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Energy and Virtuality Scale Dependence in Quark and Gluon Jets


LU TP 98-11 September 1998

arXiv:hep-ph/9805228v2 1 Sep 1998

Energy and Virtuality Scale Dependence in Quark and Gluon Jets
Patrik Ed?n1 , G¨sta Gustafson2 e o Department of Theoretical Physics Lund University

Abstract: We discuss some important issues concerning multiplicities in quark and gluon jets in e+ e? annihilation. In QCD the properties of a jet in general depends on two scales, the energy and virtuality of the jet. Frequently theoretical predictions apply to a situation where these scales coincide, while for experimental data they are often di?erent. Thus an analysis to extract e.g. the asymptotic multiplicity ratio CF /CA between quark and gluon jets, needs a carefully speci?ed jet de?nition, together with a calculation of nonleading corrections to the multiplicity evolution. We propose methods to systematically study the separate dependence upon the two scales in experimental data and compare the results with theory. We present jet ?nding algorithms which correspond well to the theoretically considered jets. We also show that recoil e?ects add corrections to the modi?ed leading log approximation which are quantitatively important, though formally suppressed at high energies.
1 2

patrik@thep.lu.se gosta@thep.lu.se

1

Introduction

QCD predictions on the scale dependence of multiplicities in high energy qq systems are experimentally well con?rmed [1]. A similar multiplicity behaviour is to be expected from high energy gluon jets, the major di?erence being the di?erent colour charges of gluons and quarks. In this paper we want to address a set of problems which have to be carefully treated for a quantitative analysis of jet properties: ? At experimentally accessible energies subleading e?ects are quantitatively important. ? The jet properties depend in general on two scales, the energy of the jet and its virtuality, speci?ed by the largest possible transverse momentum of one of its subjets. Theoretical calculations frequently refer to a situation where the two scales coincide, while in experimental analyses the two scales are often di?erent. ? In contrast to the topology of an event, which is mostly determined by a few energetic particles, the multiplicities in jets are sensitive to how the softer particles are associated to di?erent jets. As will be discussed in this paper, the multiplicity of hadrons depends not only on the properties of the perturbative parton cascade, but also on the soft hadronization process. The assumption of Local Parton Hadron Duality (LPHD) [2] implies a direct relation between the number of hadrons and the number of partons, provided a proper, locally invariant, cuto? is imposed on the parton cascade. The asymptotic behaviour of the multiplicity in jets was calculated in the leading log approximation in [3]. In the modi?ed leading log approximation (MLLA), subleading terms in the evolution equations of relative magnitude √ 1/ lns are included [4, 5]. With arguments based on “precon?nement” [6] and LPHD we demonstrate in this paper that an important correction factor is expected due to recoil e?ects. Although formally suppressed by a factor 1/lns, this e?ect is quantitatively large and has an essential impact on the ratio between quark and gluon jets at accessible energies. Many jet ?nding algorithms have been presented for the study of e+ e? annihilation events [7]. Several of these have been successfully used in comparisons between data and theory for properties like the distribution in the number of jets, and how this varies with the resolution scale. We want to stress that our problem is a di?erent one, as described in the third point above, and it will be important to speci?cally consider the treatment of soft particles in the analysis. The angular ordering e?ect in QCD [8] implies that soft particles at large angles are emitted coherently from harder particles which they cannot resolve. Strictly speaking, these soft particles do not belong to any speci?c jet, and the colour factor for the emission is determined by the colour state of the combined unresolved partons. In ref [9] a method (the “Cambridge” algorithm) is proposed to associate the above-mentioned soft particles with the quark (or antiquark) jet, leaving to the gluon jet only those particles directly associated with the emitted gluon and the gluon colour charge. In this paper we study this question further and propose some modi?ed cluster algorithms.

1

If we study two-jet events obtained in e+ e? annihilation using a jet ?nding algorithm with a distance measure of k⊥ -type, the jet properties (e.g. the hadron multiplicity) depend not only on the jet energy but also on the k⊥ -cut used. For minimum bias events, one hemisphere corresponds to a quark jet where these two scales are the same (an “unbiased quark jet”). Similarly the multiplicity of an unbiased gluon jet corresponds to one half of an imagined gg system stemming from a point source. In section 5 we will discuss how this quantity is related to a gluon jet in a qqg event. We will also present methods by which both of the jet scales can be systematically examined, as well as methods designed to de?ne unbiased jets, where the two scales coincide. Some experimental results of relevance for this paper have already been presented. In [10, 11], the hemisphere opposite to two quasi-collinear heavy quark jets in e+ e? events is analyzed. √ This corresponds well to an unbiased gluon jet, with Eg ≈ s/2. In [12] the energy scale dependence of quark and gluon jets is studied. There a ?xed resolution scale is used in the cluster algorithm, which implies that the virtuality scale is held constant. In this paper, we discuss how to systematically study both scales in a jet. We also present methods to construct unbiased quark and gluon jets in a general three-jet topology, which enables a study of the scale evolution of unbiased jets. In our analysis we will for convenience work in the Colour Dipole Model (CDM) [13], which provides a geometric picture which is easily interpreted. The outline of this paper is as follows: In section 2 we discuss precon?nement and LPHD. In section 3 we discuss the CDM and the multiplicity distributions, including corrections relevant to the MLLA approximation, as presented in [14]. In section 4 we discuss recoil corrections to the multiplicity evolution. In section 5 and 6 we discuss the two di?erent scale dependences of multiplicities in jets and present cluster algorithms designed to extract these from data. In section 7 we present results obtained by MC simulations. The results are discussed in section 8.

2

Precon?nement and Local Parton Hadron Duality

It is essential to realize that the hadronic multiplicity cannot be determined from perturbative QCD alone. From perturbative QCD it is possible to calculate the parton multiplicity within a given phase space region. As the parton multiplicity diverges for collinear and soft emissions, some kind of cut-o? is needed in order to ?nd a quantity which is correlated to the hadronic multiplicity. The nature of this cut-o? depends on the properties of the soft hadronization mechanism, and is therefore not calculable within perturbative theory. The parton cascades are dominated by planar diagrams [15] (to leading log level in an axial gauge). This implies that every colour charge after a cascade has a well de?ned partner anticharge. As shown by Amati and Veneziano [6], the ?nal parton state can be subdivided in colour singlet clusters, whose masses stay limited also when the total energy becomes very large. In [16], Marchesini, Trentadue and Veneziano showed that these “precon?nement” clusters are limited not only in momentum space but also in real space-time. At least in the large Nc limit it is then natural to assume that the clusters hadronize independently from

