somewhat more suppressed in the latter case. This is indeed what we would expect:there should be more MPIs in high- p ⊥ (and high-mass) events than in low- p ⊥ ones, and thusmore MPI survival suppression. Put another way, the soft implementation overestimatesthe suppression at low p ⊥ and underestimates it at high p ⊥ . (Assuming our new model isthe right way to view the matter.)In the future it would be desirable to include such dynamical effects also in the softframework, so that the two descriptions can be made to agree in the high- p ⊥ region. Thisis not a trivial task, however. In this section we compare the new model for hard diffraction with some available data.While many results have been presented for soft diffractive processes, less is available onhard ones.At the Tevatron, both the CDF and D0 collaborations studied hard diffractive events.We have chosen here to compare with two analyses, one in which only the diffractive frac-tions are measured, the other in which also the distributions of the hard collisions arereported.At the LHC, diffraction has been studied both by ATLAS [39–41] and CMS [42–44].One key observation there is that the
Pythia default P flux shape does not describe therapidity gap distribution so well, suggesting that a new parametrization may be needed. Inother respects the model seems to do a reasonable job. For hard diffraction we will compareto the latest ATLAS study, [41], and a similar CMS study, [43].Unfortunately, neither of the studies at hand are implemented as Rivet [45] analyses,so we have tried to apply the relevant experimental cuts as best as we can. This makescomparisons with data less than reliable, and results should therefore be taken as a firstindication only. At least for LHC the intention is that the new Pythia options can bedirectly tested by the experimental community, to allow more precise comparisons in thefuture. W / Z production at the Tevatron CDF has measured the fraction of events with a diffractively produced W / Z boson at √ s = 1 . TeV [46]. The surviving antiproton was measured in a Roman Pot forwardspectrometer, and the boson decay products in the central detector. The observed fractionof events with forward antiprotons was doubled, to compensate for there being no Roman– 25 –DF cuts W sample Z sampleND SD SD × (%) ND SD SD × (%)Lepton E eT ( p µT ) >
25 GeV 670602 2827 0.84 667851 2466 0.74Missing E T >
25 GeV 595236 2490 0.84 - - -One electron in | η | < | η | < M WT = [40, 120] GeV 327671 1361 0.83 - - - M Z = [66,116] GeV - - - 36814 1397 0.76 | t | < - 1348 0.82 - 1383 0.75 x P = [0.03,0.1] - 366 0.23 - 346 0.19 Table 7 . Cuts used in [46]. Number of events listed in each of the samples are based on MonteCarlo truth obtained when generating inclusive events. A blank means that a specific cut wasnot relevant. Pots on the proton side. Only the e and µ leptonic decays of the bosons were taken intoaccount. The cuts used in the analysis are listed in table 7, along with the number of eventsthat survive after each step. To this end, the internal W - and Z -finder projections availablein Rivet [45] have been used as a starting point; these have previously been validated forother CDF analyses. In addition the diffractive properties are derived from the measuredantiproton as t = − p ⊥ (4.1) x RPS P = 1 − | p z |√ s (4.2)which has been compared to Monte Carlo truth, giving good agreement.The results in table 7 are obtained with Pythia x P cut. Therefore, although we beginwith a “Monte Carlo truth” fraction of ∼ diffractive W / Z , this is reduced to ∼ . by the x P cut. Results look better for other choices of P flux, see table 8, but even at beststill with a factor two discrepancy. Note that it is the fluxes that rise fastest in the low- x P region that gives fractions closer to data.We can compare these values to the results from [32], where no gap survival factor isincluded. The authors only show results on W production and use different integrationlimits on x P . A subset of the results are listed in Table 9. It is worth noting that theresults using the lower integration limit are of the same order as the default settings of Pythia
8, whereas the high integration limit (which is that of CDF) are higher than bothdata and our model. This we interpret as being due to the lack of suppression factor, astheir calculations do not take MPIs into account.The diffractive fraction can also be increased by changing the free parameters of theMPI framework, with the caveat that nondiffractive events will then no longer describedata as well. Table 10 shows the diffractive fractions obtained when varying some of the– 26 –
PDF P flux ( pp → p (cid:48) + W ) × pp → p (cid:48) + Z ) × ± ± ± ± ± ± ± ± ± ± ± ± Table 8 . Diffractive fractions for the W → lν and Z → l + l − , l = e, µ in √ s = 1 . TeV pp collisions. P PDF P flux x P = 0 . x P = 0 . CDF - (1.0 ± (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . Table 9 . Diffractive fractions for the W production from [32]. Parameter ( pp → p (cid:48) + W ) × pp → p (cid:48) + Z ) × ± ± p ref ⊥ = 2 . GeV (0.59 ± ± ± ± Table 10 . Diffractive fractions for the W → lν and Z → l + l − , l = e , µ in √ s = 1 . TeV pp collisions. MPI parameters. This variation is still not sufficient when combined with the default fluxand PDF in
Pythia
8. If combined with some of the fluxes in table 8 it would be possibleto obtain fractions close to the experimentally observed values, however.
