Models for total, elastic and diffractive cross sections
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Abstract
The LHC has brought much new information on total, elastic and diffractive cross sections, which is not always in agreement with extrapolations from lower energies. The default framework in the Pythia event generator is one case in point. In this article we study and implement two recent models, as more realistic alternatives. Both describe total and elastic cross sections, whereas one also includes single diffraction. Noting some issues at high energies, a variant of the latter is proposed, and extended also to double and central diffraction. Further, the experimental definition of diffraction is based on the presence of rapidity gaps, which however also could be caused by colour reconnection in nondiffractive events, a phenomenon that is studied in the context of a specific model. Throughout comparisons with LHC and other data are presented.
1 Introduction
Historically there are two main approaches to \(\sigma _{\mathrm {TED}}\) in hadron–hadron collisions, the diagrammatical and the geometrical, although both aspects may well be represented in a specific model [1, 2, 3, 4]. In the diagrammatical approach new effective particles are introduced, specifically the Pomeron(s) \(\mathbb {P}\) and Reggeon(s) \(\mathbb {R}\), with associated propagators and vertex coupling strengths. A Feynmandiagramlike expansion may be performed into different event classes, with higherorder corrections. A subset of these are shown in Fig. 1, with \(\mathbb {X}=\mathbb {P},\mathbb {R}\) and each of the couplings denoted with a g. In the diagrammatical approach, the dashed line (the cut) represents the diagram at amplitude level. A cut through a \(\mathbb {P}\) or \(\mathbb {R}\) thus represent particle formation at amplitude level, while an uncut Pomeron or Reggeon represents an area void of particle production. In a geometrical approach the impactparameter aspects are emphasized, where diffraction largely is related to peripheral collisions. The analogy with wave scattering theory here is natural, and has given the diffractive event class its name. Diffraction can also be viewed as a consequence of the interaction eigenstates being different from the mass ones [5, 6].
Neither of these approaches address the detailed structure of diffractive events. In olden days, at low energies, a diffractive system was simply viewed as an excited proton state that could decay moreorless isotropically, a “fireball” [1, 7]. This is clearly not a valid picture for highermass diffractive states, where the same kind of longitudinal structure is observed as for nondiffractive ones. The simplest partonic approach would then be for a \(\mathbb {P}/\mathbb {R}\) to kick out a single quark or gluon from a proton, giving rise to one or two fragmenting colour strings. The Ingelman–Schlein picture [8] takes it one step further and introduces an internal structure for the \(\mathbb {P}\), such that a \(\mathbb {P}{\mathrm p}\) collision may be viewed as an inelastic nondiffractive \({\mathrm p}{\mathrm p}\) (or better \(\pi ^0{\mathrm p}\)) one in miniature. Thereby also hard jet activity and multiparton interactions (MPIs) become possible within a diffractive system, as supported by data.
A key aspect of MPI modelling is the relation to colour reconnection (CR), whereby partons in the final state may be related in colour so as to reduce the total string length relative to naive expectations. This opens for another view on diffraction, where CR can generate rapidity gaps dynamically [9, 10]. Then the diffractive and inelastic nondiffractive event classes have a common partonic origin, and only differ by the eventbyevent fluctuations in colour topologies. Even in models that do not go quite as far, the dividing line between the two kinds of events may be fuzzy. This is even more so since the experimental classification in terms of a rapidity gap allows for misidentification in both directions, relative to the classification in a specific model. Highmass diffraction need not give a gap in the central detector, while nondiffractive events by chance (CR or not) can have a large rapidity gap. The classification of each event type in Pythia 8, however, is independent of CR model, such that no double counting occurs on a theoretical level. Each event type has its own specific cross section and, in a combined sample, the mix of event types is based on that. The experimental signature of the events, however, does differ depending on CR model. Thus it is likely that CR models that can give large gaps in nondiffractive events will need a suppression of the diffractive cross sections in order to describe data.
What should now be clear is that description of the \(\sigma _{\mathrm {TED}}\) physics, and especially the diffractive part, is too multifaceted to be based purely on analytical calculations. The implementation into Monte Carlo Event Generators is crucial to test different approaches. One of the most commonly used generators is Pythia [11, 12], which by default is based on a rather old diagrammatical “tune” for the \(\sigma _{\mathrm {TED}}\) issues [13], combined with an Ingelman–Schleinstyle approach to the diffractive event structure [14]. In particular the first part does not agree well with LHC data, and so needs an overhaul.
For the total and elastic cross sections we have chosen to implement two different parametrizations, the parametrization from the COMPAS group as found in the Review of Particle Physics 2016 [15] and a model developed by Appleby and collaborators (ABMST) [16]. In addition to a better fit to the integrated cross sections, these also include a more detailed description of the differential elastic cross sections.
The ABMST model also addresses single diffraction. It is in an ambitious diagrammatical approach, supplemented with a careful description of the resonance shape in the lowmass region, based on comparisons with lowenergy data. Unfortunately, as is common in such ansätze, the diffractive cross section asymptotically grows faster with energy than the total one, making it marginally acceptable already at LHC energies and definitely unacceptable for FCC ones. We therefore study possible modifications that would give a more reasonable energy behaviour. Further, while ABMST does not address double or central diffraction, we use the framework of the model to extend it also to these event classes, and in the process need to make further adjustments. Results for the ABMSTbased modelling implemented in Pythia are compared with the already existing default framework of SchulerSjöstrand (SaS) and Donnachie–Landshoff (DL) [13, 17], and confronted with LHC data.