2

a)
bg θ rg br

b)

b bg

gr rg

Figure 1: a) After a g→gg emission with a small angle θ, there is a screening of the new colour charges (rr in the picture). This implies that they give no net contribution to the emissions of soft gluons at polar angles larger that θ (the angular ordering e?ect). Their contribution to the emission at smaller angles corresponds to normal dipole emission in the rr back-to-back frame. b) After a gluon cascade, further emissions of soft gluons corresponds to a set of independently emitting dipoles. the rest of the system. This means that the nonperturbative con?nement mechanism is local in the sense that it combines partons which are directly colour connected and that it acts only locally also in momentum space. This idea is further developed in the notion of Local Parton Hadron Duality (LPHD), proposed by the Leningrad group [2]. Here a direct relation is assumed between partons and hadrons, which is local in phase space, which means that the dominant features of hadron distributions can be obtained from the parton cascades with an appropriate cut-o? which is locally invariant. A very essential feature of the QCD cascade is “soft gluon interference” and “angular ordering” [8]. Study e.g. a bg gluon emitting a rg gluon thereby changing its own colour to br in a Lorentz frame where θ is small (see Fig 1a). In this case the red and antired charges screen each other in such a way that they do not give any emissions in directions with polar angles larger than θ, the angle of the ?rst emission. For these larger angles the two gluons emit softer gluons coherently as a single bg gluon, while for smaller angles the two gluons emit essentially independently. (For emission angles close to θ there is some azimuthal asymmetry.) This implies that in a frame where the red and antired charges move back-to-back, their contribution to the emission of softer gluons is just the normal emission from a separating charge-anticharge pair. The soft gluon interference (angular ordering) and a local cut-o? are incorporated in the Marchesini-Webber formalism for the parton cascade [17]. This formalism is implemented in the HERWIG Monte Carlo [18, 19]. In [18] it is also shown that a locally invariant cut-o? in the parton virtuality is essentially equivalent to a cut-o? in transverse momentum relative to the emitting parent parton, if this is measured in a Lorentz frame where the parton energies are large and the angles small. The HERWIG MC also contains a cluster fragmentation model with local properties. In this model the gluons are at the cut-o? level split into qq

3

pairs, which are combined to colourless clusters, which ?nally decay into hadrons. A cut-o? in k⊥ measured in the fast-moving (small angle) frame is approximately equivalent to a cut-o? with the same k⊥cut in the rest frame of the emitting charge-anticharge pair. In ⊥ this “dipole rest frame” the emission is proportional to dk⊥ dy, which for a given k⊥ means k a smooth rapidity distribution within the kinematically allowed region 1 s 2 |y| < ln(?/k⊥ ), 2 (1)

where s is the squared mass of the dipole. This phase space corresponds in the fastmoving ? frame just to the region allowed by angular ordering. We see e.g. that for given k⊥ , the rapidity range in the two jets is given by (for small angles θ) ?y1 + ?y2 = ln(2p1 /k⊥ ) ? ln(cot θ/2) + ln(2p2 /k⊥ ) ? ln(cot θ/2) ≈ (2)

2 ≈ ln(2p1 p2 (1 ? cos θ)/k⊥ ) = ln(?/k⊥ ), s 2

i.e. exactly the same result as in the dipole rest frame. Thus in a gluonic cascade as in Fig 1b the emission of softer gluons corresponds to a set of independent “dipole emissions”. A locally invariant cut-o? is obtained by a k⊥cut in the individual dipole rest frames. This is the basis of the Lund Colour Dipole Model [13], which is discussed in more detail in the following section. The local feature of the hadronization mechanism is also inherent in the Lund string fragmentation model [20]. Here the hadrons which originate from the colour ?eld stretched between a colour charge and its associated anticharge is independent of what happens further away in the system, with a correlation length corresponding to a few hadron masses [21]. In ref [22] a measure, called λ, is proposed, which in the string fragmentation model is strongly correlated to the hadron multiplicity. With a local cut-o? for the cascade, the λ-measure is also strongly correlated to the parton multiplicity [23]. We note that in both cluster and string fragmentation there is a connection between a colour charge and its associated anticharge. Thus, although the parton and hadron distributions are strongly correlated, they are not exactly identical. The colour coherence and the local properties of the hadronization process are fundamental features of the models implemented in the MC simulation programs HERWIG [19], ARIADNE [24] and JETSET/PYTHIA [25]. The great phenomenological success for these programs in describing experimental data, in particular for e+ e? -annihilation, is a strong support for the local features of the hadronization mechanisms expressed in precon?nement and LPHD. We note also that the independent jet fragmentation model, which does not have this feature, has not been able to describe the data in a satisfactory way. In spite of the phenomenological success mentioned above, there are still fundamental open questions concerning the hadronization mechanism. In the large Nc limit there is a unique way to connect the partons as in Fig 1b. This is, however, not the case when Nc = 3. If two gluons have identical colours the con?nement mechanism may connect partons which are not directly connected in the cascade generated by the simulation program. These “colour

4

2 reconnection” e?ects are suppressed by 1/Nc , and some possible consequences are discussed in [26, 27, 28]. No e?ects have, however, been observed so far in experimental data [11]. Closely related to this problem is the possibility of colour reconnection between partons from the decay of di?erent W :s in a W + W ? pair at LEP2 [28, 29]. This is of special interest as it might a?ect the W mass determination, but also here no statistically signi?cant e?ects have yet been found, e.g. in form of di?erent decay multiplicity or modi?ed Bose-Einstein correlations [30].

In perturbation theory it is possible to calculate the parton multiplicity within a given region of phase space. The conclusion of this section is that this is not the whole story if we want to calculate the multiplicity of hadrons. The hadronization e?ects of a parton containing e.g. a red colour charge depends upon where in phase space its partner antired charge is located, and this dependence is determined by the soft hadronization mechanism. An e?ective cut-o? depends on the local properties of a jet and cannot be determined as a ?xed phase space region valid for the whole jet. In [31] Ga?ney and Mueller calculate the ratio of the parton multiplicity in quark and gluon jets within a ?xed narrow cone, including correction terms of order αs (Q2 ) (or O(1/lnQ2 )). In section 4 we show that, assuming a local hadronization mechanism based on precon?nement and LPHD, we expect a further correction term which, although suppressed by a factor 1/lnQ2 , is numerically important.

3
3.1

The Colour Dipole Model
The Dipole Cascade

A high energy qq system radiates gluons with the distribution dn = CF αs x2 + x2 1 3 dx1 dx3 , 2π (1 ? x1 )(1 ? x3 ) (3)

2 where x1 and x3 are the scaled quark and antiquark momenta, and CF = 1 Nc (1 ? 1/Nc ). 2 With the de?nitions

α0 =

6 , 11 ? 2Nf /Nc

αq = α0 1 ?

2 k⊥ = s(1 ? x1 )(1 ? x3 ), 2 κ = ln(k⊥ /Λ2 ),

1 , 2 Nc 1 ? x3 1 , y = ln 2 1 ? x1 2 α0 Nc αs (k⊥ ) = , 2π κ

(4)

this distribution can be written dn = x2 + x2 αq αq 3 dκdy 1 ≈ dκdy, κ 2 κ (5)

where the last approximation holds for soft gluons.

5

κ L

a)

b)
(y1 , κ1) κ2

c)

y -L/2 L/2

Figure 2: a) The phase space for a gluon emitted from a qq dipole is a triangular region in 2 the (y,κ)-plane (κ = ln k⊥ /Λ2 , L = ln s/Λ2 ). b) After one emission at (y1 , κ1 ), the phase space for a second (softer) gluon is represented by this folded surface. c) Each emitted gluon increases the phase space for softer gluons. The total gluonic phase space corresponds to this multifaceted surface. The hadron multiplicity measure, λ(L), is given by the length of the baseline.

a)
r g r g

b)
r x3 g g

r b b x2

x1

Figure 3: a) An original two-gluon system corresponds to two colour dipoles. b) After one emission, the emission of softer gluons (with lower k⊥ ) corresponds to three dipoles. The kinematical constraint k⊥ < where L is given by √ s/(2 cosh y) ≈ √ s exp(?|y|) implies that κ + 2|y| < L, (6)

L = ln(s/Λ2 ).