Another interesting measurement performed at CDF was the process pp → p + X p , X p → X + J + J , ie. SD dijet production with a leading antiproton. CDF measured this at three– 27 –DF cutsJet E , T > E T > | η , , | < ∆ R | t | < x RPS P [0.035,0.095] Table 11 . Cuts used in [21]. different energies, √ s = 630 , 1800 and 1960 GeV [21, 47, 48]. Here not only the diffractivefractions were measured, but a number of differential distributions. Large discrepancieswere found between the diffractive structure functions determined from CDF data andthose extracted by the H1 Collaboration from diffractive deep inelastic scattering data atHERA. The discrepancies are both in normalisation and shape and were interpreted as abreakdown of factorization.Our comparison focuses on the 1800 GeV data ([21]), since this also includes a mea-surement of the diffractive structure function. The cuts used in the analysis are listed intable 11. The jets are identified with the CDF cone algorithm as implemented in Rivet[45], with a cone radius of . . Jet energy scale corrections for underying-event activityare done separately for diffractive and nondiffractive events, as outlined in the CDF article,but only has a minor impact on relative rates. The momentum transfer of the antiprotonis evaluated using eq. (4.1) and the momentum loss of the antiproton using eq. (4.2).We begin by evaluating the suppression factor introduced by the MPI framework. Thisis evaluated by running two samples of events, one with and one without the MPIcriterion, both using the cuts of table 11 and the SaS flux and the H1 Fit B LO PDF.We obtain a suppression factor of 0.11, to be compared with the quoted discrepancies fromCDF of . ± .
02 (0 . ± . when using the H1 Fit 2 (Fit 3), respectively [21]. Asimilar suppression factor as for SaS is obtained when using the H1 Fit B flux, based onthe same parametrization as the H1 Fit 2 and 3 fluxes, although with different values forthe free parameters of the model. Using this flux, however, allows for approximately twotimes more events passing the experimental cuts. This is due to the fact that the H1 Fit Bflux is less restrictive in the low- x P region, where the experiment is performed. Hence weexpect better agreement with data when using the H1 Fit B flux, as compared to SaS. Weare not able to directly compare to the suppression factors obtained in [32], as these havebeen calculated with different kinematical cuts (eg. E T > GeV and . < x P < . ),but the numbers obtained are still interesting in their own right. Alvero et. al. obtainsuppression factors of 0.061 (fit B and DL flux, (cid:15) = 0 . ), 0.029 (fit D, same flux) and 0.12(fit SG, same flux), thus ranging from the measured suppression factor to our one.Results on kinematical distributions using both the SaS and the H1 Fit B flux areshown in figure 13. The SD E ∗ T distribution has a steeper falloff than the ND distribution,indicating a lower center-of-mass energy in the collision. Likewise the η ∗ distribution is– 28 –hifted towards positive η , the proton direction, indicating a boost of the center-of-masssystem. The final kinematical distribution here is the difference in azimuthal angle betweenthe two leading jets. This observable was not shown in the 1800 GeV analysis but in the1960 GeV one. The SD events there show a tendency to be more back-to-back than the NDones. This can also be attributed to the lower energy in the P p collision than in the full pp system, leaving less space for initial-state radiation.The momentum fraction of the antiproton carried by the subcollision parton can beevaluated from the jets using x = 1 √ s (cid:88) i =1 E iT e − η i , (4.3)where the sum is over the two leading jets, plus a third if it has E T > GeV. The resultis shown in figure 14, for