Monte Carlo code parjet to simulate e+e--annihilation events via QCD jets

Computer Physics Communications 31(1984) 401 —409 North-Holland, Amsterdam

401

MONTE CARLO CODE PARJET TO SIMULATE e ~e -ANNIHILATION EVENTS VIA QCD JETS -

S. RI’JTER Sektion Physik, Karl-Marx -Universitat Leipzig DDR

Received 25 April 1983; in revised form 10 September 1983

PROGRAM SUMMARY Title ofprogram: PARJET

Keywords: Monte Carlo, high energy e~e -annihilation, QCD jets, hadronization, chain decay model

Catalogue number: ACFR Program obtainablefrom: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) ESER and IBM System 370; Installation: KarlMarx-Universität Leipzig, DDR and CERN, Geneva, Switzerland Computer:

Operating system: MVT 6.1 Programming language used: FORTRAN IV

High -speed storage required: 150 Kbytes Number of bits in a word: 32

Nature of physical problem In high energy e~e collisions quark—antiquark pairs are produced, which fragment into the observed hadrons. The Monte Carlo code PARJET simulates exclusive hadronic final states produced in e~e -annihilation via a virtual photon by two steps: (i) the fragmentation of the original quark—antiquark pair into further partons using results of perturbative QCD in the leading logarithmic approximation (LLA) and (ii) the transition of these parton jets into hadrons on the basis of a chain decay model. Method of solution The fragmentation of the initial quark—antiquark pair into further partons as well as the transition into hadrons is simulated by a Monte Carlo model.

Number of lines in combined program and tests deck: 1501 Restrictions on the complexity of the problem Other programs necessary for PARJET:

Monte Carlo code BAMJET, catalogue number: ACFQ, CPC 31(1984) 393 —

Monte Carlo code DECAY, catalogue number: ACFS, CPC 31(1984) 411

The model is not optimized to describe isotropic events as they occur below ~ 5 GeV in e~e -annihilation. Bottom quarks have been neglected.

—

Typical running time

Function RNDM(DUMMY) generates uniformly distributed random numbers between 0 and I

The running time depends strongly on the centre of mass energy v~.Simulating e.g. one event at 50 GeV needs about 1.0 s of central processor time on an ESER 1040 cornputer.

—

OO1O-4655/84/$03.OO © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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LONG WRITE-UP

2. The underlying model

1. Infroduction At present neither QCD nor any other theory is able to describe the transition of partons into the observed hadrons. Therefore the treatment of parton fragmentation is usually done in two steps: (i) For large momentum transfers •the parton fragmentation into partons can be described using perturbative QCD. (ii) For small momentum transfers phenomenological models are used describing the parton fragmentation into hadrons. The Monte Carlo code PARJET simulates the hadronization of such parton jets produce in e~e-annihilationevents via a virtual photon 1*: e~+ e quark + antiquark parton jets hadrons. The program involves basically two steps corresponding to (i) and (ii): (1) Fragmentation of the original quark and antiquark into further quarks, antiquarks and gluons (QCD jets) using the frame work of the LLA. (2) Conversion of the generated partons into hadrons and hadron resonances on the basis of a chain decay model using the Monte Carlo code BAMJET [1]. The subsequent decay of the resonances into stable final hadrons can be realized in a third step using the Monte Carlo code DECAY [2]. The model has the following features: (i) Baryon as well as meson production, (ii) Energy, momentum conservation of complete e~e events, (iii) Conservation of quantum numbers like Q, B, I.,, S (iv) Parton fragmentation into hadrons via the lowest SU(4) multiplets. The code PARJET is not optimized to describe isotropic events below V~ 5 GeV. The code DECAY also treats the decay of charmed hadrons but it provides no complete description of these decays. In the following we make only some basic remarks about the underlying model. A more detailed description of the model can be found in refs. [3—5].The structure and use of the program are explained and illustrated by a test case. —,

