Models of primary interactions

SUPPLEMENTS Nuclear

Physics

B (Proc.

Suppl.)

122 (2003)

40-55

Models of primary interactions Ralph Engel”* “Bartol

Research Institute,

University

of Delaware, Newark, DE 19716, USA.

Physics assumptions and models implemented in modern Monte Carlo simulation codes for hadronic cosmic ray interactions are reviewed. Concepts of extrapolating to highest cosmic ray energies are discussed and shortcomings of the different model assumptions are emphasized.

1. Introduction Hadronic interaction models are the least understood part of cosmic ray (CR) air shower calculations. In contrast to the fairly well developed theory and calculational methods for electromagnetic and electro-weak interaction processes, hadronic particle production is still theoretically not understood. The most powerful tool of QCD calculations - perturbation theory - is only applicable to high-momentum transfer processes, a small subset of hadron production reactions. Therefore even modern interaction models simulate the bulk of hadron production on the basis of largely phenomenological models and ad hoc assumptions. Currently the most complete and sophisticated high-energy interaction models are l

DPMJET

II.5 [l] and III [2,3],

l

nexus 2.0 [4] and 3.0 [5],

l

QGSjet 98 [6] and 01 [7], and

l

SIBYLL

2. Description ing

1.7 [8,9] a,nd 2.1 [lo].

They typically cover the a lab. energy range from several 100 GeV to the highest energies observed. These models are commonly referred to as QCDinspired models since they all simulate partonic minijet production as one important element of interactions at high energy. Low- or mediumenergy models frequently used are GHEISHA *present address: fiir Kernphysik, many.

Forschungszentrum Postfach 3640,

0920-5632/03/S - see front matter doi: 10.1016/SO920-5632(03)01956-X

76021

Karlsruhe, Karlsruhe,

Q 2003 Elsevier

[II], the Hillas splitting algorithm in various implementations [12], FLUKA [13], UrQMD [14], TARGET 1.0 [15] and 2.1 [16], and HADRIN and NUCRIN [17-191. The subdivision in low- and high-energy interaction models is arbitrary. For example, DPMJET can be used to simulate particle production in hadron-hadron collisions even at energies as low as the particle production threshold. A more complete list of hadronic interaction models used in cosmic ray simulations can be found in [20]. In this article we will summarize the theoretical and phenomenological foundations of QCDinspired high-energy models, emphasizing shortcomings and open problems and limits of their predictive power. The basic physics of scattering amplitudes, some important theorems and notations used in the following are given in the Appendix.

Institut Ger-

Science

B.V.

of nucleon-nucleon

scatter-

Many of the ideas of modeling soft interactions have been developed in the framework of the Dual Parton Model (211 and the Quark-Gluon Strings Model [22,23]. The most basic building block of all models is a detailed description of nucleon-nucleon (and correspondingly pion- and kaon-nucleon) scattering processes. In a next step, using the framework of the nucleon-nucleon model, the Gribov-Glauber approach is applied to generalize particle production from single hadron-hadron collisions to nuclei (see Sec. 3). All rights

reserved.

R. Engel/Nuclear

Physics

B (Proc.

2.1. Hard interactions Nucleon-nucleon interactions are by themselves complicated processes. The perturbatively calculable part is best understood and a corresponding Feynman diagram is shown in Fig. 1. Within the

projectile +

I

jet pair

target nucleon

Figure 1. Two-gluon scattering quark-antiquark jet pair.

QCD-improved given by - ajet d2pl

=

producing

a

parton model the cross section is

K c -1/dxldzz ijkl 1+6kl I? 2 x.f,4(51, Q2)f+29

dei,j-k,l Q2> dzpL > (1)

where ci is the parton-parton scattering process and the sum runs over all possible initial and final state parton flavours. The momentum fractions of the incoming partons are denoted by xi and x2. The parton densities f(z, Q2) are universal (factorization theorem) and can be measured in deep inelastic scattering. The factor K accounts for neglected higher-order contributions and is expected to be approximately 2. The partons carry color and are therefore subthe potential energy of ject to confinement: the color field between partons increases linearly with their distance. The transition process

Suppl.)

122 (2003)

41

40-55

from asymptotically free partons to color-neutral hadrons is non-perturbative in its nature and is described in all event generators considered here by string fragmentation models such as the Lund model [24]. The QCD parton model has proved to very successful and of great predictive power, however it has also severe limitations. First of all a cutoff on the minimum momentum transfer or transverse momentum of the partons, called here pytoff, is needed to calculate the total minijet rate. The minijet rate depends sensitively on pl cutoff but the model does not specify what value should be used. Even after years of measurements and interpretations in terms of perturbative QCD its range of validity is still under debate. Second, Eq. (1) describes only the inclusive jet cross section. It does not make any statement on the total number of such parton-parton interactions taking place in a single hadron-hadron collision. For example, in central collisions (vanishing impact parameter) more than 10 minijet pairs could be produced per proton-proton interaction and in peripheral ones, maybe, less than one. Third, the parton densities are known only in a limited range of parton momentum fraction z and virtuality Q2. Extrapolations to smaller values of parton momentum fractions, 2, are one of the important sources of uncertainties. Last but not least, data from the HERA collider changed our view on low-x parton densities over the last years completely [25]. For a low-a: parton density behaviour of

one gets an integrated for example, [26]) Ujej,t

N SD.In S .