the two P fluxes used in figure 13. As expected the SaS flux,figure 14a, suppress the diffractive events too much, as the suppression factor is too largecompared to experimental value from CDF. With this flux, the PDF selected samples lieabove the CDF data, but then drop by an order of magnitude by the MPI selection, tolie well below the data, by a factor of five. There is also some discrepancy in shape.Changing to the H1 Fit B flux, figure 14b, the PDF selected sample lies above the data asexpected, with the MPI selected sample a bit below, although only by a factor of three. Thesuppression is still too large, and shapes still disagree, but not as markedly as in figure 14a.There are some aspects of the CDF article that we don’t understand, however. The keyfigure 4 of [21] is intended to show the H1 predictions for the diffractive structure functionalong with the experimentally measured one. The information provided on how the formerprediction is obtained is inconsistent with the curve shown, however, in normalization andshape. In the end we therefore put more faith in the suppression factor between CDFand HERA, already presented above, than in absolute numbers. Assuming we could havereproduced the CDF curve intended to represent the predictions of the H1 PDFs, that thenis suppressed by an average multiplicative factor of . − . in data but . in ourmodel, we should have been a factor of ∼ above data, which is inconsistent with theoutcome in figure 14. CMS has studied the diffractive contribution to dijet events at √ s = 7 TeV pp collisions[43], The cross section is presented as a function of (cid:101) ξ , an approximation to the fractionalmomentum loss of the scattered proton correspinding to the x P variable. Dijets were selectedwith p ⊥ > GeV in the | η | < . range using the anti- k ⊥ algorithm with a cone size of R = 0 . [49]. (cid:101) ξ was reconstructed using particles in the region | η | < . with p ⊥ > . GeV for charged particles as well as particles in the range . < | η | < . with E > GeV.To enhance the diffractive contribution additional requirements was introduced, such thatthe minimum rapidity gap was of 1.9 units (no particles was allowed in the region | η | > ).Finally a cut on (cid:101) ξ < . was introduced.With these cuts, rapidity gap survival probabilities are in the range . ± . (NLO)to . ± . (LO), where the NLO gap survival probability was found using PomPyt and– 29 –
10 20 30 40 50 E ∗ T = ( E T + E T ) / − − − / N d N / d E ∗ T NDSaS flux, PDFSaS flux, MPI (a) E ∗ T = ( E T + E T ) / − − − / N d N / d E ∗ T NDH1 Fit B flux, PDFH1 Fit B flux, MPI (b) − − − − η ∗ = ( η + η ) / . . . . . . . . / N d N / d η ∗ NDSaS flux, PDFSaS flux, MPI (c) − − − − η ∗ = ( η + η ) / . . . . . . . . / N d N / d η ∗ NDH1 Fit B flux, PDFH1 Fit B flux, MPI (d) . . . . . . . φ = | φ − φ | . . . . . . . / N d N / d ∆ φ NDSaS flux, PDFSaS flux, MPI (e) . . . . . . . φ = | φ − φ | . . . . . . . / N d N / d ∆ φ NDH1 Fit B flux, PDFH1 Fit B flux, MPI (f)
Figure 13 . The mean E T of the leading jets in both SD and ND events using (a) the SaS and(b) the H1 Fit B flux. The mean η of the leading jets in both SD and ND events using (c) the SaSand (d) the H1 Fit B flux. – 30 – − − − x i/ ¯ p d σ / d x i / ¯ p [ p b ] NDSaS flux, PDFSaS flux, MPI (a) − − − x i/ ¯ p d σ / d x i / ¯ p [ p b ] NDH1FitB flux, PDFH1FitB flux, MPI (b) − − − x i/ ¯ p − − − − − R a t i o S D / N D SaS flux, PDFSaS flux, MPI CDF data (c) − − − x i/ ¯ p − − − − − R a t i o S D / N D H1 Fit B flux, PDFH1 Fit B flux, MPI CDF data (d)
Figure 14 . The antiproton momentum fraction carried by the parton entering the hard collision,for
Pythia
Pythia is run with the H1 Fit B LO PDF and (a) theSaS or (b) H1 Fit B flux. (c) and (d) shows the ratio SD to ND using (a) and (b).