—‘

—*

The parton fragmentation into partons is treated in the LLA, according to a procedure developed in refs. [4,5]. The LLA allows a simple probabilistic interpretation of the parton evolution like a branching process. This suggests the simulation of the process on the basis of a Monte Carlo model. The branching process stops, if the invariant masses of the decaying partons become less than a nonperturbative cut off parameter ~sof the order of a typical hadronic mass scale. Details of the procedure as well as the developed Monte Carlo model simulating such parton jets are described in ref. [5]. The parton fragmentation into hadrons is necessarily based on a phenomenological model. For this we use the hadronization scheme of independent relatively high-energetic partons (ft 3 Gev/c2) via a Monte Carlo chain decay model developed by Ranft and Ritter [1,3]. This model is similar to the model of Field and Feynman [6] but includes beside the meson production also the production of baryons using a method developed by Ilgenfritz, Kripfganz and Schiller [7]. The fragmentation of gluons we handle via the fragmentation of a quark—antiquark pair with the probability Pqq or a diquark—antidiquark pair with the probability ~dd 1 Pq~ 0.375 carrying the energy fraction z and 1 z of the gluon, respectively. A more detailed description of the model used and the comparison with data is given in ref. [5]. The use of a final parton mass ~ 3 GeV/c~ provides a good fit to the data but implies that the four jet contribution in the eke-annihilation becomes important only above ~ 40 GeV. The independent quark or diquark fragmentation does not include exact energy, momentum and quantum number conservation, since each quark Jet leaves behind one or two (anti-)quarks which carry a small four momentum and quantum numbers. In order to guarantee energy, momentum and quantum number conservation for cornplete eke-events in a simple manner, we reduce the quark constituents of the remaining system to a small number, neglecting quarks and antiquarks of the same flavour but conserving the four =

—

=

—

=

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momentum of the remaining system. Thus, we do not violate quantum number conservation. The remainder constructed in this manner contains in general not more than three final low energetic particles. If the remainder contains only one partide, we add an earlier created particle to the remaining system and try its decay according to two or in case of three particles according to three body phase-space. Events with more complicated remainders are very rare and can be neglected. Of course this procedure is not physical at very high energies, since the energy of the remainder may become large. Thus, one gets final particles carrying large four momenta. In order to avoid this, one has to replace our procedure by another one, which allows one to handle the decay of the remaining system also into four or more low energetic particles. 3. Description of the code PARJET Subroutine PARJET(NPART, KFA, QO, ALAMB, AMUE) generates hadronic final states in eke-annihilation originated by an initial quark—antiquark pair in two steps: (1) Fragmentation of the initial quark—antiquark pair into further partons and (2) Transition of the partons into hadrons. The result is a table of final stable hadrons or resonances stored in COMMON/FINPAR/.

PARJET calls subroutines

QCDJET, GLURES, DRELAB, CHOIN

INTERQ,

BAMJET,

Remarks

(i) Before the first call of PARJET a call of subroutine DATAR3 is necessary. (ii) The properties of all particles used are stored in COMMON/PART/ and they are defined in BLOCK DATA1 used by the Monte Carlo code DECAY [2]. (iii) Calling subroutine TESTH contained in the card deck the user gets the output of a test run of the code PARJET (see test run output, case 1). 3.1. Subroutines calledfrom PARJET (1) Subroutine QCDJET (NPART, ALAMB,AMUE,IOPT)

KFA, AEO,

Subroutine QCDJET simulates the fragmentation of single partons (quarks, antiquarks or gluons) or of a quark—antiquark two jet system into partons. The initial parton goes into the +z direction. In the case of an initial quark—antiquark pair the quark goes into the +z direction and the antiquark into the —z direction. The result is a table of final partons of mass AMUE contained in COMMON/FINALP/ (see section 3.2). Input variables

Input variables

QO KFA ALAMB AMUE

centre of mass energy ~‘i~ of the e~e collision in GeV, initial quark flavour, 1, 2, 3, 4 means uti, dd, s~,c~quark—antiquark pair, QCD scale parameter in GeV/c2, minimal allowed final parton mass /.L in GeV/c2.

Output variables

AEO

KFA

initial energy of the parton jet or centre of mass energy ~/i~of the primary quark—antiquark pair in GeV; characterizes the initial parton: 0, 1, 2, 3, 4, 7, 8, 9, 10 means: gluon, u, d, s, c, ii, d, ë quark, In case of e~e-annihilation KFA has to be 1, 2, 3 or 4, which means uti, dd, s~or cë pair, 1: single parton jet generation, 2: quark—antiquark two jet generation; ~,

IOPT

NPART number of final hadrons or resonances. The properties of the final hadrons and resonances are stored in COMMON/FINPAR/ (see section 3.2).

ALAMB QCD-scale parameter A in GeV/c2 AMUE minimal allowed final parton mass ~t in GeV/c2.

COMMON blocks used

NPART number of final partons. The properties of the final partons are stored in COMMON/FINALP/.

