Monte Carlo approach to QCD jets in the next-to-leading-logarithmic approximation

PHYSICAL REVIEW D VOLUME 36, NUMBER 1 1 JULY 1987

Monte Carlo approach to QCD jets in the next-to-leading-logarithmic approximation

Kiyoshi KatoPhysics Department, Kogakuin University, Shinjuku, Tokyo 160, Japan

Tomo MunehisaFaculty of Engineering, Yamanashi Uniuersity, Kofu, Yamanashi 400, Japan

(Received 29 December 1986)

We present a Monte Carlo model for the parton generation in the next-to-leading-logarithmic(NLL) approximation. This model provides us with the natural introduction of cross sections fornontrivial hard scatterings and it enables us to determine the scheme for the fundamental parameterA in QCD. The implementation of the NLL effects into the Monte Carlo approach is studied exten-

sively. The stochastic treatment requires the positivity of functions for hard scatterings, two-bodydecays, and three-body decays. This leads to the condition of angular ordering which restricts thephase space. We apply this model to e+e annihilation and incorporate clean three-jet events intothe QCD cascade.

I. INTRODUCTION

Some time ago the jet structure was discovered in e+eannihilation. ' Since then many important studies havebeen performed about the jet phenomena and revealedproperties of quantum chromodynamics (QCD) for jets.For the study of jets the event generator, the Monte Carlosimulation of jets, is an indispensable tool. Although wemust use a phenomenological model for hadronization,the essence of jet structure is given by perturbative QCD.The event generators ' for the e+e annihilation in theearly days used a simple technique, i.e., the conventionalperturbative expansion to generate partons, where thecross sections are calculated up to order o, Later, amore complicated technique has been presented in Refs.5—9, based on the renormalization-group equation. It issometimes called the QCD cascade or parton shower. Inthis method it is possible to generate multijets within thelimited approximation. For this reason the QCD cascadeis important for the generator of hadron-hadron collisionssuch as at CERN Spp S or the future SuperconductingSuper Collider (SSC), where the event rate for multijetswill be very high. '

Although the present QCD cascade is useful, it fails togive the correct prediction for the production of partonswith large pz to the jet axis since it is based on theleading-logarithmic (LL) approximation. For example, inthe e+e annihilation the exact cross section for threejets is not included in this method. The purpose of thispaper is to present an algorithm for the parton generatorin the next-to-leading-logarithmic (NLL) approximation,where the above defect is removed by the theoreticallyconsistent treatment of the contribution in the NLL order.Using this algorithm we have developed a Monte Carloparton generator for the e+e annihilation.

In the next section we would like to make our motiva-tion clear through the discussion of characteristics of twomethods to generate partons, the conventional perturba-tion expansion and the QCD cascade. In addition we list

the merits of our NLL approximation. It is worthwhileemphasizing here that we can fix a scheme for the QCDparameter A. In Sec. III we present our algorithm for theNLL cascade model. This is divided into three subsec-tions. In Sec. III A a brief review of jet calculus is given,noting the differences between the LL-order cascademodel and the NLL-order model. In Sec. III B we showthe analysis of the scale breaking in the inclusive crosssection. In IIIC we define various functions for theMonte Carlo approach. In Sec. IV we will apply ourmodel to the e+e annihilation. Some results of ourmodel are given in Sec. V, where one can see the con-sistency of our model. Section VI is devoted to con-clusions. Explicit formulas for the Altarelli-Parisi func-tions and the vertex functions are collected in the Appen-dix.

II. GENERATION OF PARTONS

In order to simulate the jet phenomena in a high-energyreaction, we have to generate a system of partons (quarksand gluons) based on the theory of strong interactions, i.e.,QCD. There are two methods to do this. For the sake ofa concrete description, we consider the e+e annihilationprocess hereafter. The extension to other hard processeswill be possible. The first one is based on conventionalperturbation. We calculate the matrix element of e+eto N partons assuming the partons are almost on massshell. The generation of N partons' final states requiresthe computation of exact matrix elements at least of(a, )+ ~ order. A jet is defined by an appropriate cutoffto avoid the mass singularity. This method is used in pro-grams by Ali, Pietarinen, Kramer, and Willrodt andLund group. Another one, the QCD cascade, is based onrenormalization-group equation. Partons are generated byan iterative method using approximated cross sections sothat it is possible to implement this algorithm into a

36 61 1987 The American Physical Society

KIYOSHI KATO AND TOMO MUNEHISA 36

Monte Carlo program. We study characteristics ofthese two methods in the following.

We consider the production of a quark, an antiquark,and a gluon in e+e annihilation as an example. In thecenter-of-mass frame (CMF) the exact cross section forthis process in the lowest order of a, (Ref. 11) is

30.

X"D

20.

x=.5

Exact

Appr ox.

x +x=0-(0) ' C

dxq dx 2rr (1—xq)(1 —x )(2.1)

where xq (x-) is the energy fraction of the quark (anti-qua«) xq=2&~/(Q ) rr' ' is the lowest-order crosssection for production of a quark and an antiquark, anda, is the strong coupling constant. CF is a color factor 3

for QCD. En order to relate this cross section with the ex-pression of the Altarelli-Parisi equation, ' we make use ofthe infinite-momentum frame (EMF). The momentum ofantiquark goes toward the negative z axis and the direc-tion of boost is the positive z axis. In the IMF, momentaof the quark and the gluon are expressed as

0.00.0

2.0—t5X

UJ

1.5

0.2

(b)

0.8

p =(xP'+pr /2xP*, pr, xP*),(2.2)

p, = [(1—x)P*+p, '/'2(1 —x)P*,—p, (1—x)P*],

where I'* is the infinite momentum. In this frame x isthe momentum fraction of the quark in z direction. pT isthe transverse momentum of the quark to z axis. The re-lation between (xq, xG) and (x,pr) is given by

x, =x+p, '/Q'x, xG ——(1 x)+pr'/Q—'(1 x) . —

Using these variables we have the following cross section:

1.0

0.50.0 0.2 0.6 0.8 1.0

FIG. 1. (a) Exact and approximated cross sections for qqG inthe EMF for x=0.5. Here t =pr /[Q~x(1 —x)] and the crosssection is measured in units of C =o., /2~CFo' '. (bjRatios of the exact and approximated cross sections. The solid,long-dashed, and dashed curves represent the ratios for x=0.2,0.5, and 0.8, respectively.

der" (0) &s 1=o. CFdX dpT 277 pr

21++ pT 21 —x Q 1 —x+

2 21 pT 1 1

x (1 —x) Q (1 —x)x (1—x)+ z 2'+ (2.3)

For a fixed x, the dominant contribution of Eq. (2.3) is in the sma]]-pr region. So the cross section can be we][ approxi-mated by the term with (pr~)

CF~ 1 1+~2(

d+ dp~ 2' pT 1 —+ (2.4)

e prescription of the LL summation or the «normalization-group equation is to apply the approximated cross section(2.4) for any emission of a gluon from a quark. So it is possible to generate mu]tipa~ons using the cross section (24)Note that it is not a good approximation for the production with large Pr . Figure ], where we compare Fq. (2.4) witthe exact cross section (2.3), shows that the cross section (2.4) overestimates the production rate for large p

On the other hand, in the usual perturbative method the exact cross section is used for generation of three jets. Since itis very large at small pT, the perturbative expansion in such a region becomes doubtful. In order to make clear thispoint we demonstrate the ratio of the cross section with a cutoff of invariant masses (p;+pj) )M where i =q, q, G, tothe total one at the order of a, in Fig. 2:

a, (Q /A )R = CrI2[]n(Q /M2)] —3]n(Q /M ) —rr /3+ —, +O(M /Q )I(]+a,/ir) (2.5)

36 MONTE CARLO APPROACH TO QCD JETS IN THE NEXT-TO-. . . 63

4.0

3.0

tA

2.0

CK

1.0

0 010 10 10' 10'

0 (IeV)10

FIG. 2. Fraction of three-jets cross section vs squared c.m.energy [Eq. (2.5)] with A=0.2 CxeV. The solid, long-dashed,and dashed curves represent the ratios for M=2, 5, and 8 GeV,respectively.

all order summation of a, partially. For example, the an-gular ordering for the soft gluon results from the summa-tion of all order of a, /x [a, ln (1/x)]" terms. This angu-lar ordering can be implemented in the Monte Carlomethod in the LL-order approximation. This impliesthat the Monte Carlo approach includes a part of contri-bution in any order of a, .

In our approach the principle that the perturbationshould be valid gives a strong constraint on definitions offunctions. For example, this constraint leads to a newdefinition of the angular ordering. ' This means that theMonte Carlo method will be useful for a study of softgluons beyond the leading order. The present approachmakes many points clear which are ambiguous in the LLorder because gauge invariance requires the consistent cal-culation of all quantities. Those almost uniquely deter-mined are the argument of the running coupling constant,the definition of dimensionless variable to represent the"momentum fraction, " and the upper limit of the virtualmass of the parton. They will be discussed carefully inthe following sections.