PowHeg [50]+
Pythia
PomPyt and
PomWig .Implementing the same cuts in
Pythia
8, using the SaS flux and the H1 Fit B LOPDF gives a rapidity gap survival probability of 0.06, compatible with the CMS results.Changing from the SaS flux to the H1 Fit B flux gives the same suppression factor, butallows for more events to pass the experimental cuts. We thus see the same trend as in theCDF analysis, where the SaS flux is too restrictive at low x P . Recently, the ATLAS collaboration published a study of dijets with large rapidity gaps in √ s = 7 TeV pp collisions [41]. Dijets were selected with p ⊥ > GeV in the | η | < . range,and the cross section was measured in terms of ∆ η F , the size of the observed rapidity gap,as well as in (cid:101) ξ = (cid:80) p i ⊥ e ± η i / √ s , the estimate of the fractional momentum loss deduced from– 31 –et cutsJet E , T >
20 GeVJet | η , | < k ⊥ ∆ R | p | >
200 MeV | η | < | p | >
500 MeV or p ⊥ >
200 MeV | η | < Table 12 . Cuts used in [41]. charged and neutral particles in the ATLAS detector (the sign on η depends on where inthe detector the largest gap is located). Cuts used in the analysis are listed in table 12.Experimental results were compared with the Pythia
PomWig generator [13], on the other hand, needed an additionalsuppression of S = 0 . ± . (stat) ± . (sys) in order to describe data.In this section we use the new model for hard diffraction to study the same cross sec-tions. The new model currently only includes the SD contribution, hence we will not beable to describe all aspects of data, especially in the high- ∆ η F and low- (cid:101) ξ -regions, where theSD and DD contributions are comparable in size, at least according to the soft diffractionmodel available in Pythia
8. We could also expect the normalisation of the SD eventsobtained with the hard diffraction framework to be lower than in the soft one and thusin data, because of the difference in normalisation between the two frameworks (cf. sec-tion 3.4). The ND contribution should not differ from the ATLAS analysis, however, sinceno changes have been implemented in this framework.The ND distribution was normalized to data, where the normalization factor was foundusing the first bin of the ∆ η F distribution. This approach has also been used in our analysis,although when generating an inclusive sample (e.g. the purple distribution in figures 15band 15d) this normalization is applied to the full sample, unlike in the ATLAS paper. Inthis sample, no classification of events occurs, hence the normalization cannot be performedonly on the ND sample. In the exclusive samples, the distinction between ND and SD isperformed, and we can apply the normalization to only the ND sample (cf. the blackdistribution in figures 15b and 15d).In figure 15 we show the results obtained with the model for hard diffraction. Threesamples are compared: ND, PDF-selected SD and MPI-selected SD. Note that the MPI-selected sample lies about a factor of 10 below the PDF-selected one, as usual, and that– 32 – ∆ η F − d σ / d ∆ η F [ n b ] NDPDF MPIATLAS data (a) η F d σ / d ∆ η F [ n b ] ND + MPIInclusive sample ATLAS data (b) − . − . − . − . − . − . . e ξ d σ / d l og e ξ [ n b ] NDPDF MPIATLAS data (c) − . − . − . − . − . − . . e ξ d σ / d l og e ξ [ n b ] ND + MPIInclusive sample ATLAS data (d)
Figure 15 . The dijet cross sections as a function of the size of the rapidity gap (a), (b) andthe fractional momentum loss of the proton (c), (d). Compared to the hard diffraction model of
Pythia
Pythia
8. Only statistical errors are included in the ATLAS errorbars. the suppression due to the MPI-framework is constant over both intervals. The new modelundershoots the data in the regions where the DD contribution is non-negligible ( ∆ η F > and log (cid:101) ξ < − . ). When this contribution is included in the framework, a betteragreement with data should be possible, and overall the picture should be consistent withthe soft diffractive framework. In this article we have studied hard diffraction by combining two concepts, the Ingelman–Schlein picture of a Pomeron and the