Output variables

/FINALP/, /FINPAR/, /PART/, /FININT/, /REMAIN/, /PRINT/, /INPDAT/

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COMMON blocks used

/FINALP/, /INPDAT/, /DATEN/, /PRINT/

/RMASD/,

Q CDJET calls subroutines (i) AMASK samples the parton masses during the branching process according to distributions 2)

WN(k~ k

=

exp!

x

aS(k)

—

1,

2’rr

k~

if ~

1—c

dk2 P~(z)dz]_~_J

=

exp{

—

k2

L~

a(h~2)I 2~

dk2 +N~p~(z))dzI j k~ 1

}

Remarks

4 and

for an incoming quark q or antiquark W~(k~, k2)

(1)

i

(i) Before the first call of QCDJET a call of

—

(p~(z)

subroutine DATAR3 is necessary. (ii) The code PARJET calls subroutine QCDJET automatically with IOPT 2. (iii) One can use subroutine QCDJET separately to generate only single parton jets or quark—antiquark double jets on the parton level. In test run output, case 2 a correspond=

(2)

for an incoming gluon G. a~(k2) is the strong coupling constant, Nf is the number of quark flavours and the P~(z) are the Altarelli—Parisi functions. The functions J’J’~(k~,k2) give the probability, that an incoming parton a with an invariant mass k~does not emit any further parton during its evolution down to the invariant mass k2 ~ k~.z is the longitudinal momentum fraction and is a cutoff parameter proportional to the kinematic limits for z. More details are explained in ref. [5]. AMASK needs the program GQUAD for numerical integration of expressions (1) and (2) given in functions FQ and FG. (ii) VERTXX samples the actual decay vertex q —s q + G or G G + G, G q +4 depending on the incoming parton using the probabilities: -+

needs subroutines QFLAV and function BETA to sample the quark flavour of the daughter partons in case of G -s q + 4. (iii) LONGIP samples the longitudinal momentum fraction z for the decay products depending on the actual vertex using the Altarelli—Parisi functions P~(z). (iv) ROTPTO provides a rotation of the momentum of the incoming parton into a system, where it carries no transverse momentum. rotates the parton moment back into(v)theROTLAB lab system.

-~

ing output is shown obtained by a call of subroutine TESTP (contained in the card deck). (2) Subroutine glures

GLURES handles the gluon transition into neutral meson resonances of SU(3) pseudoscalar and vectors mesons if the gluon mass is less than 1.2 GeV. For AMUE greater than 1.2 GeV each gluon goes into a quark—antiquark pair with probability DIQ or into a diquark pair with probability 1 DIQ (see section 3.3). The hadronization of quarks and diquarks is treated by subroutine BAMJET. —

(3) Subroutine INTERQ q+ G

=

1,

WG_.q+~

=

CJm~Nfp~(z)dz,

-‘

INTERQ samples the quark flavour in the case (3)

of gluon fragmentation into hadrons via a quark—antiquark pair. INTERQ needs function BETA.

WG~G±G=CJPG(z)dz~

(4) Subroutine BAMJET BAMJET simulates the fragmentation of the quarks and diquarks into hadrons on the basis of a

with 1 —

=

P~(z)dz

fzm~x

Zmin~ Zmax

+J

N~P8(z)dz.

are the kinematic limits for z. VERTXX

chain decay model. The Monte Carlo code BAMJET is described in ref. [1]. The subroutines needed by BAMJET are listed in fig. 1.

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QCDJET _________

before the first call of PARJET, BA:.:JET or ~CDJET

AI.ASK~

GQUAD~—-~FQ

-

VERTXX.

CPI~AV

—

ROTFO ROTLAB LONGIP

405

e — - annihilation events via QCD jets

\~~\

BAIIJET ABBRCH

For the generation of random numbers uniformly distributed between 0 and 1 the function RNDM(DUMMY) is used. The subroutine SFECFE makes a fast selection of the sine and ing to a random angle between 0 and 2rr. cosine without using these functions correspond3.2. COMMON blocks (i) COMMON/FINPAR/ PXF(150), ICHF(150), PYF(150), IBARF(150), PZF(150), HEF(150), AMF(150), ANF(150),

H FLAVOR H}aSSS

‘PARJET

E:JERGI ~iPULS

SFECFE

samples the sine and

cosine of an angle

y-~ uni—

~

0 and 2

NREF(150) REAL ANF *8 tides and resonances.

VEREITT ENDLI

L

~GL~

DRELAB

generates random nun—

formly distributed betseen hers ur,iforrnly between 0 and 1 distributed

IOTREO after

______ ______

DECAY calls

of or BALJ.~T

before the firot

I.