Thus the allowed phase space for gluon emission is approximately a triangular region in the κ, y-plane, cf. Fig 2a. After the emission of a gluon at κ1 , y1 , the distribution for emissions of softer gluons corresponds to two independently emitting dipoles, one between the quark and the gluon, the other between the gluon and the antiquark. The available rapidity range 2 2 for a gluon at κ2 < κ1 is then ln(sqg /k⊥2 ) + ln(sgq /k⊥2 ) = L + κ1 ? 2κ2 . Thus the phase space for further emissions can be represented by a folded surface as in Fig 2b. This can be generalized for several emissions and a multi-gluon event corresponds to a picture with many folds and sub-folds as in Fig 2c. In this language an initial gg system corresponds to two dipoles. If the gluons are e.g. rg

6

B

A B A B C D E

E

C+D

Figure 4: After one gluon emission, the di?erent regions in the folded κ-y phase space approximately corresponds to the angular regions shown in the right ?gure. The angular directions in which the gluon and (anti)quark emits coherently approximately corresponds to region B, and the emission density there is proportional to CF . Corrections to these region identi?cations are discussed in section 6.2. and gr, we have a rr and a gg dipole as illustrated in Fig 3a. If one of these dipoles emits a gluon, the emission of softer gluons corresponds to three dipoles, as illustrated in Fig 3b, and after n emissions we get a closed chain of n + 2 dipoles. The phase space of an original gg system corresponds to two triangular regions as in Fig 2a, glued together along the outer diagonal lines. The emission from a dipole stretched between two gluons is however not exactly the same as from a dipole stretched between a quark and an antiquark. For a gg dipole the emission can in analogy to Eq (3) be described by the distribution [13] Nc αs x3 + x3 1 3 dn = dx1 dx3 . 2 2π (1 ? x1 )(1 ? x3 ) (7)

For soft and collinear emissions this goes over into the standard g→gg splitting function [13]. For soft emissions we have x3 + x3 ≈ 2, which implies 1 3 dn ≈ α0 dκdy. κ (8)

This result agrees with the soft emission from a qq dipole in Eq (3) apart from the colour suppressed di?erence between αq and α0 , i.e. between CF and Nc /2. After the ?rst emission in a qq dipole, the di?erent phase space regions for further emissions can be associated to di?erent angular regions. Region A of Fig 4 roughly corresponds to emissions with negative rapidity in the overall CMS-frame, and region B to particles with positive rapidity and a larger angle to the qq direction than the ?rst gluon. Emissions from region E have larger rapidity (smaller angle) than the ?rst gluon. This is also the case for region C + D, with the rapidity measured in the gluon direction. The ?rst emitted gluon and the (anti)quark will radiate coherently with the colour charge of the parent (anti)quark in region B. This argument can be generalized to a situation with several gluon emissions. For a cascade strongly ordered in k⊥ , this implies that the colour factor is CF in the original

7

?

Figure 5: The distribution P can be subdivided into distributions for di?erent rapidity intervals ?. qq phase space triangle and Nc /2 on all extra folds. The identi?cation of regions presented in Fig 4 is however only approximately true and we will in section 6.2 study the corrections and their consequences in more detail.

3.2

Multiplicity Distributions

Assuming LPHD, the hadron multiplicity N h is closely related with the parton multiplicity np . We will here brie?y describe how the parton distribution P (n = np , L = ln(s)) is derived in the dipole formulation [13], in order to more easily discuss the e?ects of recoil corrections in the next section. To ?nd the parton distribution P (n, L), we ?rst look at the distribution P? (n) in a small rapidity interval ?, c.f. Fig 5. The Laplace transform P? (γ) ≡ exp(?γn)P? (n)
n

(9)

has the property P?1 +?2 = P?1 P?2 . Thus lnP(γ, L) =
i

lnP?i (γ).

(10)

P? (n) depends both on the width and the height of the interval ?. We denote the phase space height at rapidity y by l(|y|) and de?ne R(γ, l) ≡ lim Eq (10) can then be written lnP(γ, L) =
ymax ?ymax L

lnP? (γ) . ?→0 ?

(11)

dyR(γ, l(|y|)) = 2

κc

dlR(γ, l)

dy . dl

(12)

In [22], it is shown that R also is related to P by d (i) αi (g) P (γ, L) ? 1 , R (γ, L) = dL L i = q, g, αg = α0 . (13)

8

cq=3/2

cg=11/6

q

q

g

g

Figure 6: The hyperbolic shape of the true phase space limits and the inequality x3 + x3 < 1 3 x2 + x2 < 2 are both neglected in the leading order result. This can be corrected for by 1 3 cutting o? a strip at the triangle edges, thus reducing the available phase space. The di?erent heights of the strips re?ects the di?erence between x3 + x3 and x2 + x2 , appearing in the 1 3 1 3 emission density for a gg– and qq dipole, respectively. The di?erent magnitudes of phase space reductions implies n(q) (L) ? n(g) (L + cg ? cq ). The boundary condition at some cut-o? scale κc for the cascade P? (n, κc ) = δn0 ? R(i) (γ, κc ) = 0, then implies R(g) (γ, L) = α0 (q) R (γ, L). αq
(i)

(14)

(15)

Combining Eq (12) and Eq (13) gives the following di?erential equation for P, valid in the LLA: αi (g) d2 P (γ, L) ? 1 . (16) lnP (i) (γ, L) = 2 dL L

3.3

MLLA Corrections

Since the approximate triangular (κ, y) region is somewhat larger than the true hyperbolic shape of the phase space, and the inequality x3 + x3 < x2 + x2 < 2 was neglected, the 1 3 1 3 emission density is overestimated in LLA (especially in a gg-dipole). In the √ Modi?ed Leading Logarithmic approximation (MLLA) [4], corrections of relative order 1/ L are included. √ In [14] it is shown, that evolution equations correct to relative order 1/ L are obtained if we maintain the approximation x3 + x3 ≈ x2 + x2 ≈ 2 and instead reduce the available phase 1 3 1 3 space by cutting out a strip at the edges, as illustrated in Fig 6. The height of the strip is cq = 3/2, cg = 11/6, (17)

for a qq- and gg-dipole, respectively. The constant phase space reduction modi?es |dy/dl| to 2 |dy/dl|(i) = Θ(L ? ci ? l), (18)

9

which implies d lnP (i) (γ, L) = αi R(i) (γ, L ? ci ), dL αi d2 P (g) (γ, L ? ci ) ? 1 . lnP (i) (γ, L) = 2 dL L ? ci (19) (20)

For the gluon case, the equations are modi?ed by the possibility of g→qq splittings [14]. We do not reproduce the algebraic details here, but refer to refs [4, 13]. Extracting moments in γ of Eq (20) leads to di?erential equations for n(g) and n(q) . Their asymptotic behaviours are n(q) ≈ (αq /α0 )n(g) ? Lρ exp(2 α0 L), (21) α0 1 3 ? (2Nf /Nc + 11) (22) 4 12 √ in MLLA. Thus the introduction of 1/ L-suppressed terms in the evolution equations changes the asymptotic behaviour of n(i) . ρ=
3 The contribution to ρ from the g→qq process is the term 2Nf /Nc in the parenthesis, which is clearly a small contribution compared to the other term 11. The contribution from this process to the ratio Nq /Ng is also very small. Using numerical calculations we have found that it only modi?es the ratio by less than 2% for all energies above 4 GeV. For this reason we will neglect the splitting process in the analytical calculations presented below. However, when we compare our analytic results with MC simulations, the latter will include also the g→qq process.

where

Neglecting the g→qq process for these reasons, we ?nd from Eq (20) for the ratio Nq /Ng the MLLA result α0 (q) n (L) ≈ n(g) (L + cg ? cq ), (23) αq √ which essentially is a 1/ L correction approaching the LLA result Eq (21) for large L. We also want to point out that the distribution in the multiplicity measure λ, mentioned in section 3.2, satisfy exactly the same evolution equations as the distribution in n. Therefore the asymptotic increase is also the same. The boundary conditions at threshold are however di?erent, which implies some deviations at lower energies.