Pythia model for multiparton interactions. ThePomeron fluxes and PDFs are mainly extracted from HERA data, while the MPI picture– 33 –and several other relevant physics components) makes use of a broader spectrum of Teva-tron and LHC data. This combination allows us, in principle, to predict all physical quanti-ties of hard diffractive events, from rapidity gap sizes to charged multiplicity distributions,but most importantly the fraction of diffractive events for any hard process.Reality is not quite as simple, however. In this article we have studied the differentassumptions that go into a detailed framework, and explored the inherent uncertainties.One part concerns the assumed Pomeron flux and PDFs, where particularly the latter isdominated by one source only, namely the H1 analyses, making it difficult to assess tofull range of uncertainty. Another part concerns the MPI framework, which enters twice.When used the first time, to determine the diffractive MPI survival, it involves parametersalready tuned to nondiffractive data, so narrowly constrained in principle. There could stillbe leeway, e.g. if we were to use other parton showers that give less/more activity at small p ⊥ scales, the average number of MPIs would have to rise/drop to compensate. Thus ourstudies focus on the sensitivity of some key parameters of the framework. When the MPIsare used the second time, inside the diffractive subsystem itself, the level of uncertainty isconsiderably higher. A key example is the impact-parameter picture of the P p subcollision,notably how impact parameters are related between the pp and P p steps of an event.Our studies puts the finger on our still limited understanding of diffraction, also whenrestricted to the Pomeron framework, which is only one model class for diffraction. Fur-ther, we provide computer code that can be used to compare with data for hard diffractiveprocesses at the LHC. It thus can be used as a “straw man” model, where differences be-tween predictions and data can help pave the way for a deeper understanding and moreaccurate models. Specifically, with a generator it is possible both to emulate the experi-mental diffractive trigger and to compare the resulting event properties, both of which areconsiderably more complicated for analytical models.Comparisons with data have shown qualitative agreements in many respects, but maybeless so than one could have hoped for. For the Tevatron we face the problem of trying tounderstand 15 years old analyses, with uncertain results. The main message probably is thatthe overall Tevatron suppression factor of ∼ − , relative to HERA-based extrapolations,agrees well with what our model gives from the MPI selection step. For the future it willtherefore be more interesting to compare with LHC studies, in particular those available inRivet.It is well known that the existing Pythia model for soft diffraction is not fully describ-ing the existing LHC data; at places the difference can be up to a factor of two. Similarlywe have seen less-than-perfect agreement for the hard diffractive processes studied in thisarticle. There is therefore room for improvements in both areas, and also for work to bringthe two approaches in closer contact. As one simple example, the soft model currentlydoes not involve a MPI survival step, and therefore the Pomeron flux does not have to benormalized in the same way in the two cases. The intention is to study such issues closer,and to provide an improved description of diffractive cross sections, both integrated anddifferential ones. – 34 – cknowledgments
We thank Marek Tasevsky for detailed descriptions of the ATLAS analysis and providingthe data used in figure 15. Work supported in part by the Swedish Research Council,contract number 621-2013-4287, and in part by the MCnetITN FP7 Marie Curie InitialTraining Network, contract PITN-GA-2012-315877. This project has also received fundingfrom the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreement No 668679).
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