T—

—

call of 0 T~~TS

_______ ________

1 cFi~r.i~

~

L—TWOPAD j—_THREPD

II

FINPAR contains the properties of ~the final parPXF(I) PYF(I) x-momentum y-momentum (GeV/c) PZF(I) z-momenturn HEF(I) energy (GeV) (GeV/c) of the Ith AMF(I) ICHF(I) IBARF(I)

mass (GeV/c2) electric charge baryonic charge

II final particle

ANF(I) NREF(I)

name index

)I

XLALB

L—RLOCK DATA 1 and 2

(ii) COMMON/PART/ Fig. 1. All subroutines necessary for a call of the code PARJET and the code DECAY.

(5) Subroutine DRELA B

DRELAB rotates the particle momenta into the centre of mass system of the primary quark—antiquark pair. (6) Subroutine CHOIN

CHOIN combines the remaining system and the created hadrons and hadron resonances together, taking into account energy, momentum and quantum number conservation for the cornplete e~e event. CHOIN may be replaced by any other procedure combining all created hadrons and the remainder. Eliminating the call of CHOIN in subroutine PARJET provides events which do not conserve energy, momentum and quantum numbers,

ANAME(180), AM(180), GA(180), TAU(180), ICH(180), IBAR(180), K1(180), K2(180) REAL ANAME *8 PART is used as a label to characterize a given particle or resonance. ANAME(K) AM(K) GA(K) TAU(K) ICH(K) IBAR(K) K1(K) K2(K)

name mass (GeV/c2) full width (GeV/c2) mean life time (s) electric charge of the Kth baryonic charge particle number of the first decay channel number of the last decay channel

I I

I

I I

I

I

)

K is the index of the particle. The fields of this COMMON block are defined in BLOCK DATA subroutine 1.

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(iii) COMMON/INPDAT/ IMPS(6,6), IMVE(6,6), 1B08(6,21) IBIO(6,21), 1A08(6,21), IA1O(6,21), Al, Bi, B2, B3, ISU, BET, AS, B8, AME, DIQ The fields IMPS, IMVE, 1B08, IB1O contain the number labels of the considered particles depending on the quark flavours of their constituents. Beside the hadrons of the lowest SU(4)-multiplets: IMPS pseudoscalar meson l6-plet, IMVE vector meson 16-plet, 1B08 baryon 20-plet for spin 1/2 baryons, IB1O baryon 20-plet for spin 3/2 baryons, 1A08 antibaryon 20-plet for spin 1/2 antibaryons, IAJO antibaryon 20-plet for spin 3/2 antibaryons. We consider also leptons (em, e, ~ ~s, i~, T) and their neutrinos and the photon y. The variables Al, Bi, B2, B3, ISU, BET, AS, B8, AME, DIQ are decleared in section 3.3. All fields and variables of COMMON/INPDAT/ are defined in subroutine DATAR3.

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annihilation events via QCD jets

PX(I) PY(I) PZ(I) EN(I) AM(I) IQFL(I)

x-momentum (GeV/c) y-momentum (GeV/c) z-momentum (GeV/c) energy (GeV) mass (GeV/c2) parton kind; 0, l, of the Ith 2, 3, 4, 7, 8, 9 10 final parton means: gluon, u, d, s, c, U, d, ë quark longitudinal momentum fraction program internal label to follow the evolution of each parton ~,

IXFRAC(I) INDEX(I)

(vi) COMMON/ FININT/ PYZ(150), PZZ(150), HEZ(150), AMZ(150), ICHZ(150), IBARZ(150), ANZ(150), NREZ(150) REAL ANZ *8 FININT stores the properties of the final particles and resonances after the chain decay of each considered parton. FININT has the same structure as FINPAR, only the last letter F of each variable is replaced by Z. PXZ(150),

(iv) COMMON/REMAIN/ RPXR, RPYR, RPZR, RER, KR1R, KR2R REMAIN contains the RPXR x-momentum (GeV/c), RPYR y-momentum (GeV/c), RPZR z-momentum (GeV/c), RER energy (GeV), KR1R quark flavour 1, KR2R quark flavour 2, of the remaining jet after cutoff of the chain decay process. (v) COMMON/ FINALP / PX(100), PY(lOO), PZ(l00), AM(lOO), EN(100), IQFL(l00), XFRAC(100), INDEX(l00) FINALP contains the properties of the final partons (quarks and gluons) after the perturbative fragmentation process by subroutine QCDJET.