minijet

cross section (see,

(3)

Previous to the HERA measurements commonly A = 0 was assumed, implying a very moderate rise of the minijet cross section at high energy. The old version of SIBYLL and to some degree also QGSJET are models that are based on this assumption. The HERA measurements clearly favour A = 0.3.. .0.4. The large value of A

42

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Physics B (Proc. Suppl.) 122 (2003) 40-55

has far-reaching theoretical consequences. One of them is the expectation of non-linear effects at very high energy. The higher the collision energy is the smaller are the values of z that are are important. However, in a nucleon of given size only a geometrically limited maximum number of partons can exist as independent objects. If the parton density reaches this limit non-linear effects are expected to become important, leading to the fusion of neighbouring partons. These non-linear effects are currently subject of intense research [27] in high energy physics.

h,o ( E1.b1d’ ) 12

13

IMxxx)k

14

I5 16 17 I I total cmm section: DL tit EHLQ set 1. p,c!mm =“,“g;

100

loo0

Figure 2. Inclusive as predicted by the model. For comparison proton/antiproton cross the Donnachie-Landshoff

19 ,

20

-____--

mooo

%I

18 I

mcwo

(GeV)

minijet cross section QCD-improved parton also the total protonsection is shown, using (DL) parametrization

PI * The physics of Eq. (1) is implemented in all models. The only, but very important differences arise from assumptions on the minimum momentum transfer (PI cutoff) needed to ensure applicability of (1) and the treatment and extrapolation of parton densities. To illustrate the problem we show in Fig. 2 the inclusive minijet cross section calculated with different transverse momentum cutoffs and the GRV98 parton density parametrization [29]. An overall factor of about 4

is found between the cross sections for pytoff = 2 and 4 GeV, reflecting the strong dependence on the minimum momentum transfer. There are no theoretical arguments why a cutoff of pc;ltoff = 2 GeV should be preferred over any higher value. In fact, the higher the cutoff value is the more reliable should be the QCD parton model. On the other hand, it is desirable to chose a cutoff as small as possible since our understanding of hard processes is much better than that of soft ones. Perturbative QCD fits of structure function measurements at HERA give a perfect description of the data down to virtualities as low as 2 GeV2. This could indicate that the parton model can be used even at pl N 1.5GeV. However, the large flexibility of such fits makes it nearly impossible to use them as tool to test the range of applicability of perturbative &CD. Indeed, a comparisons with Tevatron measurements of inclusive charged hadron production at high pl shows that the parton model fails to describe the data if pytoff 6 2 GeV is used in the calculation of the minijet cross section [30]. This comparison is particularly interesting as the HERA data cover an important part of the range in z and Q2 that enters this calculation. For comparison a representative prediction of the minijet cross section using a parton density parametrization made before HERA data became available, EHLQ 1311, is also shown in Fig. 2. Closely connected to the question of the optimum value of the hard cutoff is that of a possible energy dependence. In general, the virtuality p2 of a hard scattering process has to be much larger than the QCD scale, p2 > A&,, N 0.28GeV. Here it is important with AQCD to note that the QCD evolution of the parton densities can be calculated in different apThe “stanproximation schema and orders. dard”, DGLAP phenomenology (colinear factorization) is based on summing up the terms with large logarithms os ln~2/h&n. In contrast, in the BFKL approximation (/cl factorization) the terms with era In l/a: are considered the most important. Because of In l/s 5 In l/z 5 In l/G one will always reach the situation a, In l/z >> CY,In p2/A&D at high energy, rendering extrapolations based on DGLAP evolution equations

R. EngellNuclear

Physics

B (Proc.

It would be natural to use the unreliable. BFKL evolution equations to overcome this problem as ultra-high energies are the major concern. However, the most important parameter, A, turns out to be very sensitive to many details of the calculation and, in contrast to the situation with DGLAP calculations, changes completely if higher order corrections are considered. The net effect are numerically unstable extrapolations. In light of this unclear theoretical situation it is understandable that not one of the models has the same minijet cross section implemented. In QGSjet the minimum virtuality characterizing a hard scattering process is 1.5 GeV*. The extrapolation to high energy with such a low, energy-independent cutoff is only possible because this model was developed before HERA data became available and hence implements a moderate rise of the parton density at low x. The DGLAP evolution of the parton densities is calculated explicitly in the code. DPMJET II and III are baaed on up-to-date parton density parametrizations from “expert” groups such as GRV and an ad hoc parametrization for the transverse momentum cutoff pytoff(&)

= pt + 0.12GeV

log,,, m

G

3 >

(4)

with pt = 2.5GeV [32]. Similarly, SIBYLL 2.1 uses the GRV98 parton densities and the cutoff parametrization cutoff (6) PL

= pt+O.O65GeVexp

{ 0.9&@}

.(5)

Suppl.)