Furthermore we study the sensitivity to CR by comparing with the Christiansen–Skands QCDbased CR model (CSCR) [18]. This model has no protection against “accidental” rapidity gaps in nondiffractive events, unlike the default CR framework. But it is also not intended to describe (the bulk of) diffraction, and therefore it requires a retuning to provide a sensible combined description. It therefore offers an interesting case study for a tuning task that is likely to become more common in the future.
The plan of the article is as follows: in Sect. 2 we begin by summarising the current status of Pythia 8, the default cross section parametrizations along with the hadronic event properties of diffractive events. In Sect. 3 we describe the new models for total and elastic (differential) cross sections. In Sects. 4, 5 and 6 we extend these to single, double and central diffractive (differential) cross sections, respectively. In Sect. 7 we provide some comparisons to LHC data and provide new tunes of the default Pythia 8 model. We end with Sect. 8, where we summarise and provide an outlook to further studies.
2 The current status of Pythia 8
Pythia 8 is a multipurpose event generator aimed at the generation of highenergy events. This includes collisions both of a perturbative and a nonperturbative character, each of which gives contributions to the total collision cross section. In perturbative collisions, the description begins with the matrix element of the hard scattering process in combination with parton distribution functions. This core is dressed up with several other elements such as multiparton interactions, parton showers and hadronisation. In nonperturbative scattering collisions, on the other hand, no standard formulation exists for the core process, and phenomenological models are needed. After the modeldependent choices of the key kinematical variables have been made, the event generation may be continued in a similar manner as for perturbative events, where relevant.
In this paper we focus on the nonperturbative scattering processes, and the generation of these. To set the stage for further improvements, the purpose of this section is to describe the current status of the event generator. This we have split into two parts, beginning with the description of the default cross section models, the SaS/DL one, and then go on to describe the event property aspects that are the same regardless of the choice of model.
2.1 Differential cross sections
2.2 Hadronic event properties
In the highmass regime it is assumed that the diffractive cross section factorises into a Pomeron flux, a Pomeron–proton cross section, and a proton form factor. Together these determine the mass \(M_X\) of the diffractive system and the squared momentum transfer t in the process. Neither the \(\mathbb {P}\) flux nor the \(\mathbb {P}{\mathrm p}\) cross section are known from first principles; therefore seven similar but somewhat different \(\mathbb {P}\) flux options are available in Pythia 8.
The internal structure of the \(\mathbb {P}{\mathrm p}\) system is then considered in an Ingelman–Schleininspired picture. Thus perturbative processes are allowed, and \(\mathbb {P}\) parton distribution functions (PDFs) are introduced like for a hadron. Standard factorization can be assumed, i.e. cross sections are given by hardscattering matrix elements convoluted with the PDFs of two incoming partons. Furthermore, the full interleaved shower machinery of Pythia 8 is enabled, giving rise both to initial and finalstate showers and to multiparton interactions in the \(\mathbb {P}{\mathrm p}\) system. This results in a more complex colour string structure than in the lowmass regime, which can also be subjected to additional colour reconnection, owing to overlap and crosstalk between the multiple subsystems.
The activity in the \(\mathbb {P}{\mathrm p}\) system, as represented e.g. by the average charged multiplicity, can be tuned to roughly reproduce that of a nondiffractive \({\mathrm p}{\mathrm p}\) collision of the same mass. This activity is closely related to the average number of MPIs per event, the calculation of which differs between the two systems by a \(\mathbb {P}\) vs. a \({\mathrm p}\) PDF in the numerator, and by \(\sigma _{\mathbb {P}{\mathrm p}}^{\mathrm {eff}}\) vs. \(\sigma _{{\mathrm p}{\mathrm p}}^{\mathrm {nondiffractive}}\) in the denominator. Given a \(\mathbb {P}\) PDF, and assuming the same MPIframework parameters as in \({\mathrm p}{\mathrm p}\), the \(\sigma _{\mathbb {P}{\mathrm p}}^{\mathrm {eff}}\) thus becomes the main (massdependent) tuning parameter. In reality the two systems can be different, however, so experimental information on diffractive mass and multiplicity distributions can be used to refine the tune. Be aware that a different choice of PDFs is likely to require a different \(\sigma _{\mathbb {P}{\mathrm p}}^{\mathrm {eff}}\) value. Ten different \(\mathbb {P}\) PDF sets are implemented [27, 28, 29, 30], plus a few toy ones for special purposes. Many of these have been fixed by some convention for the \(\mathbb {P}\) flux normalization, that in Pythia could be set differently. In principle most of the \(\mathbb {P}\) PDFs should only be used with the associated \(\mathbb {P}\) flux, as some of the experimentally provided PDFs do not assume a factorisation of the \(\mathbb {P}\) flux and PDF. In practise the two can be chosen independently, as this opens up for comparative studies. Similarly, the convention used for the \(\mathbb {P}\) flux normalisation is often dependent on the experimental limits and often normalised to unity at seemingly arbitrary phase space points. Other important aspects, such as the momentum sum rule, are also usually neglected in the PDFs provided by experiments, but often needed in phenomenological studies. Hence all \(\mathbb {P}\) PDFs are implemented with the option to be rescaled, e.g. in order to approximately impose the momentum sum rule.