3.4

Virtuality Scale Dependence

If we consider a two-jet event sample, picked from qq events using some jet resolution scale k⊥r , the mean multiplicity of the sample will be related to phase space area below κr ≡ 2ln(k⊥r /Λ). As shown in Fig 7, this implies n(q) (L, κ < κr ) ≡ n(q) (κr + cq ) + (L ? κr ? cq ) d (q) n (l + cq ) . dl l=κr (24)

10

a)

ci

b)
κr κr

α

γ

β

Figure 7: a) A two-scale dependent multiplicity can be studied in two-jet qq events, using a resolution κr = L. The multiplicity is related to the phase space below the cut-o? line at κr . The parton multiplicity is given by n(i) α+β = n(i) (κr + ci ), and n(i) γ = (L ? κr ? d ci ) dκr n(i) (κr +ci ). b) If the hardest gluon jet is found at κr , the phase space which determines the multiplicity is given by the same region α + β + γ, plus a fold. The multiplicity on this fold is given by n(g) (κr ). Here the ?rst term on the right hand side corresponds to the region α + β in Fig 7a, while the second term corresponds to the region γ. Thus the mean hadron multiplicity, N h , in qq events where no jet is resolved above κr is given by
h h Nqq (L, κ < κr ) = Nqq (κr + cq ) + (L ? κr ? cq )

d h N (κr + cq ). dκr qq

(25)

Events where the hardest gluon jet is found at κr corresponds to the phase space below the h horizontal line in Fig 7b. The hard gluon jet gives the contribution 1 Ngg (κr ) and therefore 2 1 h h h Nqq (L, κ = κr ) = Nqq (L, κ < κr ) + Ngg (κr ). 2 (26)

We note that while the κr dependence is fairly complicated, the L dependence is simply linear.

4

Recoil E?ects and Boundary Conditions

The ratio of the parton multiplicities in quark and gluon jets within a narrow cone was calculated including corrections to order αs , i.e. of order 1/L, in ref [31] (c.f. also ref [32]). The result is that the O(1/L) corrections are very small, only a few percent for energies above 30 Gev. In a parton cascade, Λ cannot be associated to a speci?c renormalization scheme, but must be treated as a free parameter. Changing Λ introduces terms of relative suppression 1/L in the evolution equations Eq (20). Within the language of parton cascades, it is therefore

11

non-trivial to systematically ?nd a complete set of 1/L suppressed correction terms to many observables. We note however that the 1/L terms induced by changing Λ are similar for the multiplicity in quark and gluon jets. Therefore these terms cancel to a large extent for the ratio Nq /Ng . This is in agreement with the result in ref [31], that the correction term of order αs to the ratio Nq /Ng is small and scheme independent. As discussed in detail in section 2, the observable hadrons are not directly determined by the partons within a ?xed phase space region. The hadrons originating from a particular parton colour charge depend also on the location in phase space of its corresponding anticharge. This implies that the necessary cut-o? in the parton cascade should depend upon the location in phase space of an associated colour anticharge. Such a dependence appears naturally in the dipole cascade formalism with a k⊥ -cut in the dipole restframe. Therefore the result can be sensitive to recoil e?ects experienced by the emitting parent partons. Assuming the local properties of the hadronization mechanism discussed in section 2, we will in the next subsection derive a correction term to the ratio Nq /Ng . Although formally of order 1/L, this recoil e?ect is numerically large, and important for comparisons between theory and experimental results. In section 7, we compare our results with data from OPAL [11] for jets with energy MZ /2. In section 6 we also discuss how to analyze LEP data at the Z pole in order to obtain the ratio Nq /Ng for a range of jet energies.

4.1

Recoil E?ects

As discussed in section 3.3 we will neglect the e?ect caused by the g→qq process. Numerical calculations have shown that this process, which is both colour suppressed and kinematically suppressed, only has a negligible in?uence on the result. In principle the splitting process could be included at the expense of more complicated expressions, which would make the result less transparent. Let us study the emission from an original gg-system. As discussed in section 3.1 this emission corresponds to two dipoles (a rr and a gg dipole in Fig 3a), and the phase space can be represented by two parallel triangular regions. If one gluon is emitted (e.g. from the gg dipole), the ensuing 3g state emits further soft gluons as three dipoles, as indicated in Fig 3b. As discussed in section 2, the emission from the separation between the b charge in the emitted gluon (2) and the b charge in the emitting (recoiling) gluon (1) is determined by the invariant mass s12 . The Landau-Pomeranchuk formation time for an emitted quantum is determined by the k⊥ of the emission. Thus there is a time ordering, such that a softer gluon (a gluon with smaller k⊥ ) is emitted after the establishment of the con?guration in Fig 3b. We must then expect that the weight of the rr and gg dipoles in Fig 3b are determined by the corresponding dipole masses. The squared mass of the “spectator” rr dipole is reduced from its initial value s to s13 = (1 ? x2 )s, which thus reduces the phase space for emissions from this dipole. We note that this recoil e?ect is very di?erent for a qq system, where the 2 weight of the corresponding dipole is negative, and suppressed by a factor 1/Nc . Therefore

12

recoils will a?ect the ratio between quark and gluon jets, and we will now estimate the size of this e?ect.
2 The density of gluons with squared transverse momentum k⊥ = s(1 ? x1 )(1 ? x3 ) emitted by e.g. the gg dipole in Fig 3a is given by Eq (7). This initial emission reduces the emission ′2 2 of softer gluons with k⊥ < k⊥ in the partner rr dipole, so that the rapidity range is reduced ′2 ′2 ′2 from ?y = lns/k⊥ to ?y = lns13 /k⊥ = lns/k⊥ + ln(1 ? x2 ). On average the reduction δy 2 caused by gluons within an interval dlnk⊥ ≡ dκ is then

δy = ? ≡

α0 dκ κ

dx1 dx3

α0 dκ · I(L ? κ), κ

x3 + x3 1 3 ln(1 ? x2 )δ(κ ? ln[s(1 ? x1 )(1 ? x3 )]) ≡ 2(1 ? x1 )(1 ? x3 ) (L = lns).