122 (2003)

43

4C55

the measured structure function in deep inelastic scattering at HERA is attributed to resolved photon contributions and hence subtracted in the conversion from structure function data to parton densities. This helps to tame the growth of the minijet cross section somewhat, however, ultimately the problem of a steeply increasing minijet cross section remains. As a consequence the range of applicability is restricted to energies below 10i7eV. The version 3 of nexus, currently under development, implements so-called enhanced pomeron graphs that describe among others gluon-gluon fusion and will allow simulations at the highest cosmic ray energies. 2.2. Soft interactions As up to now QCD predictions on hadronhadron scattering at soft scales (small momentum transfers) cannot be calculated by first principles, all models use Regge phenomenology here. It is worthwhile to recall that Regge theory is based on a minimum of assumptions which are know to hold (see [35] for a brief introduction and [36] for a full account on this subject). The scattering amplitude is taken to be analytic up to singularities and discontinuities implied by physics. This principle of maximum analyticity is then combined with the method of crossing to relate amplitudes of particles to that of antiparticles. The simple Regge-type parametrization often used in literature is the result of further, not rigorously provable assumptions on convergence of integrals and vanishing integral contributions

The latter one is motivated by the extrapolation of the gluon density in double-logarithmic approximation [33,34] xg(x,

48 Q*) - exp 11 - $z/ ln 14s i$”

Ini

*

, (6)

I

where Qz N 1 GeV*. Using this theoretically predicted extrapolation the cutoff is set such that the total transverse area of gluons, each occupying a typical area of the size rv l/p:, does not exceed that of a proton. nexus 2, being in its philosophy of hard interactions similar to QGSjet, is baaed on an energyindependent cutoff of about 2 GeV. A part of

with se = 1 GeV. The index k represents the contributing Regge trajectory ok(t) and n is the corresponding signature factor rl(ak(t))

= -

1 + 7e-i.rrak(t) sin(~cuk(t))

.

(8)

A single term in the sum (7) describes the contribution from an infinite number of particles exchanged in the t-channel. Each of these groups of particles is represented in the amplitude by a quasi-particle, called reggeon, with the same

44

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Physics B (Proc. Suppl.) 122 (2003) 40-55

quantum numbers. All functions and constants in (7) are free parameters to be determined by data. This large number of unknown parameters is not unexpected as Eq. (7) is based on very general assumptions. Knowing the amplitude one can calculate the total and elastic cross sections using Eqs. (37,38). In all Monte Carlo models considered here the soft parameters are determined by a fit to elastic and total proton-proton/antiproton cross sections. First the hard contribution is calculated according to the chosen minijet scenario and then the soft parameters are adjusted to describe the elastic and total cross sections. In QGSjet only the highest reggeon trajectory, the pomeron, is implemented. In all other models one effective trajectory with am PZ 0.5 and the pomeron trajectory OLD(O) x 1.09 are taken into account. All models use the approximation q = i, i.e. implement only purely imaginary amplitudes. Here an interesting conceptional problem arises. In principle the pomeron trajectory includes both soft and hard contributions. Thus adding to the amplitude (7) a hard contribution due to minijets is in contradiction to the Regge-pole philosophy as long as only one trajectory with an intercept o(0) > 1 is assumed. On the other hand the minijet contribution shows a nearly Regge-like energy dependence and hence could be considered as a “hard” pomeron, leading to the picture of a hard and a soft pomeron [37]. Conversely, in a model postulating only one universal pomeron it is questionable to expect amplitudes of the structure (7) for both the total pomeron (as sum of soft and hard contributions) and its soft part, being characterized by different intercepts.

one has ojjet

=

(~j-jet)%w

.

(9)

The procedure of calculating the inelastic cross section from inclusive cross sections is called unitarization. In the “pomeron language” unitarization corresponds to taking multiple pomeron exchange and and enhanced graphs into account. With the exception of nexus, all models employ variations of the eikonal approximation as unitarization. For sake of clarity we will discuss here only the simplest version of the eikonal model, The conclusions also apply to all other eikonaltype models. The previously introduced soft and hard cross sections are interpreted as corresponding to various single pomeron exchange amplitudes, so called Born graphs in field theory. If the impact parameter representation (see Appendix) is used, compact expressions for the unitarized amplitude can be obtained. Knowing the amplitude of the Born graph, Ag(s, t), the n-pomeron graph reads in impact parameter representation

&(s, 2) = - ;; fJ [2UB(S,B’)] . . i=l The total amplitude exchanges

is the sum of all pomeron

a(s,@= fyJqs,B) n=l

=

t (1 - exp { -x(s,

with the eikonal function x(s, i$ = -2iafj(s,

Z)})

fltot

=

gela

=

8).

%e

=

2 J cl22 (1 - exp{ -x(s,

J ( J (

(11)

x being

Assuming a real eikonal function 2.3. Unitarization: eikonal approximation and beyond The QCD minijet cross section rises much faster than the total cross section and exceeds the latter already at current collider energies (see Fig. 2). This can be understood by recalling that the QCD cross section (1) describes the inclusive cross section for the production of minijet pairs. If there are (ni,t) jet pairs produced per collision

,

d2B

1 - exp{ -x(s,

d2B

1 - exp{ -2x(3,

(12) it follows 2))

(13)

Z})2

(14)

B})

(15)

The eikonal approximation has many attractive features. For example it has a simple a;d intuitively clear interpretation. If A = 2x(s, B) is the

R. Engel/Nuclear

Physics B (Proc. Suppl.) 122 (2003) 40-55

mean number of individual partonic interactions at impact parameter B’ then the probability P, for having n independent partonic interactions is given by the Poisson statistics An P, = --e-X. n!

The inelastic cross section is simply the integral over all collision impact parameters where the probability of having at least one interaction is given by

cm c P, ?%=I

= 1 -e-‘.