That the perturbation is effective, i.e., R & 1.0, requires alarge value for the cutoff mass. For example, at Q =(60CreV) the minimum cutoff mass is about 5 GeV. If onewants to keep the perturbation effective, the cutoff massshould increase as the energy increases. It implies thatone cannot generate partons with small invariant mass,which are detectable as jets at high energy because the jetmass due to the hadronization is almost independent ofthe total energy.

The above discussion reveals features of two methods.The conventional perturbative method and the partonshower method are good approximations for the produc-

tion with large and small pT, respectively. If we can uni-fy these two methods, partons will be generated for anypT with good aeeuraey. This unification is accomplishedby extending the parton shower method to the NLL order.

Besides the unification of methods for small and largepT productions there are several improvements in our ap-proach. The most important one is that the scheme forQCD parameter A is fixed so that it is possible to measureA from e+e experiments. In the LL approximation thisimportant parameter cannot be obtained from experi-ments. Here we point out that A or a, would be overes-timated when one analyzes the experimental data using aMonte Carlo program with the conventional perturbationmethod. ' In this program the cutoff of jet mass is takento be small in order to generate events with small pT .Resultingly, it gives too large of a probability for theevents with small pT and too small a one for the eventswith large pT, which leads to the overestimation of u, .

Experimental determination of gluon spin has beentried through the measurement of the correlation betweenthe direction of beam axis and that of jets in e+e an-nihilation. ' This correlation can be correctly taken intoaccount in the NLL cascade.

The Monte Carlo method in the NLL order provides atheoretical tool for in depth study of perturbative QCD.This method is not exactly the same as the inclusiveanalysis in the NLL order because the former includes the

III. ALGORITHM FOR THE NLL CASCADE

A. Jet calculus

&s cgK~ 1+x~

2m K 1 —x (3.1)

where K is the virtual mass squared of the parton thatdecays. In the inclusive calculation the emitted gluon ison mass shell and only the quark has a nontrivial virtualmass. The jet calculus provides us a method to apply theabove calculation to the exclusive process. It is the sum-mation of diagrams such as Fig. 4, where the distributionfor N-body decay is

'X —1

dK &s

KV~(x), . . . , x~)dx) . . dx~

N

X5 gx; —1i=1

(3.2)

at one vertex. The above power of u, implies that thisdistribution is at least of order of (a, ) '. The integra-tion of the virtual mass at each vertex gives one logarithmdue to the mass singularity. This branching process is

Here we present a brief review on the jet calculus'emphasizing differences between the approximation in theLL order and that in the NLL order. As demonstrated inthe previous section the large logarithmic behavior ori-ginates from the integration by the transverse momentumor the virtual mass of the parton in the decay process.This large logarithm due to the mass singularity threatensthe simple perturbative expansion. If we can sum upterms such as (a, Ing )", reliable predictions are obtainedfrom the perturbative QCD. This summation is possiblethrough the renormalization-group equation. ' Figure 3represents an example of the LL summation on the in-clusive process in e+e annihilation. For each branchingpoint, we assign the transition probability

64 KIYOSHI KATO ANDND TOMO MUNEHISA 36

elusive quantities thful. " he introduction of P f

h ~ e olution of(Q,x) is given by

the decay func-

Of/ liO jf+Ooo

A I i') OfAgoooti

t $ ( I i A o~g

551~Tlt I I Kti if I I 6~

Q2dI ( Q,x)/dQ =P(x,a, )I ( tQ,x

where g denotes the coe convolution integral;

P( )G( )= F y)G(x/y) .

In the Monte Ce Carlo approach the rdo ot di t t lifo

r

IINE(K i,K2 ) =exp —f dx P(x,a, )

(3.3)

(3.4)

FIG. 3. Exam lernp e of a Feynman dia rae inclusive cross s

iagram which contrib tq

s e on-mass-shell cut.

named the parton shower or ~CDd de ermines arton

e is

pcontrols the distribution ofo primary par-

In the LL order the QCD casca

G G G+6 d G q+q. yin e Appendix. I

unc-ix. n the analysis of in-

CX~

E 2C +

1 —x 2~V2" (x) dx . (3.5)

Second, three-bod de'

e in- o y cayisoneof thor . era orm of theorder. The gen 1 f

e ingredients in th

y or the three-bod dy ecay ise transition prob b 1a 11-

2

z x~dx2 V3(x~,x2, 1 —x —xdx (3.6)

where e=Qo /E In t.he LL order th b d- o y- ecay functio s.

e approximation includesh, [ I ( )"S S

e transition proba gss. ese terms are

F t th ere is a correcour

o-bod d . For example, the t p- o ydeca

a iityd y q~q+G is replaced by

2

There are four kindsq~q+G+G

in s of three-bod dy ecays:G+G+G and

FIG. 4. Examxample of a diagram forh ffective propagator of a u

ine

h f *

"""""""""'"es e e fective vertex.

e

FICs.. 5. A rainbow dibod d

diagram which con

2x ~ +G (x2)+ G (x3). It hasas a color factor


G~G+q+q. Decay functions for them are found inRefs. 17, 18, 20, and 21. Third, there is the correction tothe P function. In the NLL order, however, the P func-tion is the sum of inclusive distributions for two-body de-

cays and those for three-body decays. So we can calculatethe corrections to the two-body-decay functions by sub-tracting the contribution of the three-body decay from thecorrection to the P function. The fourth a, lngcontribution comes from a restriction on phase space fortwo-body decays. This restriction must be defined con-

I

(0)

(a, /2vr) (lnK )

Pq (x) +x2) x,p(O)X) +X2 X) +X2

(3.7)

where K is the virtual mass squared of the initial parton.If one changes K by G (x; )K, it is

sistently in the NLL order. For example, we consider theleading contribution from the rainbow diagram shown inFig. 5:

p(p) + )(a /2w) [ (lnK ) + 21nK lnG (x; ) + [lnG (x; ) ] I Pqq

X& +X2 X] +X2(3.8)

Although the above change has no influence on the LLbehavior, it induces the additional contribution

(0)

(a, /2m) 21nK lnG(x;) Pqq (x) +xz) (()) xPqq

X] +X2 X ) +X2

(3.9)

(A) two-body decay,

(B) three-body decay,

(C) P function,

(D) restriction on phase space,

(E) coefficient function (cross section for hardscattering) .

(3.10)

B. Scheme dependence

The analysis based on the perturbative QCD usuallytreats the inclusive quantity while the Monte Carlo ap-proach treats the exclusive one. Our basic principle forthe implementation of the NLL correction to the MonteCarlo approach is that results from the Monte Carloshould coincide with those given by the inclusive analysis.

I

in the NLL order. This term should be included in thethree-body-decay functions.

In the NLL order one must include the correction oforder a, in the coefficient function, which corresponds tothe cross section for the hard-parton scattering. In e+eannihilation this correction is the cross section fore+e ~qqG and the virtual correction for e+e ~qq.Thus the consistent treatment of the QCD cascade in theNLL order naturally leads us to the use of the exact ma-trix element for the production of clean three-jet events.Here we summarize five ingredients of our model:

I

Under this principle, some degrees of freedom are still leftfor us. In our approach they are scheme dependencesamong ingredients given in (3.10). Separation of the fol-lowing pairs is not unique: [(A),(B)], [(B),(C)], [(B),(D)],and [(C),(E)]. We use the above freedom to make the sto-chastic treatments of the NLL corrections possible. Inthis paper we will present explicit formulas that are suit-able to parton generation in the e+e annihilationthough our algorithm can be applied to any QCD process.

We start from the QCD analysis on one-particle in-clusive cross sections. QCD enables us to calculate Qdependence of the fragmentation function M(Q, x) ine+e annihilation. ' ' For a while, we only considerthe nonsinglet part for simplicity:

M(g, x)=C(g /p, x)I (p, gp, x)

D(gp, x), (3.11)

where x =2p), g/Q, p), is the momentum of observablehadron and Q is the momentum of e+e system.C(g /)((, ,x) is the coefficient function which is used inthe conventional inclusive analysis. I is the evolutionfunction that satisfies the renormalization-group equationwith the P function Pqq(x, a, ):

p2dl (p~, gp~, x)/dp =Pqq(x, a, )I (p, gp, x) . (3.12)

D (Qp, x) is the decay function that describes the hadron-ization of quark.