(vii) COMMON/PRINT/ ISYS ISYS number for line printer. (viii) COMMON/QMASD/ QMASS(12) QMASD contains the masses used for the different quarks. In subroutine QCDJET they are defined as follows: QMASS(1) u quark mass 0.3 GeV/c2, QMASS(2) d quark mass 0.3 GeV/c2, QMASS(3) s quark mass 0.5GeV/c2, QMASS(4) c quark mass 1.6 GeV/c2, QMASS(5) b quark mass 4.7 GeV/c2, QMASS(6) t quark mass 20.0 GeV/c2. The fields QMASS(I + 6) QMASS(I), I 1 to 6, = = =

= = =

=

=

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Simulation of e

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contain the masses for the corresponding antiquarks. If the minimal allowed final parton mass AMUE is larger than QMASS(I) (I 1 to 12), QMASS(I) is replaced by AMUE.

Name

Default value

Meaning

AS

0.25

AME

0.93

ratio between particles of the pseudoscalar and vector meson 16-plet in case of SU(4) probability for sampling a meson vertex, (1 AME) for sampling a baryon vertex in case of an incoming quark free parameter in the distribution for sampling the quark flavours during the chain decay process probability for a gluon to fragment into hadrons via a quark—antiquark pair (1 DIQ probability to fragment via a diquark—antidiquark pair)

=

3.3. Variables

_____________________________________________ Name Default Meaning value QO

—

KFA

—

ALAMB

—

AMUE

—

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e — - annihilation events via QCD Jets

centre of mass energy ‘~/~of the e~e collision in GeV flavour in the initial quark—antiquark pair; 1, 2, 3, 4 means till, dd, s~,cë pair QCD scale parameter A in GeV/c2 (between 0.1 and 0.5) minimal allowed final parton mass in GeV/c2 AMUE has to be greater than ALAMB, the best fit to data one obtains for AMUE 3.0 1 u, ii quarks are considered 2 u, U, d, d quarks are considered 3 u, ii, d, d, s, ~ quarks are considered 4 u, U, d, d, s, c, ë quarks are considered free parameter in the energy sharing function free parameters used for sampling the cutoff energy for the chain decay free parameter in the transverse energy distribution ratio between particles of the (anti-)baryon 20-plet (spin 1/2) and particles of the (anti-)baryon 20-plet (spin 3/2) in case of SU(4)

—

BET

8.0

DIQ

0.375

—

=

=

ISU

4

=

=

Acknowledgements

=

=

Al

0.8

Bi B2

8.0 8.0

B3

6.0

B8

0.33

~,

The author wishes to thank Prof. J. Ranft and Dr. R. Kirschner for numerous stimulations and useful discussions during the course of this work. References [11S.

Ritter, Comput. Phys. Commun. 31(1984) 393. [21 K. Hanssgen and S. Ritter, Comput. Phys. Commun. 31 (1984) 411.

[31J. Ranft and S. Ritter, Acta Phys. Pol. BlI (1980) 259. [4J R. Kirschner and S. Ritter, Phys. Scripta 23 (1981) 763. [5] S. Ritter, Z. Phys. C16 (1982) 27.

[6] and R.P. Feynman, Nucl. B136 Acta (1978)Phys. 1. [7] R.D.Field E.-M. Ilgenfritz, J. Kripfganz and A.Phys. Schillei-, Pol. B9 (1978) 881.

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TEST RUN OUTPUT Case I INITIAL ENERGY~ 50.000 FINAL PARTON MASS MUE~

NUMBER 33 13 34 32 33 8 32 14 31 31 35 38 111

NAME RH0077 P1+ RHO—77 RHO+77 RH0077 NEU RHO+77 Fl— ETA55O ETA55O DM0783 KM—892 A5IG~0

NUMBER 23 18 14 25 23 14 13 7 14 13 23 23 23 14 23 13 8 14 13 23 13 23 14 13 14 13

NAME PlO ALAN P1— AKO PlO RI— P1+ GAM P1— P1+ PlO PlO PlO P1— PlO P1+ NEU RI— P1+ PlO P1+ PlO P1— P1+ P1— P1+

INITIAl QUARK FLAVOUR~ 3.000

MASS

CHARGE BARYONIC CH. X—

0.773 0.139 0.773 0.773 0.773 0.940 0.773 0.139 0.480 0.480 0.783 0.892 1.385

0 1 —1 1 0 0 1 —1 I I 0 —1 0

MASS

CHARGE

0.134 1.115 0.139 0.497 0.134 0.139 0.139 0.0 0.139 0.139 0.134 0.134 0.134 0.139 0.134 0.139 0.940 0.139 0.139 0.134 0.139 0.134 0.139 0.139 0.139 0.139