(17)

Eq. (16) can be derived in a field-theoretical context [38] using the Abramovski-Gribov-Kancheli (AGK) cutting rules [39]. The main shortcoming of the eikonal approximation is the lack of explicit energy-momentum conservation. This can be seen by analyzing, for example, multiple jet pair production. The cross section for the Born graph (1) is calculated taking energy-momentum conservation into account. In the eikonal approximation the amplitude for npomeron exchange is simply proportional to the Born graph amplitude raised to the power n. This ignores the fact that the energy of the scattering particles has to be shared between all interactions. In particular it could happen that the first jet-pair is produced by partons with a large momentum, leaving almost no energy for further interactions. In the eikonal model all interactions are treated as completely independent and the full primary particle energy is assumed to be available for each of them. Of course, in the Monte Carlo generation of events energy-momentum is conserved. However, this is done by rejecting events where the energies of the individual partonic interactions add up to more than the primary energy. Rejection algorithms of this type are highly model dependent. In the nexus model the unitarization is done by explicitly constructing individual partons that share the full momentum of the primary particle [40,4]. For example, the product of the eikonal expression for the two-pomeron exchange is changed

x [UB((l - x)(1 - Y)S,B)] ,

45

(18)

where z and y are the momentum fractions of the partons engaged in the first interaction. As one would expect, the energy-sharing leads to a significant suppression of graphs with large numbers of pomerons. Although this unitarization method is theoretically more sound it has two major weaknesses. First the momentum sharing functions, in our example f(z, y), are of crucial importance and not known. Second the series that can be resumed to an exponential function in the eikonal approximation has now to be calculated numerically. Large terms with alternating sign have to be summed and numerical stability of the result is very difficult to achieve. 2.4. Diffraction dissociation Diffraction dissociation is characterized by events with fast secondary particles and low multiplicity. It is of particular importance to cosmic ray simulations since diffraction is one of the reasons for large fluctuations in the early development stages of hadronic air showers. In hadron-hadron interactions typically more than 15% of the inelastic cross section can be attributed to diffraction dissociation. Measurements of cross sections and distributions of single, double and central diffraction dissociation exist for a number of projectile and target combinations. Due to the steady transition between diffractive and non-diffractive events there is no generally agreed upon, precise definition of diffraction and the experimental data scatter correspondingly. The cross section for diffraction dissociation is, despite its importance, theoretically not understood and only poorly described in contemporary models. One of the main difficulties is the experimentally observed large cross section at low energy and its very moderate rise at high energy, as can be seen in Fig. 3. The models available in literature can be classified as s-channel approaches (Good-Walker model

46

R. Engel/Nuclear

Physics B (Proc. Suppl.) 122 (2003) 40-S-5

[41]), t-channel approaches (triple-pomeron approximation, partially including pomeron flux renormalization [42]) and various hybrid models utilizing both s- and t-channel approximations. In the following we will discuss the Good-Walker model in detail since it is to date the best and most general model of diffraction. 2.4.1.

Good-Walker

model

In the Good-Walker model [41], it is assumed that the scattering particles ar? not eigenstates of the scattering amplitude a(s, B). The wave functions of the incoming hadrons can be expressed as superposition of eigenstates ]di) of a(s, g) di =~+TT,~+TT+T,...

lh) = .&d,),

(19)

i=l

If some of the eigenvalues gi (s, 2) (diJa(s, d)ldi)

= iai(s, S)

(20)

are different, the superposition of eigenstates changes due to the scattering process. Using Eq. (42) the cross sections for fixed impact parameter can be written as Qot(% @

=

4Sm(h(a(s, 2)(h)

=

4g

,c&i(S,~)

.

(21)

i=l

The total diffractive cross section (elastic scattering and diffraction dissociation) is given by

=

c

lailya&,2))2

.

(22)

Since xi JcQ]~= 1 one can interpret Eq. (21-23) in the following way [43]: the total cross section is given by the average of oi and the diffractive cross section is given by its variance. Obviously, if all oi are equal the cross section for diffraction dissociation vanishes. Applying this model one can easily deduce the high-energy behavior of inelastic diffractive cross sections. In the black disk limit all states Idi) have the same eigenvalues oi -+ l/2, corresponding to total absorption. In the high-energy limit, the only grey region of the sca_ttering amplitude is the edge of the black disk, JB( w R. With R IV Ins one gets otot N ln2 s

C7& - In2 S

(24) It should be emphasized that the Good-Walker model satisfies s-channel unitarity as long as the partial amplitudes obey the unitarity bounds. However, since almost nothing is known about the wave functions of complex diffractive states as well as their cross sections, the predictive power of the model is limited to the lowest-mass diffractive states. Furthermore, the kinematics of the diffractively produced final states are not specified. The two-channel eikonal model is a straightforward implementation of the Good-Walker model in the eikonal framework, restricting the number of new parameters to a minimum. Only two states of each incoming hadron are distinguished, the hadron under consideration (A) and a generic, excited state (A*) that is a superposition of different diffractive final states. Assuming factorization one can define the following matrix elements of the Born graph amplitude g(s, g)

Wlz?IA~) = x (ABkIA*B)

i=l

To obtain the contribution from diffraction dissociation the elastic cross section has to be subtracted

(ABIt(AB*)

(A*B(~(A*B) (AB*Jg(AB*) (A*B*(g(A*B*) Within by

(23)

=

PAX

=

,c,x

= = =

(1 - 2&4)x (1 - 2~~)x (1 - 2aA)(1 - 203)x.