In the MPI framework [31] the joint probability distribution for extracting several partons from a Pomeron needs to be defined. This is done in the same spirit as for protons [32]. MPIs are ordered in a sequence of decreasing \(p_{\perp }\) scales, and for the hardest interaction the normal PDFs are used. For subsequent ones the x value is interpreted as a fraction of the then remaining \(\mathbb {P}\) momentum, thereby ensuring that the momentum sum is not violated. Pomerons are assumed to have no valence quarks; thus the \(\mathbb {P}\) PDFs initially only contain gluons and a quark–antiquarksymmetric sea. If a quark is kicked out of the beam, however, flavour conservation requires that an explicit “companion” sea antiquark must also be present in the leftover \(\mathbb {P}\), and vice versa. Such a companion is introduced as an extra component of the \(\mathbb {P}\) PDF, similar to a valence (anti)quark, with normalization to unity (just like the d valence in a proton). Overall momentum is preserved by scaling down the gluon and ordinary sea quark distributions to compensate. If the companion is selected for a subsequent MPI, then that “valence” component is removed, and the gluon and sea components of the \(\mathbb {P}\) PDF are scaled back up.
Also initialstate radiation (ISR) requires special attention in the MPI framework. ISR is generated starting from the hard interaction and then evolving backwards, to lower scales and larger x values [33]. Such ISR branchings are combined with the MPI generation into one interleaved sequence of falling \(p_{\perp }\) scales. As above special consideration has to be given to branchings that change the flavour of the incoming parton, and that can either induce or remove a companion (anti)quark.
Similar to a proton [32], the Pomeron will leave behind a remnant after the MPIs and showers have removed momentum and removed or added partonic content. To begin, assume that only one gluon is kicked out of the incoming \(\mathbb {P}\). The remnant will then be in a net colour octet state, which means that two colour strings eventually are stretched to the outgoing partons of the hard collision (or to the other beam remnant). The remnant could only consist of gluons and sea \({\mathrm q}\overline{\mathrm q}\) pairs, since the \(\mathbb {P}\) has no valence flavour content, so the simplest representation is as a single gluon or a single \({\mathrm q}\overline{\mathrm q}\) pair. From a physical point of view the two options would give very closely the same end result, since the hairpin string via a gluon remnant eventually would break by the production of \({\mathrm q}\overline{\mathrm q}\) pairs. For convenience, the choice is therefore made to represent the remnant as an octet \({\mathrm u}\overline{\mathrm u}\) or \({\mathrm d}\overline{\mathrm d}\) pair with equal probability. In the general case, further unmatched companion quarks are added to represent the full flavour content needed in the remnant. Most MPI initiators are gluons, however, which carry colour that should be compensated in the remnant. This is addressed by attaching the gluon colour lines to the already defined remnants, which implicitly introduces colour correlations between the initiator partons. Such initialstate correlations can be further enhanced by colour reconnections in the final state. The final colour topology decides how strings connect the outgoing partons after the collision, and thereby sets the stage for the hadron production by string fragmentation.
2.3 Hard diffraction
The noMPI requirement introduces a gap survival probability determined on an eventbyevent basis, unlike other methods used in the literature. As MPIs only occur in hadron–hadron collisions, the framework provides a simple explanation of the differences between the diffractive event rates obtained at HERA and Tevatron. Diffractive fractions and survival probabilities obtained with the new framework show good agreement with experiments, while some distributions show lessthanperfect agreement, see [34] for a discussion. The model is currently only available for single diffraction; future work would be to extend this to both double and central diffraction.
3 Total and elastic cross sections
The parametrizations of the total and elastic cross sections are related through the optical theorem. The elastic cross section has historically been well described in the framework of Regge theory, with varying complexity based on the number of exchanges included in the model. Up until the LHC era the simple ansatz of DL [17] using only a Pomeron and an effective Reggeon has described the total cross section surprisingly well. With a simple exponential t spectrum, the SaS parametrization [13] extended this to the elastic cross section, and here at least the lowt data was well described. But with the higher energies probed at the LHC it has become obvious that these simple parametrizations fail. More complex trajectories have to be introduced in order to describe both the rise of the total cross section and the t spectrum of the elastic cross section.
We have chosen to implement two additional models in Pythia 8. One, the model from the COMPAS group as presented in the Review of Particle Physics 2016 [15], is of great complexity, using six different single exchanges as well as some combinations of double exchanges, along with the exchange of three gluons, the latter becoming important at high t. The other, the newly developed ABMST model [16], is somewhat simpler, extending the original DL model to four single trajectories and all possible combinations of double exchanges between these, along with the triplegluon exchange for high t values.
Recent TOTEM collaboration data on elastic scattering hint that none of the traditional models describe all aspects of their data. Specifically, TOTEM obtains a decreasing \(\rho \) parameter, and observes no structure in the hight region (unpublished, but see e.g. [35]). There is an ongoing discussion in both the theoretical and experimental community on how to describe all data simultaneously. None of the models implemented here do that, specifically they do not predict a decreasing \(\rho \) value. Further, the ABMST model does not show any sign of structure at high t, while the COMPAS one does. Models could be extended to include a maximal odderon, similar to the work of Avila et al. [36, 37] (AGN) and Martynov et al. [38] (FMO), which would be able to describe the decrease in \(\rho \). At the time of writing the former has not been fitted to the new TOTEM data and the latter has not been extended to \(t \ne 0\). Thus, for now, we have chosen not to implement either in Pythia 8, but we show the FMO model in the relevant figures for completeness. Below we will give short descriptions of each of the fully implemented models.
3.1 The COMPAS model
3.2 The ABMST model
A somewhat simpler scattering model was proposed by Appleby et al. describing \({\mathrm p}{\mathrm p}\) and \({\mathrm p}\overline{\mathrm p}\) data from ISR to Tevatron energies [16]. The model is based on work by Donnachie and Landshoff [39, 40] describing both elastic scattering and single diffractive scattering, but includes new and more sophisticated fits compared to the ones from Donnachie and Landshoff. In this section the details on the elastic scattering will be given, while the single diffractive scatterings are presented in Sect. 4.