In order to calculate the Q dependence of M(g, x) bythe Monte Carlo method, C(Q /p, ,x) and Pqq(x, a, ) arerequired to be positive. In the LL order C(g /)((, ,x) isthe 5 function of x and the P function is positive definite.Therefore, it is straightforward to realize the Monte Carlomodel for Eqs. (3.11) and (3.12). On the other hand, weface a more complicated problem in the NLL order.C( l, x) has been calculated by several authors in the NLLorde .22, 24, 25

C(l,x)=5(1—x)+ C" (x),2n

2C' "(x)=CF [ln(1 —x)+2 ln(x) —

4 ]—(5+9x)/41 —x

(3.13a)

(3.13b)

Note that C'"(x) is negative for any x. In order to recover the positivity of C( l,x), we make use of the scheme depen-

66 KIYOSHI KATO AND TOMO MUNEHISA 36

dence between the NLL-order P function Pqq"(x) and the NLL-order coefficient function C'"(x), where Pqq (x) is de-fined as

Pqq(x, a )=(a /2n)Pqq (x)+(a /2rr) P (x)

Here we introduce the moment M(Q, N), which is defined as the integral of M(Q, x) in the form]

M(Q, N) = f dx x 'M(Q, x) .

The full correction to the moment in the NLL order is given by

(3.14)

—J' '(X)/2PM(Q, N)=M(QO, N)[a, (Q )/a, (QO )] " '[1+[a,(Q ) —a, (QO )]A(N)/4m],

3 (N) 2C (N)+2 Pqq (N) 4Pqq (N) If3O0

(3.15)

and po and p, are the first and the second coefficients ofthe P function, respectively. They are given in the Appen-dix. In the above equation, only the combination ofPqq'(N) —Poc"'(N) l2 appears. We can replace C"'(x) byany function C'"'(x) without changing the observablequantity if one replaces Pqq'(x) by

P,',"(x)—P,[C"'(x) —C' "'(x)]/2

at the same time. This is a dependence between the in-gredients (C) and (E) in (3.10). Such a scheme dependenceis extensively used in this paper in order to assure the po-sitivity of functions in the evolution equation and also tokeep the higher-order contribution small enough to holdthe perturbative expansion effective.

C. Exclusive process

Next we discuss an extension from the inclusive processto the exclusive process. Though the extension is not

unique, the difference due to this ambiguity is beyond theNLL order. Since the calculation of the three-body-decayfunctions has been performed in the lightlike axialgauge, ' ' ' ' ' we consider the evolution in this gaugeand in the IMF. In this frame x =2pqQ/Q is equivalentto the fraction of light-cone variable (E+p, ). The propa-gator of gluon in the lightlike axial gauge is

DG»(k)= z [ g„—+(k„n,+k,n„)lkn],k2(3.16)

with n =0. We choose the direction of the gauge vectorn parallel to that of the initial antiquark, so that the anti-quark does not radiate a gluon.

In the Monte Carlo model, the hard-parton scattering isgenerated in the first stage. After that the cascade processoccurs. It is the qq event in the lowest order and qqGevent in the next order. The qq event corresponds to the 5function of C(l,x) in Eq. (3.13). C"'(x) is associateddirectly with the qqG cross section. As was given in Sec.II, the cross section in the IMF is

t

do [o) 1 1+x PT 2

dx dpr 2n pr 1 —x Q 1 —x~F +r

2 2

1 Prx(1 —x) Q

+ 2

1 1+ 1

(1—x)x (1—x)(3.17)

Usually, p2 in Fq. (3.11) is set to be Q2. However, another choice for p is possible as long as its value is of the order ofQ2. In order to use the exact cross section (3.17) for production of primary three jets with large pr, we take C "(x) «(3.13a) to be proportional to

der~~

"/dx, which is obtained by integrating (3.17) over pr .

do~ "/dx = J dpi' do'"/dx dpi'

12

=cr'0'CF ln[Q x (1—x)/pr;„]+2~ 1 —x 1 —x1 — [Q x (1—x) —pr, „]/Qx(1—x)

1 1+ 21+

(1 —x)x (1—x)—,'

t [Q x (1 —x)] —(pr;„) ] /Q (3.18)

where (pr~),„=x(1 —x)Q is the kinematical upper lim-p, in Eq. (3.11) is the maximum virtual mass squared

of a parton in the cascade. When the primary qqG systemis produced according to Eq. (3.17), the relation betweenpz- and the virtual mass squared KqG for the system of

the quark and gluon is

pz. /[x(1 —x)]=KqG2 .

The energy of the initial antiquark is calculated by theabove virtual mass squared. In the CMF, it is


E =(Q )'i (1—KqG /Q )l2 .

When the hard-parton scattering is the qq event, this vir-tual mass squared Kq& is determined in the first stage ofcascade. Therefore, Kq& is the invariant mass of thequark and gluon in the qqG event or the virtual masssquared of the quark at the first branching in the qqevent. In order to get a smooth distribution for the energyof the initial antiquark, i.e., that for the invariant mass ofthe quark and gluon, we introduce the minimum pz forthe hard-parton scattering and the maximum virtual masssquared of the parton in the cascade:

(p7. );„=x(l—x)5Q, p =5Q (3.19)

As a result our new coefficient function isr

12

C''"(x) =CF 1n(1/5)l —x

[x (1 —x) —l](1—&)2

1 —x

+ [1+(1—x)'] —, (1—13')1 —x

(3.20)I

As discussed in Sec. III B, the NLL-order P function forq ~q +X is modified as

P' ''(x)=P (x)— [C"'(x)—C" '(x)] .2

(3.21)

The next subject is the three-body decay and the restric-tion on phase space for the two-body decay, i.e., (B) and(D) in (3.10). As discussed in Sec. IIIA, the three-bodydecay is characteristic of the NLL approximation. Unfor-tunately these functions are not positive definite, so thatthey cannot be used as a probability to generate the three-body decay. To avoid this negativity, we make use of thescheme dependence between the three-body-decay func-tions and the restrictions on the phase space for the two-body decay. To illustrate the relation between them, weconsider the rainbow diagram (Fig. 5) where two gluonsare radiated from a quark. Virtual mass squares of theexternal quark and the intermediate quark are denoted asq and r, respectively. If Qo is the cutoff mass, the con-tribution from this diagram is

dq f&"~+"2~~ dr P(x3)

Pq

2Q

2P X) +X2 X] +X2

dx&dx3+ 2 R(x2,x3),q

(3.22)

R(x2,x3)=(1—xz —x3 x2x3)/( x2) 2 (3.23)

dq3 P(x3)ln(q /Qo )

q X&+X2 X)+X2dx~dx 3, (3.24)

while the NLL term is

x& is a fraction of a light-cone variable of the quark andx 3 x 2 are those of the first and the second emitted gluons,respectively. By momentum conservation, x

& +x 2

+x3 —1. Here we omit the factor ( CFa, /2m ) . The LLterm is

dq P(x3)ln(x ) +x2) P

X) +X2 X) +X2dX]dx2

+R (x2,x3) (3.25)

that contributes to the three-body-decay function,V~GG(x&, x2,x3). We should notice that the upper limit ofthe integral is not q but (x, +x2)q due to the kinemati-cal constraint described below. Momenta of three partonsare expressed as

po =(P*+ q'/2P*, 0,P '), p]3 = ((x~+x2)P*+ (r'+pr')/2(x~+x3)P', pr, (x ~+x2)P'),

p3=(x3P +pr /2x3P*, pr, x3P") . — (3.26)

Here po, p]2, and p3 are momenta of the external quark,the intermediate quark, and the emitted gluon, respective-ly. The momentum conservation po ——p]2+p3 impliesthat

q =r /(x&+x2)+pr Ix3(1 x3)2 2

which determines the upper limit of r .Since ( x

~ +x 3 ) is smaller than one, the first term of Eq.(3.25) is always negative. As xz goes to zero,

P(xql(x (+x2))=2x ) Ixp,so that the absolute value of this term is very large and it

breaks the positivity of the three-body-decay function forqGG. Imposing the restriction that the upper limit of ris (x&+x2)q =(1—x3)q we can eliminate this termfrom V~GG. Before going further, we would like to makea comment here. While a Monte Carlo model in the LL-order consists of only two-body decay, one can modify themodel by imposing a kinematical constraint without af-fecting the LL-order formulas. Such a modification ismeaningful since the introduction of kinematical con-straint results in the multiplicity distribution that agreeswith that given by the analytic calculation without the in-terference. In other words, we can improve the LLmodel by studying the higher-order contribution carefully.


P(x3)RqGG(x1xz x3) CF ln(X1+xz) I'

X)+X2 X)+X2

In this case it is to set the upper limit of r to be(x, +xz)q . When one adopts this upper limit in the cas-cade, one can eliminate the unfavorable term from thethree-body-decay function.

The modified decay function is'

VUGG(X1XZ X3) EGG(X1&xz X3) qGG( 1~ 2 3

(3.27)

by Marchesini and Webber in the LL order. En this im-plementation they introduced an angular variable insteadof the virtuality. This variable makes the program sim-ple. However, to obtain an accurate relation between theangular ordering and three-body-decay functions, we usethe conventional variables: the virtual mass squared andthe light-cone variable of a parton.