(25)

this model the total cross section is given

-(2l.i12ci(s,g))2 * not =s i=l

odjff - Ins .

2

d2B’ (ABI (1 - e-%) IAB).

(26)

R. Engel/Nuclear

Physics

Analogously, the cross section for single difliaction dissociation of particle A follows from gSD,A

d2B’ j(A*Bl

=

(1 - eski) (AB)12.

(27)

J

The full expressions for all cross sections can be found in [44,45]. The two-channel eikonal model is implemented in DPMJET and SIBYLL 2.1 for the description of low-mass diifraction dissociation. In both models the unitarized amplitude approaches the black disk limit at high energy and the cross sections follow the energy dependence (24). The main weakness of the two-channel eikonal model is its failure to reproduce the fast increase of the diffractive cross section at low energy, where threshold-like effects are observed. 2.4.2.

Quasi-eikonal

B (Proc.

Suppl.)

approximation

c(2) = [l +p2J2 % 1+ E, e assuming the scattering of identical particles (PA = &). s’imi ‘1ar 1y one obtains for the threepomeron graph the factor cc31 = [1+ 2pz + (1 - 2a)Pz]2.

(29)

In the quasi-eikonal approximation a simpler relation between the different enhancement factors is assumed, namely Ccn) = (C(2))n-1. The corresponding expressions for the cross sections are particularly compac :t =

4

d28&(

Uela =

4

gdiff

=

(C

-

Gle

=

2

J

J

d2ii& J

1

1-

exp{--Cxl>

1 - exp{-Cx})

(30)

2

(31) (32)

1)%3

d21?&

41

40-55

grey disk scattering with the opacity of 1/2C instead of the black disk expectation of l/2. Another important consequence is the proportionality of the elastic and diffraction dissociation cross sections clearly being disfavored by collider data (see Fig. 3). Furthermore, similarly to the twochannel eikonal model, the quasi-eikonal model does not explain the fast increase of the diffractive cross section at low energy.

One important effect of including diffraction dissociation in the eikonal model is the enhancement of multi-pomeron graphs. In a two-channel model the two-pomeron exchange amplitude is increased by the factor

ctot

122 (2003)

(1 - exp{-2Cx)).

(33)

It is important to notice that the quasi-eikonal model predicts in the high energy limit always

I 1 10

.

....,’

. . .....I

100 &

*

IO00 (c&V)

.,...’ loo00

Figure 3. Experimental data and model predictions on single diffraction dissociation in protonproton and proton-antiproton collisions. Shown are the predictions of the quasi-eikonal model, the triple-pomeron cross section and the results of the hybrid approach implemented in DPMJET III (see text).

Although the two-channel eikonal model has been used in some calculations performed with the predecessor of the QGSjet model, for example [46], only the quasi-eikonal approximation is implemented in QGSjet. Version 2 of nexus, although not being based on the eikonal approach, uses the same simplified enhancement coefficients. 2.4.3.

Triple-pomeron

description

In contrast to the Good-Walker model, Regge theory successfully predicts the kinematics of soft diffractive processes. However, it fails to describe the experimentally observed cross section.

48

R. Engel/Nuclear

Physics B (Proc. Suppl.) 122 (2003) 40-55

Fig. 4 depicts the triple-pomeron graph related to high-mass diffraction dissociation of particle A in AB scattering. All graphs with pomeronpomeron couplings are called enhanced graphs.

(b) t Figure 4. High-mass single diffraction dissociation: (a) diffractive cut of the triple-pomeron graph, and (b) corresponding particle production graph.

One of the important successes of the triplepomeron model is the prediction of a l/M: mass distribution of the diffractively produced system. Furthermore, the t-dependence of the cross section is given by the slope parameter bsb = b&./2. Both predictions are in good agreement with lowenergy measurements. However, the integrated triple-pomeron cross section rises with energy as UTP N s2A . This energy dependence is clearly in contradiction to data, as shown in Fig. 3. There have been many attempts to reduce this steep energy dependence predicted for diffraction dissociation by including the triple-pomeron graph in eikonal models and adding further enhanced graphs. Whereas the inclusion of the graph in an eikonal model helps to tame the energy dependence somehow, it still does not reproduce the experimentally observed cross sections [47]. Only the implementation of higher-order enhanced graphs leads to a satisfactory description of collider data [45]. The most sophisticated and complete description of diffraction dissociation is currently offered by DPMJET. It includes the two-channel eikonal model for low-mass diffraction dissociation, triple- and loop-pomeron graphs, and pomeron-pomeron scattering. However, due to