3.3 The FMO model
3.4 Comparisons with data
4 Single diffractive cross sections
As we proceed to the topologies of diffraction, the situation is more complicated than for total and elastic cross sections. The experimental definition of diffraction is based on the presence of rapidity gaps, but such gaps are subject to random fluctuations in the hadronization process, and therefore cannot be mapped onetoone to an underlying coloursingletexchange mechanism. Also the separation between single, double and central diffraction is not always so clearcut. Some singlediffractive data is available at lower energies, but much of it is old and of varied quality. This will of course affect any model trying to describe these topologies, as usually there are model parameters that have to be fitted to data. To the best of our knowledge, only a few models actually try to fit data fully differentially in both s, \(M_X^2\) and t. The normal ansatz is instead to define an sindependent \(\mathbb {P}\) flux, with factorized \(\xi \) and t distributions, e.g. of the form \(({\mathrm d}\xi / \xi ^{1+ \delta }) \, \exp (b \, t) \, {\mathrm d}t\) [27, 43, 44, 45] where \(\delta \) is a small number. The tintegrated \(\xi \) distribution is then directly mapped on to an \(M_X^2 = \xi s\) spectrum.
The COMPAS group has not made any attempts to describe other topologies than the elastic, neither has the FMO model. Hence, in addition to the already implemented SaS and MBR models, we are left with the ABMST model as a new alternative, that gives a full description of the single diffractive topologies. This model has been fitted to differential data in the energy range \(17.2< \sqrt{s} < 546\) GeV and in the t range \(0.015< t < 4.15\) GeV\(^2\), and is thus expected to give a reasonable prediction in this range. The model, however, has some unfortunate features, which we will discuss in a later section. But first an introduction to the basics of the model itself.
4.1 The ABMST model
4.2 Comments on the ABMST model
In Fig. 4a, b we show the different components of the ABMST model at an energy of \(\sqrt{s}=7\) TeV along with the integrated cross sections in Fig. 4c, d. We have several comments to these distributions, as they show some unexpected features.
To begin, consider the differential distribution in Fig. 4a. Here the cross section (multiplied by a factor of \(\xi \) for visibility) is shown as a function of \(\xi \), displaying both the lowmass resonances and the highmass Regge terms. Note, however, the dip between these two regimes, a decrease of a factor of 10. This is a feature of the background modelling, whereas one would expect a more smooth transition between the two regimes. There is no physical motivation as to why the Regge trajectories should have a quadratic behaviour at low masses, since none of the terms show this behaviour at higher masses. One could imagine a simple continuation of the highmass background to lower masses, with the resonances added on top. But this would likely cause too high a cross section in the lowmass region, hence requiring a remodelling of the background description to avoid too high a lowmass cross section.
Similarly unexpected is the increase of the cross section at higher masses (\(\xi \sim 1\)), induced by the tripleReggeon and pion terms. The larger the mass of the system the smaller the rapidity gap between the diffractive system and the elastically scattered proton. The rule of thumb is that \(\varDelta y_{\mathrm {gap}} \approx \ln (\xi )\), so for large \(\xi \) there will essentially be no gap at all. The diffractive system will simply look like a nondiffractive one, making it impossible to distinguish between the two experimentally. The rise at \(\xi \sim 1\) also introduces a vast increase with energy in the integrated cross section, making the singlediffractive cross section dominate at large energies, which leaves little room for other processes, see Fig. 4d. The authors themselves have tried to dampen the increase of the cross section by allowing the mass cut, separating the low and highmass regimes, to vary with s, Eq. (23). Unfortunately the introduced dampening gives rise to a kink in the integrated cross section where the dampening kicks in, at \(\sqrt{s}\sim 60\) GeV, and does not dampen the cross section sufficiently at high energies.
In Fig. 4b we show the ABMST model differential in t. Noteworthy are the tindependent terms \(C_{kki}\) and the sharp cutoff at \(t=\,4\) GeV\(^2\), both of which are unphysical on their own. That is, if the sharp cutoff is disregarded, then all but the pion and triplePomeron terms become constant at large t, lacking any form factor suppression for scattering a proton without breaking it up. The choice of t parametrization shape was based on the goodnessoffit, and not on any physical grounds. The authors note that the parametrization as such gives too large a cross section at high energies, hence the modification of the Pomeron coupling, as this dominates at high energies. The t ansatz may also cause problems if used in other diagrams, e.g. in the extension to double and central diffraction that we will introduce later.
As Pythia 8 aims to describe current and future colliders, the need for a more sensible highenergy behaviour of the ABMST model is evident. It is not realistic to have a model where single diffraction and elastic scattering almost saturates the total cross section at FCC energies (at \(10^5\) GeV \(\sigma _{\mathrm {tot}}  \sigma _{\mathrm {el}}  \sigma _{\mathrm {SD}} \approx 145  45  80 \approx 20\) mb). At the same time we want to make use of the effort already put into the careful tuning to lowenergy and lowdiffractivemass data. We have thus chosen to provide a modified version of the ABMST model, addressing the problems discussed above, as described in the next section, while retaining the good aspects of the ABMST model. Both the modified and the original version of the ABMST model are made available in the latest Pythia 8 release.
4.3 The modified ABMST model
The new parametrization of the highmass background in the lowmass region does smoothen the decrease between the two regions, but in itself does increase the integrated cross section. We tame the integrated cross section by introducing a multiplicative rescaling of the highmass region, as well as a different \(M_{\mathrm {cut}}\) parametrization. Again several possibilities have been tried, and best results were obtained for a \(\ln ^2(s)\)dependent \(M_{\mathrm {cut}}\) and rescaling. That is, \(M_{\mathrm {cut}} = 3+c\,\ln ^2(s/s_0)\) GeV and the rescaling factor is \(3 / (3+c\,\ln ^2(s/s_0))\), with c a free parameter and \(s_0 = 100\) GeV\(^2\), which is also where the rescaling begins, so as to avoid kinks in the distributions.
