In hadron-hadron interactions, the total hadronic cross section
, is calculated
using the parameterization of Donnachie and Landshoff [Don92].
In this approach, each cross section appears as the sum of one
pomeron term and one reggeon one
The total cross section is subdivided according to
At not too large squared momentum transfers , the elastic cross
section can be approximated by a simple exponential fall-off. If one
neglects the small real part of the cross section, the optical
theorem then gives
The diffractive cross sections are given by
The couplings are related to the pomeron term of the total cross section parameterization, eq. (). Picking a reference scale GeV, the couplings are given by . The triple-pomeron coupling is determined from single-diffractive data to be mb; within the context of the formulae in this section.
The spectrum of diffractive masses is taken to begin 0.28 GeV above the mass of the respective incoming particle and extend to the kinematical limit. The simple form is modified by the mass-dependence in the diffractive slopes and in the and factors (see below).
The slope parameters are assumed to be
The kinematical range in depends on all the masses of the
problem. In terms of the scaled variables
scatters elastically), and the combinations
The Regge formulae above for single- and double-diffractive events
are supposed to hold in certain asymptotic regions of the total phase
space. Of course, there will be diffraction also outside these
restrictive regions. Lacking a theory which predicts differential cross
sections at arbitrary and values, the Regge formulae are used
everywhere, but fudge factors are introduced in order to obtain
`sensible' behaviour in the full phase space. These factors are:
The diffractive cross-section formulae above have been integrated for a set of c.m. energies, starting at 10 GeV, and the results have been parameterized. The form of these parameterizations is given in ref. [Sch94], with explicit numbers for the case. PYTHIA also contains similar parameterizations for (assumed to be same as and ), , , ( etc.), , , , and .
The processes above do not obey the ordinary event mixing strategy. First of all, since their total cross sections are known, it is possible to pick the appropriate process from the start, and then remain with that choice. In other words, if the selection of kinematical variables fails, one would not go back and pick a new process, the way it was done in section . Second, it is not possible to impose any cuts or restrain allowed incoming or outgoing flavours; especially for minimum-bias events the production at different transverse momenta is interrelated by the underlying formalism. Third, it is not recommended to mix generation of these processes with that of any of the other ones: normally the other processes have so small cross sections that they would almost never be generated anyway. (We here exclude the cases of `underlying events' and `pile-up events', where mixing is provided for, and even is a central part of the formalism, see sections and .)
Once the cross-section parameterizations has been used to pick one of the processes, the variables and are selected according to the formulae given above.
A formed by in elastic or diffractive scattering is polarized, and therefore its decay angular distribution in is taken to be proportional to , where the reference axis is given by the direction of motion.
A light diffractive system, with a mass less than 1 GeV above the mass of the incoming particle, is allowed to decay isotropically into a two-body state. Single-resonance diffractive states, such as a , are therefore not explicitly generated, but are assumed described in an average, smeared-out sense.
A more massive diffractive system is subsequently treated as a string with the quantum numbers of the original hadron. Since the exact nature of the pomeron exchanged between the hadrons is unknown, two alternatives are included. In the first, the pomeron is assumed to couple to (valence) quarks, so that the string is stretched directly between the struck quark and the remnant diquark (antiquark) of the diffractive state. In the second, the interaction is rather with a gluon, giving rise to a `hairpin' configuration in which the string is stretched from a quark to a gluon and then back to a diquark (antiquark). Both of these scenarios could be present in the data; the default choice is to mix them in equal proportions.
There is experimental support for more complicated scenarios [Ing85], wherein the pomeron has a partonic substructure, which e.g. can lead to high- jet production in the diffractive system. The full machinery, wherein a pomeron spectrum is convoluted with a pomeron-proton hard interaction, is not available in PYTHIA. (But is found in the POMPYT program [Bru96].)