the limited number of implemented graphs the high-energy extrapolation is still highly uncertain, where further higher-order enhanced graphs are expected to become important. Nexus 2 does not employ enhanced graphs, only version 3 will use enhanced graphs for the simulation of diffraction and related phenomena. 2.5. Particle production The basic idea of linking amplitudes to particle production topologies can be understood on the basis of the probabilistic interpretation of hard interactions in the the eikonal model. Eq. (16) describes the number of interactions per single inelastic collision. The QCD parton model provides a microscopic interpretation for each partonic collision, including the momenta of the incoming and scattered partons. In the scattering process color is transfered and the scattered partons as well as the hadronic remnants become colorconnected. These color fields, so-called strings, are fragmented and hadronized using string fragmentation models. Although the QCD parton model predicts to some degree how much energy is carried by the hadron remnants, it does not give any information on how this energy is shared between the partons forming the remnants. In case of purely soft interactions the situation is even worse as the corresponding Regge amplitude lacks any microscopic interpretation in terms of parton-parton scattering. Furthermore, a microscopic interpretation of interactions in terms of partons is only justified on the basis of asymptotic freedom in QCD and does not apply to soft processes. Nevertheless, the topological expansion of QCD for large numbers of colors and flavours allows the interpretation of soft processes, predicting the production of “chains” of hadrons. These chains of hadrons are expected to exhibit a similar distribution as the fragmentation products of strings. Therefore all models discussed here assume that string fragmentation is universal and and implement it for both soft and hard interactions. From the field-theoretical point of view, the phenomenology of particle production can be understood in terms of unitarity cuts of the scattering amplitude, whereas each cut pomeron results

R. Engel/Nuclear

Physics

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122 (2003)

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on the single-pomeron exchange model and their generalization is not fully understood. Both QGSjet and DPMJET implement distributions based on Mueller diagrams. For example, in DPMJET the leading quark distributions are sampled from (a)

(b)

Figure 5. Unitarity cut of the pomeron graph. The color flow topology of a pomeron is in leading order a cylinder (a). Correspondingly, its unitarity cut leads to two chains of hadrons, treated in the models as two strings (b).

in two strings stretched between quarks and gluons. The physics of unitarity cuts and related predictions for particle production is described in, for example, [48,35] and references therein. Here we show as a representative example only the leading unitarity cut of a pomeron in Fig. 5.

2

a

c

b

2

a c

- Im

b (a)

(b)

a

a

E

T

b

b (cl

Figure 6. Mueller diagrams for the calculation leading particle distributions (see text).

of

Whereas central particle production is qualitatively well understood, little is known about leading hadron distributions. So far dimensional counting rules and the technique of Mueller diagrams are the only theoretical tools that can be applied to leading particle production. In the Mueller diagram technique one crosses a particle from the final state (Fig. 6a) into the initial state (Fig. 6b). The resulting amplitude (Fig. 6c) is parametrized according to Regge theory and thus a prediction for the inclusive particle distribution is obtained. These calculations are solely based

fg”(z)

‘v

z,“2(1

- “$‘2

f::?(z)

-

2,1’2(1

- zc4)1’2,

(34)

for nucleons and mesons, respectively. In SIBYLL the leading quark distribution in nucleons is given by f:;;(2)

- (xi + p2/sy4(1

- c&J3 )

(35)

where p is a regularization parameter. The situation is more complex in the case of nexus. There the leading particle distribution depends explicitly on the number of cut pomerons [4]. 3. Generalization tering

to hadron-nucleus

scat-

The well-known Gribov-Glauber formalism [49] is employed by all models but nexus to generalize the simulation of nucleon-nucleon scattering to nuclei. In the case of SIBYLL this formalism is implemented completely only for hadronair interaction. In SIBYLL nuclear projectiles are treated in a simplified way as objects being composed of independent nucleons, where only the cross sections and the number of participating nucleans per nucleus-nucleus collision is calculated using the standard Glauber multiple scattering theory. The Gribov-Glauber formalism is analogous to the eikonal approximation in nucleon-nucleon scattering. The only difference is that now the nucleon density functions of the nuclei have to be folded with that of the nucleons. For example, the inelastic and production cross sections for hadron-nucleus scattering read

(Tine1 x Jdz8 [Ix Jd’B’ [l -

gprod

exp { oEyTA(ti)}] exp {~~~T~(B)}]

, (36)

where TA(@ is the transverse density of hadronic matter of the target nucleus folded with that of

50

R. Engel/Nuclear

Physics B (Proc. Suppl.) 122 (2003) 4&5S

the projectile hadron and gNN is the nucleonnucleon cross section. Cosmic ray experiments are mainly sensitive to the production cross section. This cross section is smaller than the inelastic one and describes events in which at least one new particle is produced. The difference between the inelastic and production cross section is called the cross section for quasi-elastic interactions. The latter one refers to interactions in which either the target or the projectile nucleus breaks up without producing any new particles that are not fragments of one of the scattering nuclei. It is important to notice that, by construction, the Gribov-Glauber approximations is characterized by the same shortcomings as the eikonal approximations. In particular energy-momentum conservation is not considered at amplitude level and the implementation of diffraction dissociation is an unsolved problem. Therefore it is not surprising that almost all models, including DPMJET, predicted a particle multiplicity for AuAu collisions much higher than later measured at RHIC [50,51]. It is still debated whether energymomentum conservation effects or the increasing importance of enhanced graphs (i.e. gluon-gluon fusion and saturation) are the main reason for the large discrepancy between the models and data [52]. Nexus 2, having energy-momentum conservation fully implemented, gave similar predictions as the other models. The understanding of the RHIC measurements [53] will be of great importance for developing new theoretical concepts for hadronic interaction models [54,55]. In fact, the parton density reached at RHIC in Au-Au collisions at 200 GeV CMS energy per nucleon might be comparable to that in Fe-air interactions at the highest cosmic ray energies. 4. Implications

for extensive

air showers

Extensive air showers are most sensitive to model quantities such as leading particle distributions and inelastic and diffractive cross sections. Many of the most important model features of this kind and their implications for air shower predictions have been discussed recently in [56].