While this change reduces the cross section at intermediate \(\xi \) values, it does not address the strong rise near \(\xi = 1\). This is an unobservable behaviour, as already argued, and therefore we also introduce a dampening factor \(1 / (1 + (\xi \exp (y_{\mathrm {min}}))^p)\) for the highmass region. Here \(y_{\mathrm {min}}\) is the gap size where the dampening factor is 1 / 2 and p regulates how steeply this factor drops around \(y_{\mathrm {min}}\); by default \(y_{\mathrm {min}} = 2\) and \(p = 5\).
In Fig. 5 we show the components of the modified ABMST model as a function of \(\xi \) (a) and t (b). The improvements of the modifications are clearly seen, as the dip between the low and highmass description has decreased, the high\(\xi \) region has been dampened and none of the components become constant at large t. In Fig. 5c, d the two ABMST models are compared to the SaS model available in Pythia 8 as default. We note that the modified ABMST model shows better agreement with the SaS model at intermediate \(\xi \) values, where SaS is in rough agreement with data, while retaining some features of the ABMST model, such as the detailed resonance structure.
The bulk of the modifications applied to the ABMST framework are intended to tame the highenergy behaviour of the model. One could have used an eikonal approach to the same end, e.g. in the spirit of [46]. This would require a different set of assumptions, however, such as the impactparameter shape of the different diffractive topologies, and therefore not be any less arbitrary. For now we therefore stay with the current framework and instead proceed to address other shortcomings of the ABMST model, namely the lack of double and central diffraction.
5 Double diffractive cross sections
The ABMST model only provides a description of the single diffractive differential cross section. We can extend this to double diffractive systems, by extracting the vertices and propagators from the single diffractive framework and using them in double diffractive diagrams. Figure 1e shows a double diffractive diagram, where \(\mathbb {X}\) is one of the Reggeons used in the single diffractive framework. Thus several diagrams are obtained with Reggeons i, j, k (where i, j are connected to the proton and k are in the loop). Similar as for single diffraction, in order for the unknown phases in the propagators to vanish, the requirement of equal Reggeons is enforced in the loop. The fact that there are two different mass regimes (low and high) for the two diffractive systems X and Y gives four different combinations.
To correct for the possible suppressions arising from the chosen approximation of the elastic cross section, and from the underestimation implied by the step taken in Eq. (35), we introduce a scaling factor similar to the one introduced in the single diffractive framework. A minimal double diffractive slope can also be enforced, such as to avoid any unphysical situations. As a final modification, an option to reduce topologies without a rapidity gap is applied in the region where both of the systems are of very large masses. Again, this is to be able to distinguish the doublediffractive system from the nondiffractive ones.

Pure ABMST: the original ABMST single diffractive model together with the elastic cross sections using only Pomerons, with the minimal double diffractive slope and with reduced vanishinggap topologies.

Model 1: The modified ABMST model for the single diffractive cross section, with the onlyPomerons elastic cross section. A minimal slope is used and the vanishinggap topologies are also reduced here.

Model 2: Model 1 scaled with the tuneable factor \(k(s/m_p^2)^p\), where by default \(k=2\) and \(p=0.1\).
Figure 9a shows the t spectrum of the different models compared to the SaS model. It is evident that three models vanish faster than the SaS model. This is a result of the modest falloff of the elastic tspectrum in ABMST, as this affects the double diffractive slope less than a sharply falling elastic tspectrum in SaS, through the relation \(B_{XY}=B_{AY} + B_{XB}  B_{\mathrm {el}}\). Figure 9b shows the differential cross section as a function of \(\xi =\xi _1\xi _2\). Here, the ABMST models show an approximate \(1/\xi \)behaviour, while the SaS model indicates a \(1/\xi ^{(1+p)}\) behaviour with \(p>0\), favouring highmass diffractive systems. The results of these effects are visible in the integrated cross section, Fig. 10, where both “Pure ABMST” and “Model 1” are significantly suppressed compared to the SaS model. The scaled version, “Model 2”, gives more reasonable estimates of the cross sections, around 10 mb at LHC energies, but because of the choice of the power, \(p=0.1\), in the scaling, it does not rise as steeply as the SaS prediction. Similarly to the singlediffractive case, the SaS model predicts slightly larger cross sections than measured, so one might expect that the scaling chosen in Model 2 could be more in agreement with measurements.
6 Central diffractive cross sections
The central diffractive framework has long been neglected in generalpurpose event generators. Dedicated eventgenerators exist for exclusive central diffractive processes, such as SuperChic [47] and ExHume [48], but these only work with a limited set of final states. Pythia 8 provides a description for inclusive highmass central diffraction, but does not provide any such description for the exclusive processes. As stated earlier, we stress that the framework has not been tuned and thus is not to be trusted too far.
In this work we wish to extend the present description of central diffraction to include the highmass description of the ABMST model. We have not made any attempt to include any lowmass resonances of central diffraction, as some of these are still not well established. The lowmass resonances used in ABMST are baryonic resonances, hence they cannot be extended to the central diffractive framework, as one expects scalar mesons, possibly scalar glueballs, to be produced in the collision of two Reggeons. Future work would be to extend the model to such lowmass resonances, e.g. by including a lowmass resonance description similar to what has been developed in [49]. There the central exclusive production of a pion pair is considered and data is used to fit a model of the scalar resonances using complex Breit–Wigner shapes. Lacking a model for all such exclusive states, and since some of the resonances and their decays still are not experimentally under control, we have decided not to include any of the lowmass states in this framework.
