Diffractive cross sections and event final states at the LHC ^{1}^{1}1 Presented at Forward Physics at LHC Workshop (May 2729 2010), Elba Island, Italy.
Abstract
We discuss a phenomenological model that describes results on diffractive and cross sections and event final states up to the Fermilab Tevatron energy of TeV and use it to make predictions for Large Hadron Collider (LHC) energies up to TeV and asymptotically as . The model is anchored in a saturation effect observed in single diffraction dissociation that explains quantitatively the factorization breaking observed in soft and hard and diffractive processes and in diffractive photoproduction and low deep inelastic scattering.
pacs:
14.70.Fm, 14.70.Hp, 12.40.Nn, 11.55.JyI Introduction
As we entered a new energy frontier at the Large Hadron Collider (LHC) with data collected at GeV, 2360 GeV, and 7 TeV from Fall 2009 to Spring 2010, it became painfully clear that the Monte Carlo (MC) simulations designed to represent the collective knowledge of the field on diffractive cross sections and event final states did not meet the challenge presented to them in this new higher energy environment. The most commonly used event generators, pythia PYTHIA () and phojet PHOJET (), were found to disagree not only with the data but also with each other. The latter clearly meant that the two simulations could not both be right. Therefore, an update of the MCs was urgently needed. Because of the importance of MinimumBias (MB) MC simulations in estimating trigger rates, backgrounds, and the machine luminosity at the LHC, a “diffraction” workshop was organized at CERN on 7 May 2010 Dday () that brought experimentalists and theorists together to exchange ideas with the goal of producing a reliable MC generator for the LHC. This paper is based on a talk I presented at that meeting and an expanded version presented at this workshop.
Diffraction dissociation in / interactions may be defined by the signature of one or more “large” and characteristically not exponentially suppressed Bj () rapidity gaps (regions of rapidity devoid of particles) rapidity () in the final state. The rapidity gap is presumed to be due to the exchange of a stronglyinteracting color singlet quark/gluon combination with the quantum numbers of the vacuum, traditionally referred to as “Pomeron” (). Diffractive processes are classified as single diffraction (SD), double Pomeron exchange (DPE), also referred to as central dissociation (CD), and double diffraction (DD). In SD (SD), the dissociates while the remains intact escaping the collision with momentum close to that of the original beam momentum and separated from the () dissociation products by a forward gap; in DPE both the and the escape, resulting in two forward gaps; and in DD a central gap is formed while both the and dissociate. The above basic diffractive processes are listed below, along with two additional two 2gap processes which are combinations of SD and DD and are indicated as SDD:
acronym  basic diffractive processes 

SD  , 
SD  
DD  
DPE  , 
2gap combinations of SD and DD  
SDD  , 
SDD  . 
Here, , and represent clusters of particles in rapidity regions not occupied by the gap(s). The 2gap processes are examples of multigap diffraction, a term coined by this author to represent events with multiple diffractive rapidity gaps. A special case of DPE is exclusive production, where a particle state is centrally produced, as for example a dijet system or a boson.
Ii Strategy
A phenomenology that is used to make predictions for the LHC and beyond should be based on cross sections and final states that incorporate the current knowledge in the field molded into a form that can be extrapolated to higher energies. The issues to be addressed is how to take into account saturation effects that suppress cross sections, and what formulas to use for event final state multiplicity, pseudorapidity, and transverse energy () missingET () distributions. In addition, the structure of an algorithm for implementing this knowledge into a MC simulation should also be addressed. The algorithm must be robust against changes in the collision energy, so that it may be equally well applied to simulate collisions at fixed target energies as well as at the higher energies of and colliders and in astrophysics. In this section, we outline a strategy that addresses these issues.
The following input is used for cross sections and final states:

from superball model superball ();

optical theorem (imaginary part of the forward scattering amplitude);

dispersion relations , using low energy cross sections from global fit global ();

final states: use “nesting” to describe gap processes, where a nest is defined as a region of where there is particle production, in contrast to a gap region where there are no particles corfu01 (); multigap ().
Figure 2 shows a schematic topology of an event with four rapidity gaps and three nests of final state particles. The cross section for this configuration is presented in Sec. III.1
We propose the following algorithm for generating final states:

start with a inelastic collision at ;

decide whether the collision is ND or diffractive based on the expected cross sections; if ND, use the ND final state expected at ; if diffractive, select SD, SD, DD, or DPE based on probabilities scaled to the corresponding cross sections;

for each diffractive event, check whether the region of where particles are produced, , is large enough to accommodate additional diffractive rapidity gaps: if yes, decide whether or not the event will have other gaps within this region, again using probabilities scaled to the cross sections, and branch off accordingly;