Here we examine contributions from soft gluons to thethree-body-decay functions, which are discussed in Refs.15, 21, 26, and 27. First we study the decay function forG~GGG. This function is calculated in Refs. 20 and 21:

We will apply the same consideration to other three-bodydecays so that similar terms with logarithm can be ex-cluded from the decay functions.

There is another kind of restriction on phase space forthe two-body decay. It is the angular ordering which isrequired to consider the interference effect of soft gluons.It has been implemented in the Monte Carlo model given

+M(X1 xz X3)+Ã(X1 xz, x, )],

where

(3.28a)

GGG( lixz&x3) ~~ [L(x1yxzyx3)2

L (x1,xz, x3)=4K(x, +xz)Kln(x, +x, )

+(cyclic permutations of x, ,xz, x3),X]+X2 (3.28b)

M(x1,xz,x3)= 2 X2[K(x1)+K«3)]in +(cyclic permutations of x, ,xz, x3),X2 (x1+xz)(xz+x3) (3.28c)

where

4K(x1+xz)Kln[xz/(x, +xz)]

X]+X2

So the sum of L(x1,xz,x3) and M(x1,xz, x3) in the aboveregion is represented in the form

L (x, ,xz, x3)+M(x1,xz, x3)

=4K(x1+xz)Kx ) lnx2

X)+X2 X)+X2

K(x) =1/x+1/(1 —x)+x(1—x) —2,and the term N(x1, xz, x3) is infrared finite and given ex-plicitly in the Appendix. While L(x1,xz, x3) is a contri-bution from the ladder diagrams, M(x1,xz, x3) is a con-tribution from the crossed diagrams. In a soft-gluon re-gion that x3»x1»xz, M(x1,xz, x3) is well approximat-ed by

where r is the virtual mass squared of the parton with(x1+xz) and w is the energy of the parent parton. Simi-larly the angle 813 between partons with (x1+xz) and x3is given by

22

(X1+x2)x3m 2

The above restriction X2q & r becomes 0]3 & 0~2 in thelimiting region where x 3 »x ] »x2. Analytic discus-sions ' reveal that if one imposes an order to angle ofthe two-body decays, one can sum up the leading singular-ity of soft gluons. This ordering will be easily realized inthe Monte Carlo approach. ' However, since this dis-cussion is done in the limiting region, the definition of theangular ordering is not unique. In Ref. 15 we have pro-posed the following restriction on the virtual masssquared of partons i and j:

Adding it to the LL contribution at this order, we have

2q dyz K(x1+xz)K(x1/(x1+xz))4

Qp r X)+X2(3.29)

This shows that the most singular term in the NLLcorrection can be absorbed into the evaluation of theladder diagrams through the restriction on the phasespace x2q & r . The introduction of the angular variablemakes the meaning of this restriction clear. %'hen L9]2 isthe angle between partons with x] and x2,

p2

2X]X2NFIG. 6. A rainbow diagram which contributes to the three-

body decay for q~q(xl)+q'(x2)+q '(x3).


2 2X ]X2X3

r& (qX )X2+X2X3+X3Xi

X 1+—,'

1n[x;xi. /(x;+x) ) ](x;+xi )(3.30)

when xk &x;,XJ (k&i,j). The angular ordering modifiesthe three-body-decay function as V&GO —O'GGz, whereWGGG is given in the Appendix. Equation (3.30) gives areasonable function for the decay G~GGG. For thethree-body-decay function for q ~qGG, we present its

1

definition in the Appendix because Vq&G is derived in asimilar manner. '

After the restriction on phase space for the two-bodydecay, the functions VGGQ VqQG are positive in all regionsof x; ~ However it is not the case for the other three-bodydecays G~Gqq and q~qq'q '. For these processes wepropose another method to obtain positive functions forthe three-body decay. As an example we consider thethree-body decay for q~qq'q ' where flavor of q is dif-ferent from that of q'. A function for this decay is ob-tained from Fig. 6:

2x1 (1+x, )2V, ,(x1,x2,x 3 ) =CF Tg , I [(1—X1 —x2)/(1 —x1)]'+[x2/(1 —x, )]2j

(1 —x1) (1 —x, )2

P(1—x1)+ "1x2('—x1 —x2)/(1 —x1)'+ I[X2/(X2+x3)]'+[X3/(X2+X3)]']»(X2+X3)1 —xi

(3.31)

where x&,x2, X3 are variables for quark q, quark q', and antiquark q ', respectively. For the first step, we impose themomentum conservation on the two-body decay to eliminate the last term with logarithm. After this, we have a modi-fied decay function:

2X iV', , (x1,x2,x3 ) =CF T~(1 —x, )'

(1+x1)(1 —x, )'

1 —xi —x21 —x]

2

+1 —x1

2I —Xi —X2

+4X]X2(1—x1)

We can rewrite Eq. (3.32) by introducing a variable y =x2/(x2+x3) as

8x) 1+x)Vqq 'q ' (X 1 y X2 y X3 ) —CF Tg y ( 1 —y ) — [y

2 + ( 1 y )2](1 —x1) (1—x1) (3.33)

12,(q2) TISH(x),2 dq

q

where H (x)= [x + (1—x) ], is replaced by

(3.34)

from which we can see that V', , is still negative forqq'q '

some values of x& and y. For the second step, we makeuse of the dependence between (A) and (8) in (3.10). TheLL distribution for G ~qq,

V', , = 8CF TI1 y ( 1 —J ) .qq q (1 —x )(3.37)

Clearly this function is positive definite.Next we discuss a relation between the Altarelli-Parisi

P functions and three-body-decay functions, i.e., (B) and(C) in (3.10). The integral of the decay function for thethree-body decay over one variable contributes to theNLL-order P function

12 (q2) 1 — T~H(x) .2 dq q

q qmax(3.35)

V3 (x 1,x 2, 1 —x, —x 2 )5(x —x1 )dx 1 dx 2 (3.38)

P(1 —x1)+CF T~ 0

1 —x) X2+X3(3.36)

The final form of the decay function for qq'q ' becomes

This replacement does not have influence on the LLbehavior. In the NLL order the change of (3.35) inducesan additional term for the three-body decay:

For decay functions presented in Refs. 17, 18, 20, and 21,the contributions to the P functions have been calculated.%'e must evaluate contributions from extra termsW(x 1 x2 x3 ) for the decay functions. Here one shouldnotice that the stochastic process has the probability con-servation, which implies that W(x1,x2,x3) has 5 func-tions. For example, in the case of q~q(x1)G(x2)G(x3),

WqGG(X1|X2|X3) CF WqGG. F(X1ix2ix3)+ F A WqGG, A(x1 ~X2~X3)2

(3.39a)

qGGF( 1&X2ux3 ),P(x2)

Pxi +X3 X] +X3

P(1 —x1)ln(x, +x3)—5(X3) dy P (y /x1)ln(x1) + (x2~X3),0X]

(3.39b)


, P(x2+x3)WqGG, A (x 1 ~ X 2 ~ X3 ) K

X2+X3 Xp+X3ln. XpX3

X JX2 +X ]X3 +X2X 3

3 X2X31+— ln2 (x2+x3)

X2X3

(x2+x3)

P(X3 ) (1 —x1)/26(x2)— J dy IC(y/x3)

2 X3

3 y(x3 —y)Xln 1+—— lnI [y(x3 —y)]/X32 jy X3 y)+X3(1 X3) 2 x 2

+(X2+ x3), . (3.39c)

]. —x

0dx2 W GG(x, x2, 1 —x —x2)/2, (3.40)

taking account of a statistical factor 2. Integration of(3.39c) over x2 for fixed x1 is zero, which is due to theprobability conservation mentioned above. On the otherhand, the integration of (3.39b) gives a nontrivial contri-bution. An explicit expression of the integral (3.40) isgiven in the Appendix.

These modifications of the P function are important toobtain the correct multiplicity distribution. The mostsingular contribution of soft gluons in any order of a, istaken into account by imposing the angular ordering.This requires that in the Monte Carlo approach theNLL-order P function for 6~6 +X should not have themost singular term. The integration of WGGG(x1 x2 x3)cancels the most singular term 4C~ [ln(x)] /x, in P'G31(x)

calculated in Ref. 23. This integration is analytically per-formed for the singular term. It is

6PG"(x)= Cz I—4[in(x)) /x+ —, ln(x)/x]

+(less singular terms) . (3.41)

This result proves consistency of our algorithm.The long and rather technical discussion in this section

provides an explanation of how to build the Monte Carlomodel based on the perturbative QCD up to the NLL or-der.