(27)

For a strongly ordered cascade we have L ? κ, and in this limit we ?nd I =2 π 2 49 ? 6 72 ≡ cr , L ? κ ? 1. (28)

Deviations from this limiting value give corrections which are suppressed at high energies. We have however also checked that the value in Eq (28) is a very good approximation in the whole relevant part of phase space. Typical values of L ? κ become smaller for smaller L–values, i.e. smaller dipole energies, due to the increase of αs . Within the MLLA approximation we ?nd that for L ? 6 the average value of κ becomes ≈ 2, i.e. k⊥ ≈ eΛ, which is very close to the standard cuto? in the ARIADNE cascade MC. Thus typical values of L ? κ are always larger than 4, and for these values I deviates from the value in Eq (28) by less than 10%. Half of the e?ect in Eq (27) corresponds to positive rapidities, in which case the recoil is taken essentially by the gluon (1) in Fig 3b, while in the other half the recoil is taken by gluon (3). The emissions in the rr dipole in Fig 3a has however a similar e?ect on the gg system, and the net result on one of the gluon jets is thus given by Eq (27). Once one gluon is emitted two smaller dipoles are formed (between the pairs 3-2 and 2-1 in Fig 3b) emitting softer gluons, which give similar reductions of the “spectator dipole” phase space. Further down the emission cascade the original gg dipole in Fig 3a corresponds to a chain of dipoles. Emissions in the two end links of this chain give recoil e?ects on the spectator system originating from the rr dipole in Fig 3a. Consequently we obtain a recoil e?ect described by Eq (27) for all values of κ, not only those which correspond to the ?rst 2 emission. Summing up the e?ects from all emissions above a given value l = lnk⊥ /Λ2 , and replacing I with its limiting value cr , gives the total phase space reduction δytot (l) =
L?cg l



L ? cg α0 cr = α0 cr ln . κ l

(29)

The upper limit is given by the kinematical limit in the MLLA approximation, L ? cg . Thus the e?ective rapidity range becomes ?ye? (l) = L ? l ? cg ? α0 cr ln L ? cg , l (30)

13

where cg is the MLLA correction and the last term is the recoil e?ect. We note in particular that the recoil term goes to zero at the kinematical limit l = L ? cg , as it should. From Eq (30) we ?nd d?ye? (l) α0 cr , (31) =1? dl l which should be inserted in into Eq (12) (with 2y replaced by ?ye? ). Doing so, and taking the MLLA modi?cation in Eq (18) into account, we ?nd instead of Eq (19) α0 cr d R(g) (γ, L ? cg ). lnP (g) (γ, L) = 1 ? dL L ? cg (32)

We note in particular that the result in Eq (31) is independent of L and that the correction in Eq (29) goes to zero in the kinematical limit. These two features imply that the structure of the evolution equation is unchanged when going from Eq (19) to Eq (32). For an initial qq system the emission from a quark colour charge is lower than one half of a 2 gluon charge, CF = Nc (1?1/Nc ). For the radiation from a qqg system, this can be expressed 2 2 as a negative contribution with relative weight ?1/Nc from a dipole stretched between the quark and antiquark, corresponding to the rr dipole in Fig 3b for the purely gluonic case. If we assume that we can treat this negative dipole in the same way, we would get the following result for a quark jet α0 c(q) d r R(q) (γ, L ? cq ), lnP (q) (γ, L) = 1 + dL L ? cq c(q) ≡ ? r 1 2 Nc
1 0

(33) (34)

dx

1 + (1 ? x)2 2 ln(1 ? x) = 2 x Nc

π2 5 . ? 6 8

Combining Eqs (32), (33) and (15) now leads to d α0 1 ? α0 c(g) /(L ? cq ) d r lnP (g) (γ, L + cg ? cq ) = lnP (q) (γ, L) ≈ (q) dL αq 1 + α0 cr /(L ? cq ) dL ≈ α0 cr α0 1? αq L d lnP (q) (γ, L), dL (35)

where cr now is modi?ed by approximately 10%, to cr ≡ c(g) + c(q) = r r 10 2 3 π ? 27 2 (36)

The negative dipole in the qqg system is caused by interference e?ects due to identical colour charges. As discussed in section 2, these interference e?ects are also connected to the possibility of “colour reconnection” in the hadronization process. As these problems are still not solved, we regard the recoil e?ect for the quark jet as uncertain. Since this contribution is only 10% of the total e?ect this is however not a serious problem here. Taking the ?rst moment in the Laplace transform variable γ, we ?nd the relation Eq (35) d d between dL n(g) and dL n(q) as well. Remembering that n(g) refers to a gluon jet while n(q)

14

refers to a qq system, the relations between multiplicities in two-parton systems should be given by α0 cr d h Nc d h 1? Ngg (L + cg ? cq ) = N (L). (37) dL CF L dL qq

4.2

Boundary Conditions for Hadron Multiplicities

The relation Eq (37) has to be supplemented by appropriate boundary conditions. Extrapolating Eq (37) down to too small values of L would imply that the hadron multiplicity in a qq system would be signi?cantly larger than that in a gg system. At low values of L, the hadron multiplicity is largely determined by the hadronic phase space and thus by the total available energy. This implies that at some threshold value L0 , we expect to have the relation h h Ngg (L0 ) ≈ Nqq (L0 ) ≡ N0 , (38) Here L0 , though larger than κc , should correspond to an energy of only a few GeV. The precise value of L0 strongly depends on non-perturbative QCD e?ects. At low energies, the fact that a qq string contains two quarks while a gg string does not, may in?uence h h the ratio Nqq /Ngg . Thus the value of L0 is sensitive to details in the fragmentation of low energy qq and gg systems, while N0 to a large extent depends on how the primarily produced hadrons decay. In principle L0 ought to be determined by experimental data from charmonium and bottonium decays. In our analysis, we have instead determined L0 and N0 from Monte Carlo simulations of the Lund String Fragmentation model, using the JETSET 7.4 computer program [25]. We then get L0 ? 5.7, corresponding to an energy h h E0 = Λ exp(L0 /2) ? 4GeV for Λ = 0.22GeV. Given Nqq and L0 from MC simulations, Ngg can be derived by numerical integration of the right hand side of Eq (37). The dependence on the threshold behaviour can however be avoided if one studies how the multiplicity varies with increasing energy. This possibility is further discussed in section 7.

5

Scale Dependences in Jets

h h In the following discussion we will use the notation of [5] and [14], i.e. Ng and Nq denote h h multiplicities in jets, while Ngg and Nqq refer to multiplicities in two-parton systems. We have already introduced L and κ as logarithmic energy– and virtuality scales for two-parton systems. The multiplicity in a jet j with energy Ej will be studied as a function of the logarithmic jet energy scale (39) Lj ≡ ln (2Ej )2 /Λ2 .

The jet energy is multiplied by a factor 2 for two “cosmetic” reasons: Then the scales coincide for a gluon jet when the simple condition Lj = κ (40)