continue this process until the region is too small to accommodate another diffractive gap.
It is important to note that in our definition of a ND collision there are no diffractive gaps whatsoever in the final state of the event. In this respect, this definition differs from those of “inclusive” or “nonsinglediffractive” definitions of ND events used in the literature.
Iii Cross sections and final states
In this section, we discuss briefly the diffraction dissociation and total cross sections using information and/or excerpts from Refs. R (); lathuile04 ().
iii.1 Diffractive cross sections
In Ref. pomQCD (), the following expression is obtained for the SD cross section [quoting]:
(1) where:
(i) the factor in square brackets represents the cross section due to the wee partons in the region of particle production ;
(ii)  is the rapidity gap;
(iii) is a QCD color factor selecting colorsinglet or exchanges to form the rapidity gap;
(iv) is a gap probability factor representing the elastic scattering between the dissociated proton (cluster of dissociation particles) and the surviving proton;
(v) is the integral of the gap probability distribution over all phasespace in and ;
(vi) in is the proton form factor … ; and
(vii) is a normalization constant, whose value is rendered irrelevant by the renormalization division by . …
By a change of variables from to using and , Eq. (1) takes the form:
(2) where [ is the slope of the diffractive distribution]. Integrating this expression over and yields the total single diffractive cross section,
(3)
The remarkable property that the total single diffractive cross section becomes constant as is a direct consequence of the coherence condition required for the recoil proton to escape the interaction intact. This condition selects one out of several available wee partons to provide a colorshield to the exchange and enable the formation of a diffractive rapidity gap.
Details are presented in Ref. pomQCD (), where this formulation of the cross section is used to derive the ratio of the intercept to the slope of the Pomeron trajectory. Good agreement with the ratio extracted from measurements is obtained, providing support for the renormalization approach used in the phenomenology.
A similar expression may be use for DD, DPE, and multigap processes, as discussed in Refs. corfu01 (); multigap (). For example, the differential cross section for the process displayed in Fif. 2 is derived in Ref. multigap () as [quoting]:
(4) 
where the term in square brackets is the total cross section at the reduced value, defined through , (one for each gap) is the QCD color factor for gap formation, the gap probability is the amplitude squared for elastic scattering between two diffractive clusters or between a diffractive cluster and a surviving proton with form factor , and is the (re)normalization factor defined as the gap probability integrated over all 10 independent variables , , , and .
The renormalization factor is a function of only. The color factors are and for gluon and quark colorsinglet exchange, respectively. Since the reduced energy cross section is properly normalized, the gap probability is (re)normalized to unity. The quark to gluon fraction, and thereby the Pomeron intercept parameter may be obtained from the inclusive parton distribution functions (PDFs) lathuile04 (). Thus, normalized differential multigap cross sections at may be fully derived from inclusive PDFs and QCD color factors without any free parameters.
The exponential dependence of the cross section on leads to a renormalization factor independent of the number of gaps in the process. This remarkable property of the renormalization model, which was confirmed in twogap to onegap cross section ratios measured by the CDF Collaboration (see Ref. lathuile04 ()), suggests that multigap diffraction can be used as a tool for exploring the QCD aspects of diffraction in an environment free of rapidity gap suppression effects. The LHC with its large rapidity coverage provides the ideal arena for such studies.
iii.2 The total cross section
In Ref. superball (), an analytic expression is obtained for the total cross section using a parton model approach and exploiting a saturation effect observed in the SD cross section. The abstract of Ref. superball () reads [quoting]:
The singlediffractive and total cross sections at the LHC are predicted in a phenomenological approach that obeys all unitarity constraints. The approach is based on the renormalization model of diffraction and a saturated Froissart bound for the total cross section yielding for , where the parameters and are experimentally determined from the dependence of the singlediffractive cross section.
The following strategy is used in Ref. superball () [quoting]:
Use the Froissart formula as a saturated cross section rather than as a bound above :
This formula should be valid above the knee in vs. at GeV (Fig. 3) and therefore valid at GeV.
Use in the Froissart formula multiplied by 1/0.389 to convert it to mb.
Note that contributions from Reggeon exchanges at GeV are negligible, as can be verified from the global fit of Ref. global ().
Obtain the total cross section at the LHC:
For a numerical evaluation of we use as input the CDF cross section at GeV, mb, the Froissart saturation energy GeV, and the parameter .
…The resulting prediction for the total cross section at the LHC at is:
For TeV, the predicted cross section is:
The result for TeV is in good agreement with obtained by the global fit of Ref. global (), where the uncertainty was estimated from and the dependence from which the value of the parameter was obtained.
Iv Conclusions
We briefly discuss a phenomenological model that describes available results on diffractive and cross sections and event final states up to the Fermilab Tevatron energy of TeV and refer the reader to previous publications for further details. We also outline a procedure to be used to implement the predictions of the model into a Monte Carlo simulation that is robust against changes in the collision energy, so that it may be equally well applied to simulate collisions at fixed target energies as well as at the higher energies of the Tevatron, the LHC, and beyond. The model is anchored in a saturation effect observed in single diffraction dissociation that explains quantitatively the factorization breaking observed in soft and hard and diffractive processes and in diffractive photoproduction and low deep inelastic scattering.
V Acknowledgments
I would like to thank my colleagues at The Rockefeller University and my collaborators at the Collider Detector at Fermilab for providing the interactive environment in which this work was made possible, and the organizers of the Workshop for putting together a comprehensive program of presentations at the threshold in time of the opening of a new era in particle physics at the Large Hadron Collider.
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