High Energy Model

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 $\pi^+$, $\pi^-$, $K^+$, $K^-$, $K^0_S$ and $K^0_L$ mesons, and for $p$, $n$, $\Lambda$, $\Sigma^+$, $\Sigma^-$, $\Xi^0$, $\Xi^-$, and $\Omega^-$ baryons and their anti-particles.

Initial Interaction

In a given implementation, the generation of the final state begins with the selection of a nucleon from the target nucleus. The pion multiplicities resulting from the initial interaction of the incident particle and the target nucleon are then calculated. The total pion multiplicity is taken to be a function of the log of the available energy in the center of mass of the incident particle and target nucleon, and the $\pi^+$, $\pi^-$ and $\pi^0$ multiplicities are given by the KNO distribution.

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.

Intra-nuclear Cascade

The cascade of these particles through the nucleus, and the additional particles generated by the cascade are simulated by several models. These include high energy cascading, high energy cluster production, medium energy cascading and medium energy cluster production. For each event, high energy cascading is attempted first. If the available energy is sufficient, this method will likely succeed in producing the final state and the interaction will have been completely simulated. If it fails due to lack of energy or other reasons, the remaining models are called in succession until the final state is produced. If none of these methods succeeds, quasi-elastic scattering is attempted and finally, as a last resort, elastic scattering is performed. These models are responsible for assigning final state momenta to all generated particles, and for checking that, on average, energy and momentum are conserved.

High Energy Cascading

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 roughly by

$$N_m = C(s) [A^{1/3} - 1] N_{ic}$$ (20.2)

where $A$ is the atomic mass, $C(s)$ is a function only of $s$, the square of the center of mass energy, and $N_{ic}$ is approximately the number of hadrons generated in the initial collision. This can be understood qualitatively by assuming that the collision occurs, on average, at the center of the nucleus. Then each of the $N_{ic}$ hadrons must traverse a distance roughly equal to the nuclear radius. They may therefore potentially interact with a number of nucleons proportional to $A^{1/3}$. If the energy-dependent cross section for interaction in the nuclear medium is included in $C(s)$ then Eq. 20.2 can be interpreted as the number of target nucleons excited by the initial collision and its secondaries.

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, $N_n$, pions, $N_\pi$ and strange particles, $N_s$. 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:

• the original projectile, or its substitute if charge or strangeness exchange occurs, is assigned to the forward hemisphere and the target nucleon is assigned to the backward hemisphere;

• the remainder of the particles from the initial collision are assigned at random to either hemisphere;

• pions and strange particles generated in the intra-nuclear cascade are assigned 80% to the backward hemisphere and 20% to the forward hemisphere;

• nucleons generated in the intra-nuclear cascade are all assigned to the backward hemisphere.

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 $p_T$ from an exponential distribution:

$$exp [-a {p_T}^b ]$$ (20.3)

where

$$1.70 \le a \le 4.00 ; \ 1.18 \le b \le 1.67 .$$ (20.4)

The values of $a$ and $b$ depend on particle type and result from a parameterization of experimental data. The value selected for $p_T$ is then used to set the scale for the determination of $x$, the fraction of the projectile's momentum carried by the fragment. The sampling of $x$ assumes that the invariant cross section for the production of fragments can be given by

$$E \frac{d^3 \sigma}{dp^3} = \frac{K}{(M^2 x^2 + {p_T}^2)^{3/2}}$$ (20.5)

where $E$ and $p$ are the energy and momentum, respectively, of the produced fragment, and $K$ is a proportionality constant. $M$ is the average transverse mass which is parameterized from data and varies from 0.75 GeV to 0.10 GeV, depending on particle type. Taking $m$ to be the mass of the fragment and noting that

$$p_z \simeq x E_{proj}$$ (20.6)

in the forward hemisphere and

$$p_z \simeq x E_{targ}$$ (20.7)

in the backward hemisphere, Eq. 20.5 can be re-written to give the sampling function for $x$:

$$\frac{d^3 \sigma}{dp^3} = \frac{K}{(M^2 x^2 + {p_T}^2)^{3/2}} \frac{1}{ \sqrt{m^2 + {p_T}^2 + x^2 E_i^2} } ,$$ (20.8)

where $i = proj$ or $targ$.