15

~k⊥

Figure 8: The transverse resolution scale k⊥ between the softer jets in a three-jet event also speci?es the maximal allowed transverse momentum of unresolved particles within the jets. The multiplicity of the jets thus depends both on the jet energy and on k⊥ . is ful?lled. (I.e. the relevant energy scale of this gluon jet is Eg = k⊥ /2, which is the MLLA result presented in [5].) Furthermore, the multiplicity in a one-scale dependent jet is simply given by 1 h h Np (Lj ) = Npp (Lj ), p = q, g. (41) 2 In general, the multiplicity of a jet depends on two scales, energy and virtuality. Reducing the maximal allowed transverse momentum within a jet reduces the multiplicity, even if the energy remains constant (c.f. Fig 8). As discussed in section 3.4, the virtuality scale dependence of the multiplicity in quark jets can be easily studied in two-jet qq events varying the resolution scale. To compare quark and gluon jets we can use a three-jet event sample. The multiplicity in the jets will also in this sample depend both on the jet energy and the resolution scale k⊥r . Keeping k⊥r ?xed, the energy scale dependence can be studied [12]. However, the restriction implied by a ?xed k⊥r is rarely considered in theoretical predictions. Rather, most calculations apply to the very forward region of jets [5, 31]. There coherence e?ects are negligible, and energy-momentum conservation restricts k⊥ of emissions to such an extent that k⊥r introduces no new bias. We will here examine the possibility to study unbiased jets at moderate energies, using the broadest possible cones to de?ne the contents of the jets. To ?nd the jets of an event, cluster algorithms are used. In general, these contain a de?nition of a distance measure d between the jets, combine the two closest jets into one, and continues until all distances are above some resolution scale dcut . In CDM and other approaches [19], the transverse momentum speci?es the resolution. Thus it is appropriate to use a k⊥ -type of distance measure, as e.g. in the Durham [33], LUCLUS [25] or DICLUS [24] algorithms. We will use an approach, where jets are combined in the order given by the chosen algorithm until only three remain. The topology of the jets then speci?es the transverse momentum, k⊥ , of the event (c.f. Fig 8). This approach enables the de?nition of unbiased jets, whose evolution with k⊥ can be studied, while the energy scale dependence can be examined in an event sample with ?xed k⊥ . To construct unbiased jets, we note that a cone-like surface used to de?ne the forward contents of a jet corresponds to a perpendicular plane in some other frame. In this frame the jet resembles one hemisphere of a full event, and if the energy and virtuality scales in this frame are equal, the jet is unbiased. We will refer to this observation as the “One-Scale Criterion” (OSC), since the mean multiplicity of the jet in this case indeed only depends on

16

one scale – the energy of the corresponding full event.

6

Jet Algorithms

Soft gluon coherence can be approximated by an angular ordering (AO) constraint [8]. The multiplicity in a jet de?ned by AO will thus depend on one colour factor only. According to LLA, the jet will also be unbiased and depend on one single transverse energy scale, provided no extra cut-o? in transverse momentum is imposed by a ?xed jet resolution scale. We will present results from two algorithms based on this AO observation: The Cambridge algorithm presented in [9], and a previously undiscussed “Mercedes” algorithm, where the multiplicities of the jets are de?ned in the symmetric “Mercedes” Lorentz frame of the event. The main reason for studying both is to look for similarities, which point at general properties in the AO approximation. Corrections to the AO algorithms can be found by using the OSC explicitly. We will here discuss one such algorithm, which has similarities with the Cambridge algorithm, called the “Cone Exclusion” algorithm, and one corrected “Boost” algorithm where the jets are analyzed in a frame similar to the Mercedes one. The two OSC algorithms give the same gluon jets, but the de?nition of quark jets di?er. With the Cone Exclusion algorithm all three jets are unbiased, while the Boost algorithm constructs quark jets with well-de?ned but di?erent energy and virtuality scales. Thus the Boost algorithm is suitable for a study of the separate dependence of the two scales, while the Cone Exclusion algorithm is suitable for a comparison of quark jets vs. gluon jets.

6.1

Angular Ordering Algorithms

The Cambridge Algorithm The Cambridge algorithm is designed to construct gluon jets uncontaminated by coherently emitted particles. Thus the gluon jet properties depend only on the transverse momentum to the nearest harder jet, and on the gluon colour factor Nc . More speci?cally, the Durham k⊥ -distance 2 min{Ei2 , Ej } dij ≡ 2 (1 ? cos θij ) (42) 2 Evis is used to resolve jets, but the particles and sub-jets are merged in inverse angular order (those closest in angle are combined ?rst). Once a soft jet is resolved, it is “frozen out”, i.e. it gets no extra multiplicity contribution. Thus the contents of the soft jet is con?ned to a cone given by the smallest angle to any harder jet. This implies that the treatment of a gluon jet depends on wether its energy is higher or smaller than the quark jet energy. The Cambridge algorithm is therefore suitable for gluon jet analyses primarily when the hardest gluon jet is signi?cantly softer than the quark jets.

17

In the Cambridge algorithm, dcut not only determines when the clustering stops, but also at what stage di?erent jets are frozen out. Producing a ?xed number of jets is therefore not a completely trivial task. If we put dcut arti?cially large and pursue the clustering down to three jets, no freezing will occur. This would then correspond to a strict angular ordered clustering, similar to a cone algorithm. Fixing the number of jets is therefore better achieved by changing dcut in every event to a value which produces three jets. This procedure faces two problems: There may not be any dcut giving three jets, and when there is a large range of dcut values giving three jets, the multiplicities in the jets could depend on the choice of dcut . However, both of these situations occur very rarely (at percentage level), and reliable conclusions may therefore be drawn from events clustered by the Cambridge algorithm, varying dcut to ?x the number of jets to three. The Mercedes Algorithm In conventional cluster algorithms the bisectors between jets roughly separate the contents of them. In the speci?c case of a completely symmetric three-jet event, commonly referred to as a Mercedes event, such a jet de?nition will satisfy angular ordering [5]. The kinematical constraint implies a ?xed scale, but the scale evolution can still be studied by boosting a general three-jet event to its Mercedes frame. Since particles are in general shu?ed from one jet to another under a Lorentz transformation, the mass of a jet is not invariant. Instead the direction of a jet, corresponding to a parton in the cascade, approximately transforms as a light-like vector. Using a k⊥ -based algorithm to ?nd the jets, the Mercedes algorithm constructs gluon jets similar to those in the Cambridge algorithm.

6.2

OSC Algorithms

The Cone Exclusion Algorithm The “Cone Exclusion” (CE) method combines k⊥ - and angular distances in a way that has similarities with the Cambridge algorithm. After the construction of three jets using a k⊥ distance, a cone-like region is de?ned around the gluon jet. Only particles assigned to the jet that lie inside the region are then contributing to the multiplicity. Note that in spite of the name “Cone Exclusion”, the method is using a k⊥ -based cluster scheme to ?nd the jets. The “Cones” are used to assign soft particles to the correct jet. With this method it is simpler to ?x the number of jets to three than in the Cambridge algorithm, and it provides a better treatment of hard gluon jets. It is however speci?cally designed for studies of multiplicities in jets in three-jet events, and does not share the bene?ts of the Cambridge algorithm as compared to other k⊥ -algorithms in other respects. E.g., the Cambridge algorithm is designed to avoid the formation of “junk jets” (when soft particles from di?erent jets which happen to be close in phase space are combined and may be resolved as a jet if the resolution scale dcut is small). Since we in this analysis always pursue the clustering until only three jets remain, these “junk jets” will in general be absorbed into

18

a)
κr 1 2

1

y=0

2

b)

2

1
1 2

Figure 9: a) If the partons of a dipole are going apart back-to-back, particles from the triangular region of height κr in the direction of jet j all have a rapidity y > ln(2Ej /Λ) ? κr /2, which implies cot(θj /2) > 2Ej /k⊥r . This holds also if the dipole is not in its CMS. b) If the dipole is in a general frame, the triangular phase space areas corresponds to two “egg-shaped” regions which after a boost along the bisector to the back-to-back frame become the cones of ′ ′ ′ ′ (a), i.e. which satisfy cot(θj /2) > 2Ej /k⊥r , where θj and Ej are measured in the collinear frame. “proper jets”, and we are therefore not very sensitive to this problem addressed by the Cambridge algorithm. To ?nd the proper cones to use, we will start our discussion with two-jets events obtained with some resolution k⊥r . We look at the events in a frame where the original partons are going out back-to-back, though not necessarily in the overall CMS. In the κ, y phase space picture (with folds), an unbiased jet corresponds to a triangular region. When the partons of a dipole are moving apart back-to-back, the relation 4E1 E2 = s implies L = ln( 2E1 2E2 1 ) + ln( ) = (L1 + L2 ), Λ Λ 2 (43)

where L1 and L2 are the logarithmic jet energy scales in this frame. A particle p stemming from a triangular region of height κr = 2ln(k⊥r /Λ) in the direction of e.g. jet 2 has an angle θp2 to the jet given by (c.f. Fig 9a) θp2 1 ln cot2 2 2 =y> L2 ? κr 2E2 . = ln 2 k⊥r (44)

19

An equivalent expression for this constraint is
2 ln(sp1 ) ? ln(sp2 ) > ln(s12 ) ? ln(k⊥r ).