IV. APPI, ICATION TO e+e ANNIHILATION

The algorithm discussed in Sec. III is directly appliedfor the parton generation in e+e annihilation. Howeverwe have to introduce some approximation. The most im-portant approximation is that we limit the P function forGAG+X to the first-order one. This is reasonable be-cause the gluon sector is less important ' than the quarksector since a gluon has no direct coupling to a photon.Another reason is that the major contribution to P func-tions has already been accounted for through the angularordering and the argument of a, . A study of the gluonsector in the NLL approximation will be possible if acolor-singlet current that consists of gluons is consideredinstead of the charged current. Because the second-orderP function for 6~6 +L is not used here, there is ambi-guity in three-body decay for G~GGG. In our model we

A contribution to Pqq'(x) from WqGG(x1, x2, x3) is yielded

by

Pqq (x) =Pqq (x)+AC(x)+b Vg(x) . (4.1)

In Eq. (4.1) b C(x) is due to the replacement of the coeffi-cient function by the exact cross section for qqG. EVg(x)comes from the modification of the decay functions forq~qGG. This function is given in the Appendix. Theleading behavior of Pqq' (x) as x goes to one is

(CF/3o/2)[ —21n(1 —x)/(1 —x)] .

While the first-order behavior of Pqq(x, a, ) is

a, (Q ) 2CF

2~ 1 —x

(4.2)

(4.3)

where it is explicitly shown that the argument of n, is thevirtual mass squared. Clearly the large logarithmicbehavior of (4.2) threatens the perturbative expansion.Because the integration of (4.2) on x is of order of[ln( Q /Qp )], the second-order contribution will be com-parable to the first one. However, this dangerous termcan be included in the first-order term if the argument ofa, is replaced by pr ——x(1—x)Q . Final Pqq is

Pqq (x) =Pqq' (x)—CFPp/2

and

X [ —ln[x(1 —x)](1+x2)/(1 —x)l (4.4)

Pqq(x, a, )= la, [x(1—x)Q ]/27r]P' '

+ Ia, [x (1 x)Q']/2~]'Pqq' (—x) . (4.5)

Although the argument of a, in the second order cannotbe fixed, we use the same argument as that in the first-order one.

Next, we consider the argument of a, in the coefficientfunction, which is just the cross section for the hardscattering. It cannot be fixed completely within the NLLapproximation. The requirement that the energy distribu-tion for the antiquark should be smooth forces us to

l

use the function given in the Appendix. Similar approxi-mation is also done for the P function for G~qq.

We will discuss the argument of a, in the evolution ofthe cascade. In Ref. 32 it is concluded that the argumentshould be pT instead of the virtual mass squared fortwo-body decay. The same conclusion is also obtained byone of our principles, i.e., to hold the perturbation effec-tive for the P function for q~q+X. In our model thesecond-order P function is given by


choose the same argument of a, for both the cascade andthe hard scattering. This is well understood from Eq.(3.19). We conclude that the argument of ct, is pr forthe coefficient function.

V. MONTE CARLO RESULTS

We have written a program for a parton generator ine+e annihilation according to our algorithm. In thissection we show some results of our Monte Carlo genera-tor. Except for the case when the value is explicitlynoted, the results of Monte Carlo simulation in this sec-tion are achieved by the following parameters: the c.m.energy, Q=100 GeV, AMs ——0.2 GeV (where MS denotesthe modified minimal-subtraction scheme), Qo

——4 GeV,Nf ——4, and 6=0.25. When we study the energy depen-dence, the value of parameters 6 is tuned as 0.2, 0.15,0.17, 0.25, 0.27, 0.30, 0.33, and 0.35 for Q=10, 31, 60,100, 200, 310, 600, and 1000 GeV, respectively.

We will compare results from our model with thosefrom LL models. There are several kinds of LLmodels. The model presented by Odorico is suitablefor the comparison because the gauge is the same in bothhis and our model. If this gauge differs, the quantitywhich is not predicted by QCD, e.g. , the absolute. valuesof multiplicity, might not be the same. The model givenby Odorico includes a part of the higher-order contribu-tion through the kinematical constraint. By setting a flag,our Monte Carlo generator produces events based on theNLL theory as well as those based on the LL theory. Theresults of the latter are very similar to those in Ref. 7.When we compare the LL model with the NLL model, Ais taken to be 0.8 GeV. As is discussed later, the LLmodel with this value produces a similar distributiongiven by the NLL model.

We display typical events in Figs. 7 and 8. In these fig-ures, a diamond with a solid line represents a quark and across with a dashed line represents a gluon. Figure 7shows the evolution of cascade in the IMF where the hor-izontal length of each line is the fraction of the light-conemomentum and the vertical length is the sum of the trans-verse momentum of a parton and that of its parent par-ton. The leftmost diamond represents the initial anti-quark which does not evolve due to our choice of gauge.The event in Fig. 7(a) is an example of two jets, and thethree-body vertex, which is characteristic to the NLLtheory, is found in the right part of the tree. The event inFig. 7(b) starts its evolution from hard three jets for whichwe use the exact matrix element. Figure 8 shows theconfiguration of final partons in the CMF for thesame events as those in Fig. 7. These figures representL- Y, Y-Z, Z-X projections of momentum vector in theCMF, where the Z axis is the thrust axis and the Y axis isperpendicular to the acoplanarity plane.

The next subject is on the symmetry between the quarkand the antiquark. Although we distinguish betweentreatments of the primary quark and the antiquark, theresulting parton distribution in the CMF must be thesame for the quark and the antiquark. This point is ex-tensively studied in Ref. 7 within the LL approximation.In Fig. 9 we demonstrate pT distributions of the quark

CL 0

gL0

100 I

(

I I

)CL 0

-+

+

—50

I I I I I I !

FIO. 7. Typical event of QCD cascade obtained by theMonte Carlo simulation. The horizontal length of each linerepresents the energy fraction x in the IMF and the verticallength represents the pT at the branching.

and the antiquark to show the restoration of the symme-try, where pT is defined as a transverse momentum to thethrust axis. One can see good agreement between bothdistributions. Figure 10 shows rapidity distributions forthe quarks and the antiquarks, where the thrust axis isused as the jet axis and the positive direction is defined tobe parallel (antiparallel) to that of the quark (antiquark)which is the most energetic fermion in the event. Agree-ment of these distributions is satisfactory. A rapidity dis-tribution of a gluon is useful as a check of the symmetry.The good left-right symmetry of the distribution, which isshown later, implies the consistency of our model.

Energy distributions for the valence quark and the anti-quark provide a good place to check the symmetry. Herewe should notice that the energy distribution of thevalence antiquark is sensitive to 6, introduced in Sec. IIIfor the separations of two jets and three jets in the pri-mary vertex. As discussed in Sec. III the energy of thevalence antiquark is given by the invariant mass of the in-itial quark and gluon KqQ.

E = —,Q(1 —KqG IQ ) .

The above equation means that for the antiquark with

KIYOSHI KATO AND TOMO MUNEHISA

Q =- 100.0(Gev)

QUARK = 4

GLUON = 3

Rapidity10

0.8

Distribution ( Quark and Antiquark )

(z=Thrust Axis,y=Acopl. Axis)

0.6

0.4

0.2

FIG. 10. Rapidity distribution of the quark (solid line) andantiquark (dashed line). For comparison, the distribution of an-tiquark is drawn vs —y.

+/

//

//

ip

//

'T,+

1

g =- 100.0(GeV)

QUARK = 2x

GLUQN = 7

(z=Thrust Axis,y=Acap1. Axis)

E ~ —,Q(1 —5) the cascade initiated by two jets deter-mines the energy distribution, while for that with

E~ & —,Q(1 —5), the hard scattering does. The energy dis-tribution of the antiquark should be continuous atKzG ——6Q . Therefore too small or too large a value of 6is not allowed, which can be understood in the energy dis-tribution of the valence antiquark for various 5 (Fig. 11).In these figures we find the gap at (1—5) of x =2E /Qfor too small and too large 6. On the other hand, the dis-tribution for the quark is insensitive to 6. We have tuned6 in order to have better symmetry between the quark andthe antiquark and to obtain a smooth distribution for theantiquark. Figure 12 shows the energy distribution forthe valence quark and antiquark. The difference between

FICx. 8. Momenta of produced partons in the events

displayed in Fig. 7. X- Y, Y-Z, and Z-X projections are shownin the CMF.

pr Distribution ( Quark and Antiquark )

Distr ibu t ion ('v aicncc Antiqrrnrk)1 I

]1 1 1 [

)

T [ 1 1

f

1 I I 1

[1 1~ Tg

(&)

1.00 /

f

/ & & /

[

& i / /

J

/ / & /

0.50

0. 10{b)

0.05

0.010 1512.5102.5 5 7.5

pT (GeV)

FIG. 9. Distribution of pT of quark and antiquark. Thetransverse momentum is defined with respect to the thrust axis.The solid line and the dashed line represent the distribution ofquark and antiquark, respectively.

p0 0,2 0.4 0.8

FIG. 11. Energy distribution of the valence antiquark for (a)6=0.1 and (b) 6=0.5.


x Distribution (Valence Quark and Antiquark)I I

/

I I I

N=2

.9

8AlC5

.7

00 0.2 0.4 0.6 0.8

.61 02 10 104 10'

Q (GeV )

10'

FIG-. 12. Energy distribution of the valence quark (solid line)and antiquark (dashed line).