$x$-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 $p_T$ and the integral of the $x$-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 $p_z$ components of their momenta. The $p_T$ components selected by $x$-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 kinetic energy

$$40 < T_{nuc} < 600 \rm {MeV}$$ (20.9)

if the kinetic energy of the original projectile, $T_{inc}$, is 5 GeV or more. If $T_{inc}$ is less than 5 GeV,

$$40 ( T_{inc}/5 \rm {GeV} )^2 < T_{nuc} < 600 ( T_{inc}/5 \rm {GeV} )^2 .$$ (20.10)

In each range the energy is sampled from a distribution which is skewed strongly toward the high energy limit. In addition, the angular distribution of the nucleons is skewed forward in order to simulate the forward motion of the cluster.

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 $T_1 / T_2$. $T_2$ is the total kinetic energy in the lab frame of all the final state candidates generated assuming a projectile-nucleon center of mass. $T_1$ 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 cascade.

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 $A$:

$$E_B = 25\rm {MeV} \left( \frac{A-1}{120} \right) e^{-(A-1)/120)} .$$ (20.11)

Another correction reduces the kinetic energy of final state $\pi^0$s when the incident particle is a $\pi^+$ or $\pi^-$. 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 $\pi^0$s is re-distributed among the final state $\pi^+$s, $\pi^-$s and $\pi^0$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:

$$T_{evap} = 7.716 \rm {GeV} \left( \frac{A-1}{120} \right) F(T) e^{-F(T) - (A-1)/120} ,$$ (20.12)

where

$$F(T) = \rm {max} [ 0.35 + 0.1304 ln(T) , 0.15 ] ,$$ (20.13)

and

$$T = 0.1 \rm {GeV} \quad \rm {for} \quad T_{inc} < 0.1 \rm {GeV}$$ (20.14)

$$T = T_{inc} \quad \rm {for} \quad 0.1 \rm {GeV} \le T_{inc} \le 4 \rm {GeV}$$ (20.15)

$$T = 4 \rm {GeV} \quad \rm {for} \quad T_{inc} > 4 \rm {GeV} .$$ (20.16)

The mean energy allocated for proton and neutron emission is $\overline{T_{pn}}$ and that for deuteron, triton and alpha emission is $\overline{T_{dta}}$. These are determined by partitioning $T_{evap}$ :

$$\overline{T_{pn}} = T_{evap} R \quad , \quad \overline{T_{dta}} = T_{evap} (1-R) \quad \rm {with}$$

$$R = \rm {max}[ 1 - (T/4\rm {GeV})^2 , 0.5 ] .$$ (20.17)

The simulated values of $T_{pn}$ and $T_{dta}$ are sampled from normal distributions about $\overline{T_{pn}}$ and $\overline{T_{dta}}$ and their sum is constrained not to exceed the incident particle's kinetic energy, $T_{inc}$.

The number of proton and neutrons emitted, $N_{pn}$, is sampled from a Poisson distribution about a mean which depends on $R$ and the number of baryons produced in the intranuclear cascade. The average kinetic energy per emitted particle is then $T_{av} = T_{pn}/N_{pn}$. $T_{av}$ 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 $(Z/A)N_{pn}$ and $(N/A)N_{pn}$, respectively.

A similar procedure is followed for the deuterons, tritons and alphas. The number of each species emitted is $0.6 N_{dta}$, $0.3 N_{dta}$ and $0.1 N_{dta}$, 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 $\simeq$ 65 GeV.

• If the final state particle is a nucleon or light ion with a momentum of less than 1.5 GeV/c, its momentum will be set to zero some fraction of the time. This fraction increases with the logarithm of the kinetic energy of the incident particle and decreases with $log_{10}(A)$.

• If the final state particle with the largest momentum happens to be a $\pi^0$, its momentum is exchanged with either the $\pi^+$ or $\pi^-$ having the largest momentum, depending on whether the incident particle charge is positive or negative.

