The high energy model is valid for incident particle energies from 10-20 GeV up to 10-20 TeV. Individual implementations of the model exist for , , , , and mesons, and for , , , , , , , and baryons and their anti-particles.
From this initial set of particles, two are chosen at random to be replaced with either a kaon-anti-kaon pair, a nucleon-anti-nucleon pair, or a kaon and a hyperon. The relative probabilities of these options are chosen according to a logarithmically interpolated table of strange-pair and nucleon-anti-nucleon pair cross sections. The particle types of the pair are chosen according to averaged, parameterized cross sections typical at energies of a few GeV. If the increased mass of the new pair causes the total available energy to be exceeded, particles are removed from the initial set as necessary.
As particles from the initial collision cascade through the nucleus more
particles will be generated. The number and type of these particles are
parameterized in terms of the CM energy of the initial particle-nucleon
collision. The number of particles produced from the cascade is given
Some of these nucleons are added to the intra-nuclear cascade. The rest, especially at higher momenta where nucleon production is suppressed, are replaced by pions, kaons and hyperons. The mean of the total number of hadrons generated in the cascade is partitioned into the mean number of nucleons, , pions, and strange particles, . Each of these is used as the mean of a Poisson distribution which produces the randomized number of each type of particle.
The momenta of these particles are generated by first dividing the final state phase space into forward and backward hemispheres, where forward is in the direction of the original projectile. Each particle is assigned to one hemisphere or the other according to the particle type and origin:
It is assumed that energy is separately conserved for each hemisphere. If
too many particles have been added to a given hemisphere, randomly chosen
particles are deleted until the energy budget is met. The final state
momenta are then generated according to two different algorithms, a cluster
model for the backward nucleons from the intra-nuclear cascade, and a
fragmentation model for all other particles. Several corrections are then
applied to the final state particles, including momentum re-scaling, effects
due to Fermi motion, and binding energy subtraction. Finally the
de-excitation of the residual nucleus is treated by adding lower energy
protons, neutrons and light ions to the final state particle list.
Fragmentation Model. This model simulates the fragmentation of the highly excited hadrons formed in the initial projectile-nucleon collision. Particle momenta are generated by first sampling the average transverse momentum from an exponential distribution:
The values of and depend on particle type and result from a parameterization of experimental data. The value selected for is then used to set the scale for the determination of , the fraction of the projectile's momentum carried by the fragment. The sampling of assumes that the invariant cross section for the production of fragments can be given by
-sampling is performed for each fragment in the final-state candidate list. Once a fragment's momentum is assigned, its total energy is checked to see if it exceeds the energy budget in its hemisphere. If so, the momentum of the particle is reduced by 10%, as is and the integral of the -sampling function, and the momentum selection process is repeated. If the offending particle starts out in the forward hemisphere, it is moved to the backward hemisphere, provided the budget for the backward hemisphere is not exceeded. If, after six iterations, the particle still does not fit, it is removed from the candidate list and the kinetic energies of the particles selected up to this point are reduced by 5%. The entire procedure is repeated up to three times for each fragment.
The incident and target particles, or their substitutes in the case of charge-
or strangeness-exchange, are guaranteed to be part of the final state.
They are the last particles to be selected and the remaining energy in their
respective hemispheres is used to set the components of their momenta.
The components selected by -sampling are retained.
Cluster Model. This model groups the nucleons produced in the
intra-nuclear cascade together with the target nucleon or hyperon, and
treats them as a cluster moving forward in the center of mass frame. The
cluster disintegrates in such a way that each of its nucleons is given a
Momentum Re-scaling. Up to this point, all final state momenta have been generated in the center of mass of the incident projectile and the target nucleon. However, the interaction involves more than one nucleon as evidenced by the intra-nuclear cascade. A more correct center of mass should then be defined by the incident projectile and all of the baryons generated by the cascade, and the final state momenta already calculated must be re-scaled to reflect the new center of mass.
This is accomplished by correcting the momentum of each particle in the
final state candidate list by the factor . is the total
kinetic energy in the lab frame of all the final state candidates generated
assuming a projectile-nucleon center of mass. is the total kinetic
energy in the lab frame of the same final state candidates, but whose momenta
have been calculated by the phase space decay of an imaginary particle.
This particle has the total CM energy of the original projectile and a
cluster consisting of all the baryons generated from the intra-nuclear
Corrections. Part of the Fermi motion of the target nucleons is taken into account by smearing the transverse momentum components of the final state particles. The Fermi momentum is first sampled from an average distribution and a random direction for its transverse component is chosen. This component, which is proportional to the number of baryons produced in the cascade, is then included in the final state momenta.
Each final state particle must escape the nucleus, and in the process reduce
its kinetic energy by the nuclear binding energy. The binding energy is
parameterized as a function of :
Another correction reduces the kinetic energy of final state s when
the incident particle is a or . This reduction increases
as the log of the incident pion energy, and is done to reproduce
experimental data. In order to conserve energy on average, the energy
removed from the s is re-distributed among the final state s,
s and s.
Nuclear De-excitation. After the generation of initial interaction
and cascade particles, the target nucleus is left in an excited state.
De-excitation is accomplished by evaporating protons, neutrons, deuterons,
tritons and alphas from the nucleus according to a parameterized model.
The total kinetic energy given to these particles is a function of the
incident particle kinetic energy:
The number of proton and neutrons emitted, , is sampled from a Poisson distribution about a mean which depends on and the number of baryons produced in the intranuclear cascade. The average kinetic energy per emitted particle is then . is used to parameterize an exponential which qualitatively describes the nuclear level density as a function of energy. The simulated kinetic energy of each evaporated proton or neutron is sampled from this exponential. Next, the nuclear binding energy is subtracted and the final momentum is calculated assuming an isotropic angular distribution. The number of protons and neutrons emitted is and , respectively.
A similar procedure is followed for the deuterons, tritons and
alphas. The number of each species emitted is ,
and , respectively.
Tuning of the High Energy Cascade The final stage of the high energy cascade method involves adjusting the momenta of the produced particles so that they agree better with data. Currently, five such adjustments are performed, the first three of which apply only to charged particles incident upon light and medium nuclei at incident energies above 65 GeV.
As in the high energy cascade model, the high energy cluster model
randomly assigns particles from the initial collision to either a
forward- or backward-going cluster. Instead of performing the
fragmentation process, however, the two clusters are treated
kinematically as the two-body final state of the hadron-nucleon
collision. Each cluster is assigned a kinetic energy which is
sampled from a distribution
The particles produced in the intra-nuclear cascade are grouped into a third cluster. They are treated almost exactly as in the high energy cascade model, where Eq. 20.2 is used to estimate the number of particles produced. The main difference is that the cluster model does not generate strange particles from the cascade. Nucleon suppression is also slightly stronger, leading to relatively higher pion production at large incident momenta. Kinetic energy and direction are assigned to the cluster as described in the cluster model paragraph in the previous section.
The remaining steps to produce the final state particle list are the same as those in high energy cascading:
A second difference is in the treatment of the cluster consisting of particles generated in the cascade. Instead of parameterizing the kinetic energies and angles of the outgoing particles, the phase space decay approach is used.
Another difference is that the high energy tuning of the final state distribution is not performed.
If the incident energy is too small to excite the nucleus, elastic scattering is performed. The angular distribution of the scattered particle is sampled from the sum of two exponentials whose parameters depend on .