Measurements at the LHC have shown that there are sizable disagreements among Monte Carlo (MC) implementations of “soft” processes based on cross sections proposed by various physics models, and that it is not possible to reliably predict all such processes, or even all aspects of a given process, using a single model [1-3]. In the CDF studies of diffraction at the Tevatron, all processes are well modeled by the MBR (Minimum Bias Rockefeller) MC simulation, which is a stand-alone simulation based on a unitarized Regge-theory model, RENORM [4], employing inclusive nucleon parton distribution functions (PDF’s) and QCD color factors. The RENORM model was updated in a presentation at EDS-2009 [5] to include a unique unitarization prescription for predicting the total  cross section at high energies, and that update has been included as an MBR option for simulating diffractive processes in PYTHIA8 since version PYTHIA8.165 [6], to be referred here-forth as PYTHIA8-MBR. In this paper, we briefly review the cross sections [7] implemented in this option of PYTHIA8 and compare them with LHC measurements.

The PYTHIA8-MBR option includes a full simulation of the hadronization of the implemented diffraction dissociation processes: single, double, and central diffraction. In the MBR simulation used at CDF, the hadronization of the final state(s) was based on a data-driven phenomenological model of multiplicities and  (transverse momentum) distributions calibrated using SS and Fermilab fixed-target results. Later, the model was successfully tested against Tevatron minimum bias (MB) and diffraction data. However, only  and  particles were produced in the final state, with multiplicities obeying a statistical model of a modified Gamma distribution function that provided good fits to experimental data [8]. This model could not be used to predict specific-particle final states. In the PYTHIA8-MBR implementation, hadronization is perfor-med by PYTHIA8 tuned to reproduce final-state distribut-ions in agreement with MBR's, with hadronization done in the PYTHIA8 framework. Thus, all final-state particles are now automatically produced, greatly enhancing the horizon of applicability of PYTHIA8-MBR.

The following diffraction dissociation processes are considered in PYTHIA8-MBR:

SD                       (1)

or             

DD                  (2)

CD/DPE              (3)

The RENORM predictions are expressed as unitarized Regge-theory formulas, in which the unitarization is achi-eved by a renormalization scheme where the Pomeron (IP) flux is interpreted as the probability for forming a diffractive (non-exponentially suppressed) rapidity gap and thereby its integral over all phase space saturates at the energy where it reaches unity. Differential cross sections are expressed in terms of the  IP-trajectory, , the IP-p coupling, , and the ratio of the triple-IP to the IP- couplings, . For large rapidity gaps, , for which IP -exchange dominates, the cross sections may be written as,

         (4)

   (5)

  (6)

where  is the 4-momentum-transfer squared at the proton vertex,  the rapidity-gap width, and  the center of the rapidity gap. In Eq. (6), the subscript  enumerates Pomerons in a DPE event,  is the total rapidity gap (sum of two gaps) in the event, and  is the center in  of the centrally-produced hadronic system.

The total cross section () is expressed as:

  (7)

  (8)

where  and  are energy and the Pomeron flux saturation scales, respectively [7]. For  TeV, where there are Reggeon contributions, we use the global fit expression [9], while for  TeV, where Reggeon contributions are negligible, we employ the Froissart-Martin formula [10-12]. The two expressions are smoothly matched at  TeV.

The elastic cross section is obtained from the global fit [9] for  TeV, while for  TeV we use an extrapolation of the global-fit ratio of , which is slowly varying with  multiplied by . The total non-diffractive cross section is then calculated as .

In this section, we present as examples of the predictive power of the RENORM model some results reported by the TOTEM, CMS, and ALICE collaborations for pp collisions at  TeV, which can be directly compared with RENORM formulas without using the PYTHIA8-MBR simulation.

Fig. (1, left) shows a comparison of the TOTEM total, elastic, and total-inelastic cross sections, along with results from other experiments fitted by the COMPETE Collaboration [13]; the RENORM predictions, displayed as filled (green) squares, are in excellent agreement with the TOTEM results. Similarly, in Fig. (1, right), good agreement is observed between the ALICE [14] and CMS [15] total-inelastic cross sections and the RENORM prediction.

Fig. (1). (left) TOTEM measurements of the total, total-inelastic, and elastic pp cross sections at s = 7 TeV shown along with best COMPETE fits [13], with RENORM predictions added as filled squares; (right) ALICE [14] and CMS [15] measurements of the total inelastic cross section at s = 7 TeV show good agreement with the RENORM prediction (PYTHIA8-MBR).

Fig. (2). Measured SD (left) and DD (right) cross sections for ξ < 0.05 compared with theoretical predictions; the model embedded in PYTHIA8-MBR provides a good description of all data.

The uncertainty shown in the RENORM prediction of  in Fig. (1, left) is dominated by that in the scale parameter . The latter can be reduced by a factor of ~4 if  is interpreted as the mean value of the glue-ball-like object discussed in [16] and the data shown in figure 8 of [16] are used to determine its value. Work is in progress to finalize the details of this interpretation.

Another example of the predictive power of RENORM is shown in Fig. (2), which displays the total SD (left) and total DD (right) cross sections for , after extrapolation into the low mass region from the measured CMS cross sections at higher mass regions, presented in [17], using RENORM.

We reviewed our pre- LHC predictions for the total, elastic, total-inelastic, and diffractive components of the proton-proton cross section at high energies, which are based on a special parton-model approach to diffraction employing inclusive proton parton distribution functions and QCD color factors. We discuss single diffraction/dissociation, double diffraction/dissociation, and central diffraction or double-Pomeron exchange, comparing predictions with LHC measurements. Agreement between data and PYTHIA8-MBR predictions is found in all cases.