(45)

2 We now turn to three-jet events, where k⊥ = s(1 ? x1 )(1 ? x3 ) speci?es the resolution. Then all quantities in the constraint in Eq (45) are Lorentz invariant, and it can immediately be applied to the three jets in the CMS. For the qg dipole, the constraints are then

xg sin2 (θpg /2) < 1 ? xq xq sin2 (θpq /2) for the gluon jet and

(46)

xq sin2 (θpq /2) < 1 ? xq (47) xg sin2 (θpg /2) for the quark jet. Here θpq denote as before the angles between the hadron p and the jet direction q or g. The relations for the qg dipole are similar. For the q (or q) jet, Eq (47) speci?es an “egg-shaped” region, which after a boost along the bisector of the dipole to a back-to-back frame becomes a cone satisfying Eq (44) (c.f. Fig 9). The gluon jet is however attached to two dipoles, and CE thus speci?es two di?erent regions. In the directions being accepted by one dipole but not the other, most particles emerge from the “wrong” dipole and should not contribute to the multiplicity of the gluon jet. We have required particles belonging to the gluon jet to satisfy both possible restrictions from Eq (46). To summarize, the Cone Exclusion algorithm works as follows: Three jets are constructed, using a k⊥ -based cluster algorithm. A particle assigned to the gluon jet will then contribute to the multiplicity only if it satis?es Eq (46). The quark jets are treated similarly, using Eq (47). Thus the multiplicity in the forward region of every jet – corresponding to an unbiased jet – is studied, while soft central particles are simply ignored. The Boost Algorithm The OSC can also be used to improve the AO-based Lorentz transformation to the Mercedes frame. Consider a three-jet event boosted to a Lorentz frame where the angles are
′ ′ θqg = θqg ≡ θ′ ,

(48)

and the partons carry energies Ei′ . Let the bisectors de?ne the planes between di?erent jet regions. For the gluon, the logarithmic back-to-back jet energy scale from both dipoles is then 2E ′ θ′ Lg = 2ln( g sin ), (49) Λ 2 which coincides with the virtuality scale when Lg = κ. This implies x′g sin2 (θ′ /2) = (1 ? xq )(1 ? xq ),
2

2E ′ x′i ≡ √ i . s

(50)

20

The search for cone-like boundaries which de?ne an unbiased gluon jet has thus been reformulated to a search for a speci?c Lorentz frame where the jet regions are easily identi?ed. After a bit of algebra, one ?nds the general relation x′g sin2 (θ′ /2) = (1 ? xq )(1 ? xq ) Thus the requirement Eq (50) is satis?ed when cos2 (θ′ /2) = 1 ? xg . 4 (52)
2

4 cos2 (θ′ /2). 1 ? xg

(51)

In the soft gluon limit, the wanted Lorentz frame coincides with the Mercedes frame, with θ′ = 120?, but larger gluon energies give larger angles θ′ . We also note that the energy scale for the quark jet in this frame is x′q sin2 (θ′ /2) = and similarly for the q jet. The gluon jet de?ned in this way is actually equivalent to the one de?ned by the CE algorithm. If we combine Eq (50) and (53) into x′g /x′q = 1?xq and exploit the Lorentz invariance of the left hand side of Eq (46), we note that the CE condition for the gluon jet in Eq (46) ′ ′ can be rewritten as sin2 (θpg /2) < sin2 (θpq /2), which is just the bisector condition in the Boost algorithm. The treatment of massive particles, whose masses have been neglected in the discussion, di?er however in the two algorithms. Using both is a simple test of the sensitivity on particle masses. The de?nition of quark jets di?er in the two OSC algorithms. From Eq (53), we see that the quark jets in the Boost algorithm correspond to regions A + B and E in Fig 4. Thus the Boost algorithm provides means to study the two-scale dependent multiplicity for di?erent jet energy scales in a ?xed energy experiment, while the CE algorithm is better to use for the study of one-scale dependent multiplicities. To summarize, the Boost algorithm is as follows: Find three jets using a k⊥ -based cluster algorithm. Boost the event to the frame where the jet directions, assumed to be massless, satis?es Eq (48) and Eq (52). Let the jet boundaries be given by the bisectors to the other jets in this new frame and re-assign particles to the jets accordingly.
2

1 ? xq , 1 ? xq

(53)

7

Results

h h We have tested the analytic form for the ratio Nqq /Ngg in Eq (37) by comparing with multiplicity results from MC simulations and preliminary data [11]. Using MC simulations, we also examine the presented jet algorithms by comparing the multiplicities obtained in jets with the multiplicity of complete events at corresponding energies.

21

The CDM is available as a Monte Carlo simulation program, ARIADNE [24]. There distribution factors such as x2 + x2 are taken into account and energy conservation is obeyed. 1 3 Thus Monte Carlo simulations can show whether corrections other than presented above are needed to understand the model predictions on the scale evolution. As discussed in section 4, a qqg system can be regarded as three dipoles, where the qq dipole is colour suppressed and has a negative weight. Alternatively, the system can be described by two dipoles, qg and gq, where the colour factor transforms from CF in the q (q) end to Nc /2 in the gluon direction. However, dipoles with negative or non-uniform colour factors are ill suited for MC implementation. In the standard ARIADNE Monte Carlo, the solution ?2 is to neglect most terms of order Nc , using the colour factor Nc /2 in all gg– and qg dipoles. In [27], a modi?cation to the MC correcting for this approximation is presented. There non-uniform colour factors in dipoles are implemented, in a way re?ecting the discussion around Fig 4, where the emission density is assumed to be proportional to CF in all of the original qq phase space triangle, while Nc /2 applies to all extra phase space folds after gluon emissions. At moderate energies, the mean multiplicity is mostly determined by the hardest gluon. The emission of this gluon in the qq dipole is correctly given by CF in both MC versions. Thus, results from the two MC approaches are expected to deviate only at larger energies, showing 2 a relative discrepancy of at most 1/Nc . A ggg con?guration is well described by three dipoles, all with positive emission density, determined by Nc /2. This picture is implemented in default ARIADNE, which is used to obtain the simulation results from gg-systems presented below. In the simulations we have used the default tune of ARIADNE 4.08, with Λ = 0.22GeV. The parton con?gurations obtained in the cascade simulation are hadronized using the JETSET Monte Carlo [25], which is an implementation of the Lund String Fragmentation model.