1LAMBDAz

0 0 —1 0 0 —1 1 0 —1 1 0 0 0 —1 0 1 0 —1 1 0 1 0 —1 1 —1 1

0 0 0 0 0 1 0 0 I I 0 0 —1

—1.037 —0.790 —0.343 —0.223 —0.246 1.059 0.445 0.203 —0.222 0.293 —0.064 2.627 —0.702

BARYONIC CH. X— 0 —1 0 0 0 I 0 0 0 0 0 0 I 0 0 0 1 0 0 0 0 0 0 0 0 0

—0.302 —0.400 1.096 1.531 —0.104 —0.050 0.090 0.156 0.025 0.112 —0.167 —0.024 —0.031 0.203 —1.155 0.600 0.059 0.102 —0.348 0.115 —0.338 0.143 —0.486 —0.790 —0.608 —0.429

0.510

Y— 0.895 1.696 0.908 —0.081 0.145 —1.535 —1.216 —0.833 0.138 —0.004 —0.371 0.847 —0.590

Z—MOMENTA ENERGY 6.167 8.346 2.628 1.978 0.621 0.498 0.635 —0.141 —12.559 —1.974 —4.478 —2.275 0.554

6.365 8.554 2.906 2.137 1.032 1.868 1.636 0.880 12.571 2.052 4.561 3.686 1.751

Y—

Z—MOMENTA ENERGY

—0.242 —0.348 0.064 0.783 —0.340 —0.015 —0.016 0.058 —0.087 0.026 1.110 0.022 0.016 —0.833 —1.398 —0.817 —1.535 0.341 —0.196 0.086 —0.166 0.629 0.279 1.696 0.929 —1.034

0.033 0.521 —0.773 —1.502 —1.069 —2.386 —1.023 —0.420 —1.405 —1.149 —8.018 —2.300 —2.241 —0.141 0.365 0.270 0.498 0.404 0.217 1.499 0.479 1.522 1.116 8.346 4.254 1.913

0.411 1.340 1.350 2.336 1.134 2.391 1.036 0.452 0.438 1.163 8.022 2.304 2.246 0.880 0.578 1.058 1.868 0.556 0.475 1.512 0.625 1.659 1.247 8.554 4.399 1.966

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annihilation events via QCDjets

Case 2 INITIAL ENERGY~ 60.000 FINAL PARTON MASS MUE~

PARTON

KIND MASS

0 O 1 O 7 O

1.000 1.000 1.000 1.000 1.000 1.000

PARTON

KIND MASS

1 O 7 0 O 0

1.000 1.000 1.000 1.000 1.000 1.000

PARTON

KIND MASS

1 0 7 0 0 O 0

1.000 1.000 1.000 1.000 1.000 1.000 1.000

INITIAL QUARK FLAV0UR~ 1.000

X—

Y—

—0.349 —0.631 1.750 —0.770 —1.607 1.607

0.651 —0.421 —0.260 0.030 0.412 —0.412

X—

Y—

0.0 2.682 1.294 —2.186 —1.511 —0.279

0.0 0.137 0.421 —1.617 0.416 0.642

X—

Y—

—1.814 1.814 3.483 0.909 —1.975 —2.176 —0.242

—1.790 1.790 1.023 1.577 —0.023 —1.175 —1.403

1

OPTI0N~

Z—MOMENTA ENERGY 1.250 2.026 21.477 3.943 —28.024 —0.672

2

lAMBDA~

XFRAC

1.763 2.383 21.573 4.140 28.091 2.050

0.044 0.071 0.748 0.137 —0.977 —0.023

Z—MOMENTA ENERGY

XFRAC

27.988 —0.356 —2.075 —12.679 —11.186 —1.692

28.006 2.888 2.675 13.005 11.339 2.086

1.000 —0.013 —0.074 —0.453 —0.400 —0.060

Z—MOMENTA ENERGY

XFRAC

22.763 2.720 —20.121 —5.207 0.964 —1.312 0.193

22.927 3.859 20.470 5.606 2.415 2.973 1.750

0.893 0.107 —0.790 —0.204 0.038 —0.051 0.008

INDEX 2 3 1 4 1 5

INDEX 1 2 1 3 4 5

INDEX 1 2 1 4 3 5 6

0.500