• If the number of baryons produced in the cascade is a significant fraction ($> 0.3$) of $A$, about 25% of the nucleons and light ions already produced will be removed from the final particle list, provided their momenta are each less than 1.2 GeV/c.

• The final state of the interaction is of course heavily influenced by the quantum numbers of the incident particle, particularly in the forward direction. This influence is enforced by compiling, for each forward-going final state particle, the sum
  

  $$S_{forward} = \Delta_M + \Delta_Q + \Delta_S + \Delta_B,$$ (20.18)

  
  
  where each $\Delta$ corresponds to the absolute value of the difference of the quantum number between the incident particle and the final state particle. $M$, $Q$, $S$, and $B$ refer to mass, charge, strangeness and baryon number, respectively. For final state particles whose character is significantly different from the incident particle ($S$ is large), the momentum component parallel to the incident particle momentum is reduced. The transverse component is unchanged. As a result, large-$S$ particles are driven away from the axis of the hadronic shower. For backward-going particles, a similar procedure is followed based on the calculation of $S_{backward}$.

• Conservation of energy is imposed on the particles in the final state list in one of two ways, depending on whether or not a leading particle has been chosen from the list. If all the particles differ significantly from the incident particle in momentum, mass and other quantum numbers, no leading particle is chosen and the kinetic energy of each particle is scaled by the same correction factor. If a leading particle is chosen, its kinetic energy is altered to balance the total energy, while all the remaining particles are unaltered.

High Energy Cluster Production

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 $T$ which is sampled from a distribution

$$exp[-aT^{1/b}]$$ (20.19)

where both $a$ and $b$ decrease with the number of particles in a cluster. If the combined total energy of the two clusters is larger than the center of mass energy, the energy of each cluster is reduced accordingly. The center of mass momentum of each cluster can then be found by sampling the 4-momentum transfer squared, $t$, from the distribution

$$exp [t (4.0 + 1.6ln(p_{inc}) ) ]$$ (20.20)

where $t < 0$ and $p_{inc}$ is the incident particle momentum. Then,

$$cos\theta = 1 + \frac{t - (E_c - E_i)^2 + (p_c - p_i)^2}{2p_c p_i},$$ (20.21)

where the subscripts $c$ and $i$ refer to the cluster and incident particle, respectively. Once the momentum of each cluster is calculated, the cluster is decomposed into its constituents. The momenta of the constituents are determined using a phase space decay algorithm.

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:

• re-scaling of the momenta to reflect a center of mass which involves the cascade baryons,
• corrections due to Fermi motion and binding energy,
• reduction of final state $\pi^0$ energies,
• nuclear de-excitation and
• high energy tuning.

Medium Energy Cascading

The medium energy cascade algorithm is very similar to the high energy cascade algorithm, but it may be invoked for lower incident energies (down to 1 GeV). The primary difference between the two codes is the parameterization of the fragmentation process. The medium energy cascade samples larger transverse momenta for pions and smaller transverse momenta for kaons and baryons.

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.

Medium Energy Cluster Production

The medium energy cluster algorithm is nearly identical to the high energy cluster algorithm, but it may be invoked for incident energies down to 10 MeV. There are three significant differences at medium energy: less nucleon suppression, fewer particles generated in the intra-nuclear cascade, and no high energy tuning of the final state particle distributions.

Elastic and Quasi-elastic Scattering

When no additional particles are produced in the initial interaction, either elastic or quasi-elastic scattering is performed. If there is insufficient energy to induce an intra-nuclear cascade, but enough to excite the target nucleus, quasi-elastic scattering is performed. The final state is calculated using two-body scattering of the incident particle and the target nucleon, with the scattering angle in the center of mass sampled from an exponential:

$$exp [ - 2 b p_{in} p_{cm} (1 - cos\theta) ] .$$ (20.22)

Here $p_{in}$ is the incident particle momentum, $p_{cm}$ is the momentum in the center of mass, and $b$ is a logarithmic function of the incident momentum in the lab frame as parameterized from data. As in the cascade and cluster production models, the residual nucleus is then de-excited by evaporating nucleons and light ions.

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 $A$.