Here we want to concentrate on the importance of the minijet cross section. as it plays an important role in all modern interaction models. Though the QCD parton model cannot be used to predict leading particle distributions, it is the most important theoretical input for extrapolating to ultra-high energies.

log,,( E,, / ev ) 12 250 L

13 I

14 I

15 I

16 1

17 1

18 I

19 I ,:’

20 I

,,.’

,:’ ,,’ ,,,.

t

J%P GeV) Figure 7. Predictions of model I and II for total and elastic cross sections in pp and pjj scattering.

To study the impact of such uncertainties we consider in the following two variants of the SIBYLL 2.1 interaction model. In model I a minijet cross section is implemented which rises moderately with energy. This is achieved by increasing with energy the transverse momentum cutoff applied in the pert. QCD calculation. In model II a rapidly increasing minijet cross section is employed. By construction both models describe all collider data, including Tevatron measurements, reasonably well. In particular, the current total cross section measurements at fi = 1800 GeV do not allow to discriminate between a high- or low-cross section extrapolation (see Fig. 7). As discussed previously, the uncertainties in the description of pplpfi interactions directly enter the modeling of hadron-air and nucleus-air interactions. Fig. 8 shows the predicted mean mul-

R. Engel/Nuclear

Physics B (Proc. Suppl.) 122 (2003) 4&5’S

%P (GeV) loglo ( I&, I eV 1 12 600;

13 I

14 I

15 I

16 I

.G 5clcJ L 2

%

17 1

18 19 20 I I model1 . mdel II . . . . . . . . ,;: ; ,.,. :

(GW

Figure 8. Model predictions for pair interactions. The lower panel shows the mean charged particle multiplicity and the upper panel the elasticity.

tiplicity of charged particles produced in pair collisions and the corresponding elasticity. The elssticity is defined as the energy fraction carried by the most-energetic hadronic particle in the final state. Although the production of minijets does influence the energy fraction carried by fast secondaries only through energy-momentum conservation, the impact is clearly visible. A large minijet cross section increases not only the multiplicity of secondaries but also reduces significantly the elasticity. To estimate the importance of the minijet cross section for EAS we implemented the two new vari-

primary

7> 8

0.3 0.25

. .‘.“.I

. .‘.....I

-.,

energy E

. “.““I

(eV)

. “..“.I . ...n.fr model1 m&el *I . . . . . .

-

/

primary

energy E

(eV)

Figure 9. Model predictions for proton-induced EAS at 45’ zenith angle. The lower panel shows the mean charged particle multiplicity of the shower at detector level. The upper panel gives the multiplicity of muons with E,, > 300MeV. Only particle above the cutoff energy of 3 and 300 MeV for electrons and muons, respectively, are shown.

ants of SIBYLL in CORSIKA [57] and simulated showers for the altitude of the southern part of the Pierre Auger Observatory in Argentina. Fig. 9 shows the expected mean number of shower particles reaching observation level. Model II with a large minijet cross section predicts an earlier development of the showers (i.e. at higher altitude) which leads to a stronger absorption of the electron component and a higher muon

52

R. Engel/Nuclear

Physics

B (Proc.

multiplicity. The differences between model I and II are small at low energy (1014 - 1015eV). At the highest energies the differences grow to N 25% resulting in considerable uncertainty in a mass and composition analysis of showers.

CASA-BLANCA DICE fly’sEye Haverah Park HEGRA HiRes-MIA SPAS&VULCAN Yakutsk model1

1:. lOI

10’5

L--&-’ >-..A..-: ). -* ..: ?--T.-‘: L-+-4 *..+--.: t-. Q .-.: -

10’6

10”

primary energy E

Figure 10. World data on X,, [58] and model predictions.

lo’8 (eV)

lOI9

low

measurements

Fluorescence detectors such as HiRes and Auger can measure the longitudinal shower profile. Particularly well-suited for primary mass discrimination is the position of the shower maxiIn Fig. 10 a compilation of X,, mum, ha,. measurements is compared to predictions of the models I and II for proton and iron primaries. Only one representative result for iron primaries is shown. The model-dependence of the interpretation of the measurements in terms of primary cosmic ray mass is striking. Model II predicts a rather light (i.e. proton-like) composition whereas model I would correspond to a mean mass number close to that of carbon. The uncertainty of model extrapolations due to the limited knowledge of the minijet cross section is most pronounced in quantities related to the production of slow particles such as the charged particle multiplicity. Although reduced, it affects

Suppl.)

122 (2003)

significantly

40-55

predictions

for extensive air showers.

5. Conclusions

There are a number of sources for uncertainties in extrapolating hadronic interaction models to ultra-high energies. First of all, HERA measurements of the proton structure functions have revealed a rapid growth the number of partons at low x. This result makes it impossible to use straight-forward parton density extrapolations to predict minijet rates at ultra-high energy. Nonlinear effects such as gluon-gluon fusion or even saturation are expected to become important at high energy. Furthermore the range of applicability of various evolution equation formalisms is, though subject of intense research, still unclear. These questions are currently not understood and as a consequence it is impossible to decide from theoretical arguments whether a model extrapolation with high multiplicity (i.e. QGSjet) or one predicting a moderate increase of the multiplicity (i.e. SIBYLL) is more realistic. Minijet production is closely related to central particle production at colliders. Therefore one can expect that LHC measurements will ultimately allow to decide between different high-energy extrapolations. Another important aspect of hadronic interaction models is the unitarization. Both the eikonal model and the Gribov-Glauber formalism have severe shortcomings. They do not implement energy-momentum conservation at amplitude level and do not provide a natural framework for diffraction dissociation. The limited predictive power of the Gribov-Glauber formalism is most strikingly visible when model predictions for Au-Au collisions at RHIC are compared to measurements. Although detailed data on pp collisions at the same energy exist, none of the models implementing the Gribov-Glauber multiple scattering approach could predict the secondary particle multiplicities correctly. The inherent similarity between the Gribov-Glauber formalism and the eikonal model suggests that the eikonal model cannot reliably be extrapolated to ultra-high energy. On the other hand, the only model that is not based on the Gribov-Glauber approach,