Similar to the double diffractive framework, the central diffractive framework will depend on the choice of single diffractive framework, thus several options exist. Figure 11 shows three choices of models with the same name conventions as used in the double diffractive framework. Note, however, that the t spectrum is not shown, as this is exactly that of the single diffractive model. The mass of the diffractive system is shown in Fig. 11a, where the sharp cut at \(M_X=M_{\mathrm {cut}}\) is present for all ABMST variants. The SaS model has a similar sharp cutoff, but at \(M_X=1\) GeV. Lacking both model and data in the lowmass region, the cut allows for a clear distinction between what is included and not, albeit being unphysical.
7 Results
In this section the models are confronted with more recent LHC data. Several experiments have performed measurements on integrated cross sections and diffractive fractions, but not many provide results on differential distributions. We focus on the analyses available in Rivet [50], where only two analyses provide differential results. First we provide a discussion of the available data and the tuning prospects, and end with results obtained with the SaS model, the CSCR model and the ABMST models.
7.1 The 7 TeV LHC data and tuning prospects
In 2012 and 2015 ATLAS [51] and CMS [52] presented results on 7 TeV events with rapidity gaps. Both experiments measure all particles with transverse momenta larger than 200 MeV in pseudorapidity ranges of \(\eta  < 4.9 \, (4.7)\) for ATLAS (CMS), and define the measured gap \(\varDelta \eta _F\) as the largest distance between either detector edge and the particle nearest to it. The two experiments, however, obtain different results for the shape of the distribution.
Besides the above mentioned datasets measurements of the inelastic and diffractive cross sections have been performed by both ATLAS, CMS and ALICE. We include the following measurements: the inelastic cross section from ATLAS 2011 [53], the inelastic cross section from CMS 2012 [54] and the inelastic and diffractive cross sections from ALICE 2012 [55].
Figures 13 and 14 show the effects of changing the \(\mathbb {P}{\mathrm p}\) cross section on the charged particle distributions in the different \(\varDelta \eta _F\) bins compared to the 4C tune. In [51] the 4C tune generally was seen to undershoot the low clustermultiplicities, while overshooting the mid to high cluster multiplicities. In the highest \(\varDelta \eta _F\) bin, dominated by the diffractive events, Tune 4C undershoots both the low and highmultiplicity activity. Reducing the \(\mathbb {P}{\mathrm p}\) cross increases the multiplicity, and vice versa. Thus, to describe the highmultiplicity events, a smaller \(\mathbb {P}{\mathrm p}\) cross section would be preferred. This could be compensated by allowing the perturbative description to go below \(M_X=m_{\mathrm {min}}=10\) GeV, thus allowing slightly more activity in lowmass systems, possibly increasing the number of lowmultiplicity events. The effects of including a mass dependence in the \(\mathbb {P}{\mathrm p}\) cross section is seen in Fig. 14. A parametrization has been chosen as \(\sigma _{\mathbb {P}{\mathrm p}}^{\mathrm {eff}}(M_X) = \sigma _{\mathbb {P}{\mathrm p}}^{\mathrm {ref}} \, (M_X / M_{\mathrm {ref}})^p\), with \(M_{\mathrm {ref}} = 100\) GeV. Here, an increase of p slightly decreases the highmultiplicity region, albeit more subtly than with an increase of the \(\mathbb {P}{\mathrm p}\) cross section. Recall that the mass of the diffractive system is related to the collision energy, such that a value of \(p\sim 0.20.3\) is not unreasonable, corresponding to a rise of the cross section with energy of \(s^{0.1}s^{0.15}\).
The bulk of the cross section arises from nondiffractive events. These tend to only give rise to small rapidity gaps, as the phase space is more or less evenly filled by multiparton interactions. Gaps of intermediate or large size can occur, however, e.g. by colour reconnection between the partons [9, 61]. The default Pythia CR framework has been designed to avoid accidental gaps, so as to keep a clean separation between diffractive and nondiffractive topologies. In other models, e.g. the CSCR one [18], the colour reshuffling tends to give somewhat larger probability for intermediate gaps. A combination of the CSCR model and the default SaS diffractive setup then results in too large a cross section in the intermediategap range, cf. Fig. 15a, b.
Diffractive events are more likely to give rise to intermediate to large gaps. Hence, depending on colourreconnection model used, they will dominate from gap sizes of approximately two and larger. The size of the gap is closely connected to the mass of the diffractive system. Thus a model with a \({\mathrm d}M_X^2/M_X^2\) ansatz, like the SaS one (modulo some corrections), will give an approximately flat distribution of measured gap sizes. This can be modified by the recent inclusion of the masscorrection factor \(\epsilon _{\mathrm {SaS}}\), which introduces an additional \(1/M_X^{2\epsilon _{\mathrm {SaS}}}\) factor to the differential model. Depending on the sign of \(\epsilon _{\mathrm {SaS}}\), it will either increase or decrease the highmass cross section. In both the ATLAS and CMS datasets an increase of the largegap cross section is seen. Thus we expect a positive sign for \(\epsilon _{\mathrm {SaS}}\), as this will enhance the activity at low masses. For simplicity, adding the mass correction will not affect the integrated diffractive cross section.