1.0

the two distributions is small enough to be consistent withthe inclusive analysis for the scale breaking. This pointwill be discussed below.

One of the important improvements in our model is tofix the scheme for QCD parameter. It implies that theMonte Carlo results with AMs have to agree with those bythe conventional analysis using the same AMs within theNLL approximation. Since the scale breaking of the in-clusive cross section is the well-established prediction ofQCD, we analyze the scale breaking for valence quarks tocheck the consistency of our model. We generate eventswith AMs ——0.2 GeV at various energies. Next, we calcu-lat~ iiie moments of the inclusive cross section for thegenerated events where the moment is defined as

1

M(Q, N)= I dxx 'Q do(Q )/dx . (5.1)

Since @CD prediction is made on Q dependence of themoment, we plot the moment M(Q, N) normalized atQ, = 100 CxeV . Figure 13 also shows the result of the in-clusive analysis ' ' for the structure function Fq(Q, x)with various value for AMs. We neglect the longitudinalcomponent to identify F2(Q,x) as the inclusive cross sec-tion because it is small except that for small x. This fig-ure shows that the generated events are consistent withthe conventional analysis with AMs

——0.2 GeV, but notthose with AMs

——0.1 GeV and AMs ——0.3 GeV. Thedifference between the Monte Carlo results and the con-ventional calculations is smaller than the ambiguity due tothe so-called scheme dependence of QCD prediction,which can be seen in Fig. 14. Our data for the scalebreaking prove the consistency of our algorithm andmodel. In Fig. 15, we can see that the scale breaking is in-sensitive to the cutoff mass Qo. Figure 16 shows that thescale breaking of the antiquark is consistent with the con-ventional calculation. So we can say that the differencebetween the energy distributions for the quark and the an-tiquark is no serious problem, although there remains aslight discrepancy between these distributions.

When one makes a comparison between the LL model

.9

~ 410 104 10'

Q (GeV )

1.0

.9

.7

ce

.5

~ 3102 10 10

Q (GeV )

I

10' 106

FIG. 13. Moments of the inclusive cross section for thevalence quark. (a) N=2, (b) N=4, and (c) N=6. They are nor-malized at Q2=100 GeV . The stars represent the results ofMonte Carlo simulations. The parameters are described in thetext. The solid, long-dashed, and dashed curves represent thepredictions by the conventional QCD analysis with a=4 usingAMs=0. 2, 0.1, and 0.3 GeV, respectively. Here a is defined asa, (Q !aAMs ). Because of this modification —ga/2 1n(a)Pqq'(x)must be added to P~q'(x).

74 KIYOSHI KATO AND TOMO MUNEHISA

and the NLL model, one should be careful with choice ofA for the LL model because the difference between bothmodels can be made small through a suitable choice of Afor the LL model. In Fig. 17 we display the moments ofthe inclusive cross section generated by the LL model.Here we choose A for which the scale breaking for quark

(a)

1.0N=2

AJC5

~ 8

7

.9 .610 10 10 10

G (GeV )

10

~ 71.0

N=4

.610 10 104 10'

0 (GeV )

10.9

.8

1.0N=4 Al

C57

6

5

7CVC5

.6

.410 10 104 10'

0 (GeV )

10

.5

.410 10 10 10

G (GeV )

10'

1.0

9

~ 8

N=6

(c)

1.0

.9

.8

.7

5

4

.310 10 10 105

G (GeV )

10'

, 4

.310 10 104 10'

Q (GeV )

10

FIG. 14. The scheme dependence of moments. The solid(dashed) curve denotes the prediction with a = 1 (a = 10). Thestars are the same as in Fig. 13.

FIG. 15. The dependence of the cutoff mass Qo on the mo-ments in Fig. 13. Squares, stars (Fig. 13), and triangles are theresults with Qo

——3, 4, and 5 GeV, respectively. In the case ofQo =3 GeV, the parameter 5=0.21, 0.15, 0.20, 0.25, 0.25,0.30, 0.33, and 0.38 for Q= 10, 25, 50, 90, 170, 250, 500, and800 GeV, respectively. In the case of Qo =5 GeV~, 5=0.21,0.16, 0.20, 0.22, and 0.27 for Q= 10, 40, 70, 140, and 400 GeV,respectively.

36 MONTE CARLO APPROACH TO QCD JETS IN THE NEXT-TO-. . .

distribution is almost the same for the LL model and ourmodel. As a result A=0.8 GeV corresponds to AMs=0. 2CzeV in the NLL model. In Fig. 17 the moments of theantiquark distribution are also shown. The largediscrepancy between quark and antiquark is seen in theLL model.

%'e show various observables in Figs. 18—24. Figures18 and 19 show remarkable differences between the LLand NLL models in the gluon multiplicity and gluon rapi-dity. These are mainly due to the angular ordering andthe choice of the argument of a, . We present the energydependences of the total multiplicity in Fig. 20 with the

1.0

~ 8AlC5

~ 7

.8C5

.7

10 104 10'Q (GeV )

1D

.61 Q2 10

I

100 (GeV )

10' 106

1.0 i

1.0

.9

. 8

~ 7P4

.6

~ 5

. 410 10 10 10

Q (GeV )

106

.8K

C4IC5

.6

.5

1 Q2 10 104 10'0 (GeV )

10

1.01.0

.9

.8

.7.?

~ 310 10' 10'

0 (GeV )

10'

FKz. 16. Moments of the inclusive cross section for thevalence antiquark. The curves are the same as those in Fig. 13.

~ 31Q 10 10' 10'

g2 (Ge10

FIG. 17. Moments of the inclusive cross section for thevalence quark in the LL model with A=0.8 GeV. Stars andcrosses represent the moments of the quark and those of the an-tiquark, respectively. Curves are the same as in Fig. 13.


0.4

Gluon Multiplicity

I I I I I I

10.0 NL x LL

0.3

0.2

6.1

A.

z; 4.

2.

0 1 0.10 10 104 10'

u' (ceV'&

10'

0.00 2 4

N

6 8 10 FIG. 20. Dependence on Q of the average multiplicity —l

in the NLL (star) and the LL (cross) Monte Carlo models. Thesolid curve is Eq. (S.2).

FIG. 18. Gluon multiplicity distribution in the NLL (solid

line) and the LL (dashed line) Monte Carlo models.

theoretical prediction

[ln(Q /A )]~expI[c ln(Q /A )]' (5.2)

where p = ——,' —(2NF/3Po)(1 —CF/Cq ), and c =8C„/

f3o. The prediction is normalized at Q =100 GeV . Herewe plot (N ) —1 taking account of the one-jet structure.

Figures 21—23 show the thrust, the sphericity, and theacoplanarity (see Ref. 33 for the definition of D) distribu-tions, respectively. In the sphericity distribution it is dif-ficult to distinguish between the NLL model and the LLone. In the thrust distribution there is clearly a 10%%uo to

20% discrepancy. There is a qualitative difference in theacoplanarity distribution. Although we have the same Qdependence in the inclusive cross section by choosing suit-able A which is 0.8 GeV for the LL model compared with0.2 GeV for the NLL model, the above discrepancy sug-gests the overproduction of the clean three jets in the LLmodel, which is expected from the discussion in Sec. II.The different treatments of the hard scattering in the twomodels become manifest in these distributions. The sametendency is also found in the energy-energy correlation[Fig. 24(a)]. In order to check the three-jets matrix ele-ment implemented in our model, we compare the resultsof the Monte Carlo simulation for the asymmetry distri-bution of the energy-energy correlation with the conven-tional calculation. Because the higher-order contribu-

GluorI Rapidity Dist. ributIonOB I

'I'IS& l&~t DiSt Vibut. ian

06

100

E-a

10

10

0.6 0.7 O. B

Thrust0.9

FIG. 19. Gluon rapidity distribution in the NLL (solid line)and the LL (dashed line) Monte Carlo models. The jet axis isdefined by the sphericity axis.

FIG. 21. Thrust distribution in the NLL (solid line) and theLL (dashed line) Monte Carlo models.