R. Engel/Nuclear

Physics

nexus, also failed to predict the particle multiplicities measured at RHIC the same way as the other models did. Last but not least, there is no good understanding of leading particle production. The correct implementation of diffraction dissociation is important for extrapolating leading particle distributions. The conceptionally insufficient quasieikonal approximation, though it predicts compact cross section formulae, should be abandoned. In inelastic interactions all models are based on the assumption of energy-independent leading particle distributions, up to effects of energymomentum conservation. There is no experimental data to prove or disprove this hypothesis. All models considered here have many similarities, i.e. are baaed on the same concepts whether experimentally confirmed or not. Therefore the differences in the model predictions most likely do not represent the full uncertainty in the extrapolation of hadron production according to our current knowledge. Nevertheless it is of prime importance that cosmic ray experiment analyze their data with different models to estimate the modeldependence of the interpretation of the measurements and, where possible, even rule out certain model predictions. On the other hand different ideas such as the color glass condensate model should be implemented in modern interaction models and new approaches of extrapolation should be sought. All currently available hadronic interaction models rely very much on measured data. This will not change in the near future. Therefore a close collaboration between the high energy physics and the cosmic ray communities is necessary to ensure progress in the understanding of hadronic multiparticle production. Acknowledgments

The author thanks J . Alvarez-Muniz, T. Gaisser, D. Heck, J. Ortiz, S. Ostapchenko, J. Ranft, S. Roesler, T. Stanev, and T. Thouw for collaboration and discussions on various subjects related to this work. This work was supported by the US Department of Energy contract DE-FG02 91ER 40626.

B (Pmt.

Suppl.)

122 (2003)

Appendix:

53

40-55

notations

and theorems

The elastic scattering amplitude A(s, t) is defined by the following relation to the differential elastic cross section 1 dwa MS, t>12 , (37) - dt = 647r~(i)~ with k being the momentum of the incoming particles in the center-of-mass system (CMS) and s and t denoting the Mandelstam variables. Then the optical theorem reads 1 disc, (A(s, t = 0)) ~tot = mfi %

; Sm (A(s, t = 0)).

Experimental

(38)

data support the parametrization t=o

exp Ubht)

,

where &la is the slope parameter. It is convenient to discuss the asymptotic behavior of the scattering amplitude in impact parameter space. The transformation into impact parameter space is justified by angular momentum conservation (remember 1 = [B’jk with 1, k being the angular momentum and particle momentum in CMS). The impact parameter amplitude a(s, B’) is defined as Fourier transformation of the amplitude (37)

a(s,@= h

d2f..

J

ozA(~,

t)e-i’fL?

In a%alogy to geometrical optics, the function a(s, B) can be interpreted as the density function for sources of scattered waves producing interference patterns. The inverse transformation reads A(s, t) = 4s

J

d2fi a(s, B)&’

Using the impact parameter (40), the elastic and the total (37,38) can be written as

Uda(S) = 4 %t(S)

=

4

J J

.

(41) representation cross sections

d26 la(~,l?)/~, d2ti Sm(a(s, I?)) .

(42)

54

R. Engel/Nuclear

Physics

If the complex phase of a(s, 2) is independent 2, the elastic slope &la is given by [59]

d21? B2 a(s, 2) J h,(s) = J 2

B (Proc.

of

Suppl.)

(hepph/0012252), in Proc. of Int. Conf. on Advanced Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications (MC 2000), Lisbon, Portugal, 23-26 Ott 2000, A. Kling, F. Barao, M. Nakagawa, L. Tavora, P. Vaz eds., Springer-Verlag Berlin,

(43)

2000 3. 4.

5.

(44) 6.

where R is the radius of the black disc. As expected, the inelastic cross section (cross section for absorption) is given by the geometrical size of the disk Dine = nR2 whereas the total cross section is twice the disk size uttot = 21rR~. This is a result of unitarity: absorption gives rise to elastic scattering 1

uel,, = nR2 = -crtc,t . 2

(45)

At high energies the growth of the total cross section is limited by the Froissart bound [60]

7. 8. 9.

10.

11. 12.

(46)

13. This bound is often used to motivate an ln2s increase of the total cross section. However, since the factor r/m: is numerically large this bound is easily satisfied by all models considered here. The Pumplin bound follows from the assump tion %e(a(s, @)/Sm(a(s, 2)) 4 0 for diffractive and non-diffractive processes [61] 1 o’ela

+

odiff

5

2

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Eq. (43) has a simple physical interpretation: the elastic slope &la is a measure of the transverse size of the scattering objects. The special case of maximum absorption (black disk limit) one gets

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