The ABMST models show slightly better agreement with the shape of the rapidity gap distributions, although the original ABMST model overshoots both datasets. This was to be expected, as the model had trouble with the increase of the single diffractive cross section at LHC energies. The modified version of the ABMST model shows very nice agreement with both datasets, except for an undershoot of the highmass region of the doublediffractivedominated region in Fig. 15d. This behaviour closely correlates with the flatness of the \(\xi {\mathrm d}\sigma /{\mathrm d}\xi \)spectrum, Fig. 9b. Both the ABMST models have a mass spectrum shape comparable to data in the singlediffractiondominated region, unlike the SaS model, which overshoots the highmass systems.
7.2 The tuned models
The parameters used in the tunes for the different models
\(\epsilon \)  \(\alpha '\)  \(\sigma _{\mathrm {SD}}^{\mathrm {max}}\)  \(\sigma _{\mathrm {DD}}^{\mathrm {max}}\)  \(\sigma _{\mathrm {CD}}^{\mathrm {max}}\)  \(\epsilon _{\mathrm {SaS}}\)  

SaS  0.06  0.4  22.31  39.83  0  0 
SaS+CSCR  0.15  0.26  20.81  13.13  0  0 
SaS+\(\epsilon _{\mathrm {SaS}}\)  0.04  0.30  24.78  52.49  0  0.08 
\(k_{\mathrm {SD}}\)  \(k_{\mathrm {DD}}\)  \(k_{\mathrm {CD}}\)  \(p_{\mathrm {SD}}\)  \(p_{\mathrm {DD}}\)  \(p_{\mathrm {CD}}\)  

ABMST  0.58  2.45  1.0  0  0.05  0.03 
ABMST modified  0.92  1.72  1.38  0  0.1  0.04 
Figure 16 shows the three SaSbased models tuned to the abovementioned data. Neither of the three models are able to describe the shape of the gap data perfectly, Fig. 16a, b. The tune has decreased the amount of activity in the mid to largegap region by a decrease of the \(\sigma _i^{\mathrm {max}}\) values used in Eq. (9). The inclusion of \(\epsilon _{\mathrm {SaS}}\) has shifted some of the activity from intermediategaps to larger ones, while keeping the integrated cross section fixed. Unfortunately this is at the expense of an undershoot in the transition region \(\varDelta \eta ^{\mathrm {F}} \sim 2\) between diffractive and nondiffractive topologies. This is the region where CSCR does better, so a combination of CSCR with an \(\epsilon _{\mathrm {SaS}} > 0\) could provide a flatter MC/data distribution in Fig. 16a, b.
Figure 18 shows the CMS inelastic cross section obtained with two different approaches. One uses forward calorimetry (\(3< \eta  < 5\)), to measure protons with fractional momentum loss greater than \(\xi > 5\cdot 10^{6}\), corresponding to everything but lowmass diffractive systems (\(M_X>16\) GeV). The other uses the central tracker, requiring either one, two or three tracks. The SaS+\(\epsilon _{\mathrm {SaS}}\) and the modified ABMST models perform better than the others, with a maximum 5% deviation from CMS data. The SaS and the CSCR models has the same model for diffractive systems, and hence it is not expected that these differ in the measured inelastic cross section. With the SaS+\(\epsilon _{\mathrm {SaS}}\) model, however, some of the activity has been shifted to lower diffractive masses, resulting in a lower inelastic cross section. For ABMST, the reduction of the highmass systems in the modified model results in a reduction of the inelastic cross section relative to the original one.
Table 2 shows the integrated cross sections obtained with the ALICE and ATLAS 2011 analyses mentioned above. The ALICE results have been obtained for \(M_X<200\) GeV (\(\xi < 0.0008\)) for single diffraction, for gap sizes larger than \(\varDelta \eta >3\) for double diffraction, and with a van der Meer scan using diffractive events adjusted to data for the inelastic cross section. In the Rivet analysis, this corresponds to at least two tracks in the final state, i.e. effectively without any experimental cuts and hence returning the generatorlevel cross section. The SaS+\(\epsilon _{\mathrm {SaS}}\) model gives a better prediction for the single diffractive data, because of the increased lowmass cross section. The CSCR model predicts a larger double diffractive cross section, because of the larger probability for “accidental” gaps. The inelastic cross section, however, is the same for all three SaSbased models when compared with the ALICE data, as all have the same generatorlevel integrated cross section. In the ATLAS measurement of the inelastic cross section (for \(\xi >5\cdot 10^{6}\)) the SaS+\(\epsilon _{\mathrm {SaS}}\) model predicts a lower inelastic cross section, again because of the larger lowmass cross section.