Sphericity Distribution Energy —Energy Correlation

10.0

5.0

1.000.900.800.700.600 50

0.40

I I ( 1 I I I I I f I

1.0

Vl00'o

'eb

0.30

0.20

0.50. 100.090.080.070.060.05

—1

C OS/

0. 1

0 0.2 0.3Sphericity

I

0.5 Energy —Energy C'or relation (Asymmett yi1.00

FIG. 22. Sphericity distribution in the NLL (solid line) andthe LL (dashed line) Monte Carlo models.

tion for the asymmetry distribution is small, it is mean-ingful to compare the lowest-order calculation and theMonte Carlo result which includes some higher-order con-tributions. Quantitative comparison for the thrust distri-bution and the energy-energy correlation is not reliablesince the higher-order correction is large. ' ' In Fig.24(b) the LL model gives smaller asymmetry than theNLL model does, which is in contrast with the other dis-tributions. The result of the NLL model is in good agree-ment with the conventional calculation. We conclude thatthe LL model cannot provide consistent configuration for

0.50

0V

0Vcj

b

0 10

0.05

Q01 I I I

0 Q'3

C:OS)(

0.6 0.8

FICx. 24. (a) Energy-energy correlation and (b) its asymmetryin the NLL (solid line) and the LL (dashed line) Monte Carlomodels. In (b) the solid curve represents the conventional calcu-lation with A=0.2 GeV.

Acoplanarity DistributionI I T I

100

Energy Flow for Three

t

0

Jets

10- '

10

I

0.2 0.4 0.610

0I

2008(degree)

I I I

100 300

FIG. 23. Acoplanarity distribution in the NLL (solid line)and the LL (dashed line) Monte Carlo models.

FIR. 25. Energy flow for the three jets in the NLL (solidline) and the LL (dashed line) models.


the three-jets events.In Fig. 25 we show the energy-flow distribution for the

three-jet-like events. Selection of the three-jet events andthe analysis method are the same as Ref. 37, however, wetreat partons. In this figure we find the remarkablesuppression of the energy flow of partons between thequark jet and the antiquark jet due to the angular orderingof the NLL model.

VI. CONCLUSION AND DISCUSSION

We present a Monte Carlo model for the parton genera-tion in e+e annihilation including the NLL effects. Toimplement the full NLL effect into the Monte Carlomethod, we made the decay functions positive by usingthe scheme dependence. Their positivity is essential to in-troduce the stochastic treatment. The Monte Carlo re-sults on the scale breaking of the inclusive cross sectionshow that our model can fix the scheme for QCD parame-ter A. To determine AMs is one of the most importantsubjects in e+e experiments. We have tried to obtainAMs from the conventional analysis for the inclusive crosssection some years ago. ' In this analysis it is difficult toestimate the systematic error due to the hadronization.Now we are able to study this error through connectingthe QCD cascade and the hadronization model. Anotherimportant improvement is that the clean three jets havebeen incorporated into the cascade model in the theoreti-cal point of view. The unification of the cascade modeland the conventional perturbation method enables us togenerate partons precisely for both small-pz and large-pzregion.

There remain two problems that are phenomenological-ly important. One is a cascade of heavy quarks andanother is correct generation of clean four jets in e+eannihilation. The former requires mass-dependentanomalous dimensions and the latter requires three-loopanomalous dimensions. Calculations of these quantitieshave not been made and will be tedious work. In ourMonte Carlo generator the mass of the quark is only akinematical parameter which is added to the virtual masssquared and is consulted to check whether the branchingis possible kinematically. Our model may overestimatethe generation of the clean four jets if our discussion inSec. II is valid for the four-jets case. It will be interestingto compare between results from our model and the exactmatrix element of the four jets.

Because of our choice of the lightlike axial gauge, aparton structure in e+e annihilation is one jet; i.e., thevalence antiquark does not cascade. If one chooses thetimelike axial gauge, the parton structure is the two jets;i.e., the valence antiquark cascades independently as thevalence quark does. A difference between both gaugeswill be found in a back-to-back correlation. In the formermodel this correlation between the back-to-back jets existsbut not in the latter model. The differences between bothmodels should be of O(1/ln(Q /A )) in the LL order andO(1/[ln(Q /A )] ) in the NLL order. To determinewhich of gauges is favorable in a phenomenological modelis an interesting study in experiment. Because of the one-

jet structure, there remains a slight discrepancy between

ACKNOWLEDGMENTS

We thank our colleagues of the KEK TRISTANtheoretical working group for various discussions. Wealso acknowledge the constant encouragement of Y. Shim-izu and the useful advice on the programming providedby T. Kamae and T. Takahashi. One of us (T.M. ) wouldlike to thank H. Yamamoto and the PEP-4 group forwarm hospitality during his stay at LBL, where this workstarted.

APPENDIX

In this appendix we summarize functions needed forour Monte Carlo program. We use the following nota-tions for color factors:

C~ =3: f,g, fdb, =C~5,d,CF ——, . T'T'= CFI,TR= —, Tr(T T )=TRo.b

a,' '(Q') =4'/[Pain(Q /& )],P0=(11C„4NFTR )/3, —

NF. number of flavors .

(A 1)

the quark and antiquark distribution. As was seen fromthe analysis of moments, this causes no problem in theNLL order. This discrepancy would not be resolved untilwe go higher order than the NLL order. Since our modelis consistent up to the NLL order and it can determine thecorrect AMs, it would be allowed to do symmetrization inorder to absorb the remaining small asymmetry when ourprogram is applied to an analysis of experimental data.

Our algorithm for the NLL-order QCD cascade can beapplied to other processes, for example, the deep-inelasticprocess. In this case one might think that the cascade ofthe initial partons is needed before the hard scattering.However only the cascade of the final partons is enoughto have correct jet structure if one adopts the axial gaugevector as the momentum direction of the initial parton.So it is straightforward to extend our model to the deep-inelastic scattering by taking account of the kinematicsand the hard scattering. In the next ep machine HERA atDESY, such a cascade model will be indispensable forevent generations. Also three-gluon decay of the heavyquarkonium is an interesting application of the QCD cas-cade, where a detailed study for soft gluons will be possi-ble.

On the other hand, an application to hadron-hadroncollisions, which is one of the most important QCD pro-cesses for finding new physics, is not an easy task, becauseone needs a consistent construction of the initial cascadeand a consistent separation between the cascade and thehard scattering. These problems are more complicateddue to many variables relevant to this process. This com-plication leads to an uncertainty of the argument of e, .The complete construction will require a careful study ofthe contribution of order of a, .

MONTE CARLO APPROACH TO @CD JETS IN THE NEXT TO

p] ——102——", ~F . (A3)

First we present I' functions or two-body-decay func-tions in the LL order. The definition of the P function isgiven by

The expression for the running coupling constant up tothe NLL order is

1/a, (Q')+(P]/41rPo)»[a, '"(Q')]=(P /4')ln(Q /Az),

(A.2)

where we introduce

P (x) =[1+(1 —x)']/x .

{1])G~G(x)+G(1 —x):

PGG(x) =2CgK(x),where

K(x)= 1/x + 1/(1 —x)+x (1—x) —2 .

(i]i) G~q {x)+q(1 x).

PqG (x)= TRH (x),where

(A6)

(A7)

(A8)

(A9)

Q d &(Q,x)/dQ =Px(x a. )I (Q»)Px(x, a, )=(a, /2n)Px (x. )+(as/2m)Px (x) ..

(j) q~q(x)+G(1 —x):P' '(x)=C P(1 —x),

where

(i) G~G(xz)+q(x])+q(X3):

(A4)H(x)=x +(1—x)2. (A10)

dQ /Q (a, /2m) Vox(x],xz, x3) .

Here we omit terms with 6 functions.

(A 1 1)

First we show the three-body-decay functions. The def-inition of three-body-decay functions is given through thetransition probability

VG (x„xz,x, ) =4(CF —C„/2) T& [1+(x] +x3 )/xz jln[xz/(x] +xz)(x3+xz)]+ —,

'C~ T„I2[(x]2+X32)/xz —(x,z+x, z+1)/(1 —xz)+1]ln[x]x3/(x, +xz)(x3+xz)(x]+x3) ]

+2(x] —x3)ln[x](x] +xz)/x3(x3+xz)] ]

+CFT~[—(1+x])/(1 —x3)+2x]/(1 x3) —(1+x3)/(1 —x, )+2x3/(1 —x]) ]

+CONTR

I(X]+X3)[—1/(1 —xz)+6/(1 —xz) —6/(1 —xz)]—4/(1 —xz)+4/(1 —xz) I

+CFT~[H(x] )P(xz/(1 —x3))/(1 — X)3+H( X)P3( x/z(1 —x]))/(1 —x] )l

+C~Tg2K(xz)H(x]/(1 —xz))/(1 —xz) .(ii) q~q(x])+q'(xz)+q '(x3):

(a) q&q',

Vqqq(x], xz,x 3 ) =8C~T+x ]xzx 3/(x z+x 3)