The integrated cross section obtained with the three aforementioned Rivet analyses for the tuned models. For ALICE [55], the SD cross section is for \(M_X < 200\) GeV, the double diffractive for gaps larger than 3, the inelastic using a van der Meer scan using diffractive events adjusted to data. The ATLAS [53] inelastic cross section is for \(\xi > 5\cdot 10^{6}\)
\(\sigma _{\mathrm {SD}}\) (mb)  \(\sigma _{\mathrm {DD}}\) (mb)  \(\sigma _{\mathrm {inel}}\) (mb)  \(\sigma _{\mathrm {inel}}\) (mb)  

(ALICE)  (ALICE)  (ALICE)  (ATLAS)  
Data  14.9 ± 5.90  9.00 ± 2.60  73.20 ± 5.28  60.33 ± 2.10 
SaS  6.13 ± 0.01  5.72 ± 0.01  71.06 ± 0.02  66.48 ± 0.02 
SaS + CSCR  6.15 ± 0.01  6.19 ± 0.01  71.06 ± 0.02  66.43 ± 0.02 
SaS + \(\epsilon _{\mathrm {SaS}}\)  7.98 ± 0.01  5.62 ± 0.01  71.06 ± 0.02  63.69 ± 0.02 
ABMST  7.24 ± 0.01  4.69 ± 0.01  71.62 ± 0.02  67.44 ± 0.02 
ABMST mod  9.41 ± 0.01  5.09 ± 0.01  71.62 ± 0.02  63.72 ± 0.02 
In general, however, all models fail to describe the measured integrated cross sections, although some of the more sophisticated models do improve in some respects. Similarly, it seems that neither of the models describe well the transition from a nondiffractivedominated region to a diffractiondominated one. Including a colourreconnection model that allows for larger gaps in the nondiffractive events, like CSCR, is likely to improve the description in the midsizedgap range, if combined with a model that predicts a lower diffractive cross section there, like the ABMST models and SaS+\(\epsilon _{\mathrm {SaS}}\). The overall question of how to combine the descriptions of nondiffractive and diffractive topologies, however, will still exist even if the CR model “accidentally” (i.e. by “accidental” gaps) improves the description of data. All this highlights our still limited understanding of nonperturbative QCD, which forces us to work with models e.g. rooted in Regge theory. This may be good enough for an overall understanding, but still not for a precise reproduction of all relevant data.
8 Conclusions
In this paper we provide an updated description of the cross sections and hadronic event shapes in the event generator Pythia 8. The update has been required since the first results appeared from the LHC experiments, showing significant discrepancies between the models provided by Donnachie and Landshoff for the total cross section, as well as the elastic and diffractive cross sections by Schuler and Sjöstrand. By chance the DL undershooting of the total cross section and the SaS undershooting of the elastic cross section partly cancel in the inelastic cross section. Further to that, the SaS overshooting of the diffractive cross sections gave rise to a reasonable agreement between Pythia 8 and LHC measurements on the observable nondiffractive cross section, which is the relevant one for many of the measurements performed at the LHC. Thus, in spite of these shortfalls, the default Pythia 8 cross sections usually were good enough, notably when diffractive cross sections had been reduced somewhat (Eq. 9).
The discrepancies became largely evident with the precision measurements of the elastic and total cross sections performed by both TOTEM and ATLAS+ALFA. Here the exponential shape of the t spectrum in Pythia 8 is too simplistic, and other models have to be used for comparisons. Some of these models have now been implemented into Pythia 8, thereby providing a more sophisticated framework for elastic scattering and total cross sections.
For diffractive topologies the precision is less. The studies are marred by nondiffractive events mimicking diffractive ones, and vice versa, making the explicit distinction between the various diffractive and nondiffractive event topologies hard. The possibility of tagging the elastically scattered protons would greatly improve the separation of the samples, but so far no analyses on diffraction with tagged protons have appeared from CMS + TOTEM or ATLAS + ALFA. Thus we are left with measurements only using the central generalpurpose detectors. Unfortunately these do not give fully consistent answers. Notably the CMS and ATLAS rapiditygap measurements disagree in the diffractiondominated region, making it hard to compare models with data. Lacking any further guidance, we have here aimed for a middle ground between the two data sets.
The situation is even worse for of hadronic event shapes. Single diffractive data is available for very low energies, most of which goes into the ABMST model, but rather little for higher energies. This means that, even if integrated cross sections were provided for diffractive topologies from the LHC experiments, no constraints are put on the internal structure of diffractive systems. The ansatz of Pythia 8, that the diffractive system properties are similar to those of nondiffractive events, could be wrong. A future study of these event shapes, and of the different strategies underlying commonly used event generators, would help provide a guideline what would be interesting distributions to see measured at the LHC.
In conclusion, we provide an updated and extended framework for elastic and diffractive topologies, as well as an update for all parts of the total cross section. We rely on previous work provided by several other authors, but have corrected and extended the models where need be. Each of the models have been tuned to available data, thus providing an upgrade of the already present models in Pythia 8. We have discussed some of the consequences of different approaches for creating rapidity gaps, such as the CSCR model, and how this affects the predictions for LHC. Still, the lack of data or the discrepancies of present data, leaves us with imperfect descriptions and predictions, in particular for diffraction. The situation may be “good enough” for current needs, but will hopefully improve with new data in the future. At present we are not able to decide which model is “the better one” for diffraction, but in the case of total and elastic cross section the new models, COMPAS and ABMST, offer an improved description as compared to the SaS model. As the COMPAS model offers no description of diffraction, we propose to use the ABMST model for total and elastic cross section and the modified ABMST model for diffraction, with the tuned parameters as provided in this paper. We expect to change the default behaviour in the next Pythia release.
Foreseeable further work could include a lowmass description for central diffractive topologies, possibly modelling the resonances present there. Other work would be an extensive study of the diffractive event shapes as discussed above. A study on eikonalisation aspects, e.g. of events with both diffractive and nondiffractive Pomeron exchanges, could also provide more insight on both cross sections and event topologies. Finally, the diffractive framework could be extended also to other processes, such as \(\gamma {\mathrm p}\) and \(\gamma \gamma \) collisions.
Notes
Acknowledgements
We would like to thank Vladimir Ezhela from the COMPAS group for assistance with their model. Similarly we would like to thank Rob Appleby, James Molson and Sandy Donnachie for their huge effort in explaining various aspects of their model, as well as providing all of their data used in their fit. Work supported in part by the Swedish Research Council, contracts number 62120134287 and 201605996, and in part by Marie Curie Initial Training Networks, FP7 MCnetITN (Grant agreement PITNGA2012315877) and H2020 MCnetITN3 (Grant agreement 722104). This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement no. 668679).
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