(b) q =q',

Vqqq(X] yxzyx3 ) =8CFTI]x ]xzx3/(xz +x3 ) + 8C+T~x ]xzx3/(x 1 +x3 )

(A12)

(A13)

+2(CF —C„/2)C~I —2+(1+x])/(1 —xz) —2x] /(1 —xz) +(1+xz)/(1 —x])—2xz/(1 —x] )

+[2—(1+xz)/(1 —x])—(1+x])/(1—xz)+2/(1 —x])(1—xz)]In[(1 —x]){1—xz)/X3]] . {A14)(iii) G~G(x] )+G(xz)+G(x3):

VGGG{x]~xz~x3) CA [ VGGG{x]~xzix3) ~GGG(x]~xz~x3)l2

VGGG{x]~xz~ 3) {x]~xz~x3)+M{X1 ~x2~x3)++(x]~xz~x3)where

I (x ] xz x3 ) =4K(x] +xz )K (x ] /x ] +xz )ln(x 1 +xz )/(x 1 +xz ) + (cyclic permutations of x ],xz, x3 )~(x] xz X3)=(2/xz)[K(x])+K(X3)]ln[xz/(x]+xz)(xz+x3)]+(cyclic permutations of x],xz, x3)X(x]pxzpx3) =

I 2( 1/x3+ 1/x] )[—1/( 1 —xz)+2 —xz( 1 —xz)]—2[1/(1 —x])+1/(1 —x3)][1/(1+xz)—2 —xz(1+xz)]—6xz] ln[xz/(x]+xz)(x3+xz)]+[6x, /(1 —x3) —6x, (1+x, )/(1 x, )'+(1+7x]+x]')/(1 —x3)' —3(1+x])/2(1—x3)+ ]] ]

(A15a)

(A15b)

(A15c)

(A15d)

(A15e)

+[6x3 /(1 —x] ) —6x3(1+x3)/(1 —x]) +(1+7x3+X3 )/{1—x] ) —3(1+x3)/2( 1 —x])+ —, )+ —,

+(cyclic permutations of x],xz, x3),


and

~GGG(x)&X2&X3 ) ~(x3 X 1 )~(x3 X2 )4+(x) +X2 )+ [x) /(x) +X2 )]/(X 1+X2)

+ In((x)X2X3)/(x)X2+x)X3+X2X3) I + (X1X2)/(x)+X2) [(x)X2)/(x)+X2 ) ] } )

+(cyclic permutations of x),x2,x3)

(iv) q~q(x) )+G(X2)+G(X3) (see Ref. 39):

(A15f)

(A16a)

(A16b)

B(x 1 x2 x3 ) =( 1 —x2 —x3 —x2x3 )/( 1 —x2 ) + (exchange of x2 and x3 )+X2Xg

(A16c)

C(x),x2,x3) —I4P(x2+x3)/x3+2 —4x2/(x2+x3)+2[1+(1 —x3) ]/x2(x3+x2) }

2( 1+x) )ln[x) /(x)+x2)(x)+x3 )],

EGG(x )&X2&X3) qGG(X)&X2&X3) EGG(x)&X2&X3

VqGG(x )&x2&x3 ) =CF [B(x ),x2,x3 )]+—,CFC~ [C(x ),x2, x3 )+D (x, ,x2,x3 )+E(x ) &x2,x3 )]

2(1+x) )ln[x, /(x)+ x2 )(x)+x 3 )],

4P(x2+x3)K[x3/(x2+x3)]D(x),x2,x3)= ln(x2+x3 ),

X2+X~

E(X) X2 X3)=2— 4IX2X3[6—6(X2+X3)+(X2+X3) ]—X)(X2+X3)2 2

(x2+x3)4P (x2+x 3 )

WqGG(x), x2,x3) = , C„CF— I(. [x2/(x2+x3)]ln (x2x3)/(x)x2+x)x3+x2x3)X2+X&

3 x2xgX 1+— ln[(x2x3)/(x2+x3 ) ]2 (x2+x3)

(A16d)

(A16e)

(A16f)

(A16g)

Next we present equations related to the hard-parton scattering (qqG). The interaction is limited to QED. The lowestcross section ( qq ) for one flavor is

4 2(0) 477 o

( 3Q )2Q2

(A17)

where Qq is quark charge in unit of proton one. We use the infinite-momentum frame, where the system is boosted tothe positive z axis and the direction of the antiquark is negative z. In this frame x is defined as energy fraction of quarkwhile (1—x) is that of gluon. Also a transverse momentum of the quark to z axis is denoted by pT. Of course that ofthe gluon is —pT. If o'" is the cross section for q +q+ G,

2

o'0) dxdpT' 2~ pT' 1 —x Q' 1 —x

2 2

1 PTx(1 —x) Q2

1 1

(1—x)x (1—x)1+ (A18)

We need the cross section integrated over pT because it is directly related to the coefficient function. The integrated re-gion is between x (1—x)5Q and x (1 —x)Q:

)o)——CF ln(1/6)+ [x (1 —x) —1](1—6)+ [(1—x) + 1](1 —5 )/2

g dx 277 1 —x 1 —x 1 —x

We present P functions at the NLL order. In e+e annihilation, we need only that for q~q +anything:

Pqq'(x)=CF[CFP, (x)+ —,' CgP2(x)+NFTgP3(x) —(CF —C„/2)P4(x)],

P, (x)=2 ln(x)ln(1 —x)+1+x 3

1 —x 1 —x—5 —3x ln(x)+ —2 ln (x) —5(1—x),1+x 1+x

2 1 —x

12

P2(x) = [ln (x)+ —,ln(x)+ —,' n /3]+2(1+x)ln(x—)+ —, (1—x),

1 —x


P, (x) = — [—ln(x) ——,'

]—2(1—x)3 1 —x

2

P4(x)=2(1+x)ln(x)+4(1 —x)+2 [ —,ln (x) —ln(x)ln(1+x) —Sp(1/(1+x))+Sp(x/(1+x))] .1+xHere we use the Spence function Sp(x) defined as

X

Sp(x) = —f dy ln(1 —y)ly .

(A20)

The above P function is calculated by Curci, Furmanski, and Petronzio. We use the cross section of qq6 as the coeffi-cient function and the argument of the running coupling constant as x (1—x)Q . Then P function is modified by b C(x)and b, A (x), which are due to the first and the second modification, respectively. Also we impose the momentum conser-vation on the branching processes so that the three-body functions for quark~quark + two gluons andquark~quark + quark + antiquark have been modified. These modifications result in some changes b, Vs(x) of P func-tion Pqq(x):

Pqq(x, a, ) =2a, (x(1—x)Q )

Pqq (x)+(a, /2m. ) Pqq (x),2&

Pqq' (x) =Pqq'(x)+ b C(x)+ b A (x)+b, Vg(x),r

EC(x) = —(11C„4N~Tg—)/6 Cp" (x)—(])

(0) (a, /2~) ' —3CL"

2C(2" (x) = CF [ln(1 —x)+21n(x) ——,]—(5+9x)/4

1 —x

CL"(x)=CF,

1+x'(a, /2n) '=CF In(1/5)+ [x(1—x) —l](1—6)+2 1 (1—x) +1 (1—6 )

dx '1 —x 1 —x 1 —x 2

(A21)

11Cg —4NF Tgb, A (x)= CF ln[x (1—x)] .

6 1 —x

Here Cz(x) is a coefficient function given in Ref. 22 and corresponds to the structure function F2. CI (x) is a coefficientfunction for the longitudinal part. b, Vg(x) is defined as the following integral:

b, Vg(x)= —CF f dy ln(1 y)P((1 —x ——y)/(1 —y)) — ln(x) f dy P(y/x)P(y) P(1 —x) X

(A22a)1 —y x 0

b, Vg(x) has been calculated analytically:

6 Vg(x) = —CF (2P(1 —x) [ —,ln (x) —ln(x)ln[x/(1 —x)]+ 41n(x) I + —,(1+x)ln (x)+(x —1)ln(x)) . (A22b)

Next we summarize restrictions on the phase space. Here K and Q are denoted as the virtual masses squared of theparton and its parent, respectively.

(1) q(1)~q(x)+G(1 —x):

K (xQ(2) G(l)~G(xi')+G(x3)~[G(xi)+G(x2)]+G(x3):

2 2x (x2x3 3 2 2K (Q I+ —(xlx2)/(xi+x2) in[(x lx2)/(xl+x2) ]x )x2 +x]x3 +x2x3

for x3 )x „x~.

(3) q(1) q(xi)+G(xq3) q(xi)+[6(x2)+G(x3)]:

[ + (x2x3)/(x2+x3) [(x2x3)/(x2+x3) ]Ix ]x2+x]x3 +x2x3

(4) G(I)~q(x)+q(1 —x):

K (xQ


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Monte Carlo approach to QCD jets in the next-to-leading-logarithmic approximation

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