7.1

h h The Multiplicity Ratio Nqq/Ngg

From MC results we ?nd the threshold energy where the multiplicity in quark and gluon jets are the same, to be given by L0 ? 5.7. This corresponds to a CMS energy of ? 4GeV. h L0 also speci?es N0 ≡ Nqq (L0 ).
h h In Fig 10a, the simulated ratio of Nqq /Ngg is compared to the prediction of Eq (37), using h values of N0 , L0 and Nqq (L) obtained from the MC. If the 1/L recoil correction in Eq (37) is √ neglected the result di?ers signi?cantly, even at L ? 20, i.e. at s ? 5TeV. The prediction √ is further modi?ed if the MLLA correction of order 1/ L in Eq (21) is neglected.

Fig 10b shows the di?erent multiplicity ratios obtained in simulations of e+ e? annihilation and pure dd events. At energies slightly above a heavy quark thresh-hold, the mean multiplicity gets a signi?cant √ contribution from isotropically decaying heavy hadrons. This is √ seen as a peak at L ? 6 ( s ? 2mD ) and a little shoulder starting at L ? 8 ( s ? 2mB ).

22

1

MLLA + recoil MLLA LLA Lo=5.7

ARIADNE colour factor corrected: dd e+edefault: e+eLo=6.7

Nqq/Ngg

0.5

a)
6 10 L 14 18

b)
6 10 L 14 18 22

1
(dNqq/dL)/(dNgg/dL)

0.5

c)
6 10 L 14 18

d)
6 10 L 14 18

Figure 10: Multiplicities ratios of qq and gg systems, based on all stable particles. qq samples are based on pure dd events (diamonds) and e+ e? annihilation, using ARIADNE default (boxes) and the modi?cation of [27] (crosses). The statistical errors for the MC results are h h within symbol sizes. We ?nd that Nqq = Ngg at L0 = 5.7. a) Comparison with the prediction h of Eq (37), using values of N0 , L0 and Nqq (L) obtained from the MC. If neglecting the recoil correction (dashed line) and also the scale shift cg ? cq (dotted line), √ prediction is far the from MC. Thus recoil corrections of order 1/L are essential, even at s ? 5TeV. b) For e+ e? events, cc and bb threshold e?ects are seen as a peak at L ? 6 and a shoulder starting at L ? 8. The relation determined by Eq (37) ?ts well with L0 = 6.7. The di?erence between default and modi?ed MC are important at L ? 20, but negligible at Z0 energies and below. c) d The dL N(L) ratio approaches the asymptotic value CF /Nc faster, but subleading corrections d cannot be neglected. d) The ratio of multiplicity derivatives dL N(L) is very similar for dd and e+ e? simulations and thus independent of L0 , as expected from Eq (37). The details of these threshold e?ects are beyond the scope of this paper. Instead we note √ that Eq (37) still ?ts well with L0 ? 6.7 ( s ? 6.3GeV). In Fig 10b it is also shown that h the corrections to Nqq from the Monte Carlo modi?cation of [27], which more consequently implements the di?erence between CF and Nc /2 after the ?rst gluon emission, are important at very high L, but negligible at Z0 energies (L = 12) and below. The ratio between the derivatives
d N(L) dL

is presented in Fig 10c. This ratio is expected

23

to approach the asymptotic value CF /Nc more rapidly, which is also con?rmed by data [12]. This is also born out in our analysis, and we note a good agreement between MC and the analytic form in Eq (37). According to this relation, the ratio of multiplicity derivatives d N(L) is expected to be independent of L0 . Fig 10d shows the similar results of dd and dL + ? e e simulations, con?rming this expectation.
h h In [11], the OPAL collaboration has studied the Ngg /Nqq ratio via quark and gluon hemispheres with energies E = MZ /2 (L = 12). Their preliminary result h Ngg (L = 12) = 1.509 ± 0.022(stat) ± 0.046(syst) h Nqq

(54)

is in excellent agreement with our result including recoil e?ects in Fig 10b,
h Nqq 1 (L = 12) = 0.67 = . h Ngg 1.5

(55)

In [11], OPAL also present preliminary data for the ratio of multiplicities in the central rapidity region,
h Ngg (L = 12, |y| < 2) = 1.815 ± 0.038(stat) ± 0.062(syst). h Nqq

(56)

For a reasonably small central rapidity range ?y, the multiplicity N(L, ?y) corresponds well d to ?y dL N(L). It is therefore interesting to compare this result with the analytical expression for the ratio of multiplicity derivatives, which is more independent of the boundary conditions at L0 . From Fig 10c, we ?nd
d Nh dL qq (L d Nh dL gg

= 12) = 0.547 =

1 , 1.83

(57)

with recoil e?ects included. The MLLA prediction without recoil e?ects is smaller than 1/2. Thus we conclude that MLLA calculations complemented with the recoil e?ect discussed in this paper are in very good agreement with the preliminary experimental data from OPAL. The recoil e?ect gives a sizeable correction which implies that the asymptotic value CF /Nc is far beyond reach in accelerator experiments.

7.2

Determination of Multiplicities in Jets

h The quantity Ngg in the previous subsection is the multiplicity in a gg event which is hard to realize and study directly in experiments. In section 6, di?erent jet algorithms designed to investigate this quantity in normal three-jet events are presented. We have tested their performance by simulating events at Z0 energies. The events are clustered into three jets and the multiplicities in the jets from di?erent algorithms are studied as a function of κ and y. The results for the jets are compared with the full-event results presented above.

24

20 Durham Cambridge Boost 0.5*Ngg(L=7.11)

10

Cambr. Mercedes Boost CE 0.5Ngg MC 6 κ 10

Ng( κ=7.11)
10 0

20

Ng(κ)

1

y

2

Figure 11: a) Multiplicities in gluon jets as a function of κ. Results for several values of y are plotted, or the range of results are represented with shaded areas. The AO algorithms h (Cambridge and Mercedes) have similar behaviours, and the multiplicity approaches 0.5Ngg for low virtualities. The high values of the Cambridge algorithm corresponds to large y-values, where the gluon jet is not the softest one. The CE and Boost algorithms both give results h independent of y which are in very good agreement with 0.5Ngg also for higher transverse momenta. b) Multiplicities in gluon jets as a function of y for ?xed κ. The Durham algorithm assigns coherently emitted particles to the gluon jet, and the multiplicity increases with y. The Cambridge result is more independent of y, as long as the gluon jet is the softest one. h h Ng of the Boost algorithms is very close to 0.5Ngg for all y.

All generated events are considered in the MC analysis. Neutrinos are excluded from the event, all other neutral particles are treated as massless while all charged particles are treated as pions. The obtained visible system in each event is boosted to its CMS before the analysis. To tag the gluon jet, we note that the algorithms are applicable to a general jet topology. It is therefore possible to study events where one jet is much softer than the other two, when it is a good approximation to assume the softest jet to be the gluon jet. This makes the analysis simple and independent of sophisticated tagging methods. The gluon jet is softest in the phase space region κ + 2|y| < L ? ln(4). In our “soft tag” analysis, we have restricted the phase space to κ + 2|y| < L ? 2ln(4), in order to avoid events with two similar soft jets. The restriction still allows us to study scales up to k⊥ ? 20GeV. In an experimental situation, harder gluon jets can be identi?ed using heavy quark information, why it is of interest to test the performance of the presented algorithms in a larger part of phase space. We have therefore also performed an analysis where the gluon jet is tagged using information available in the MC simulation, which however is experimentally non-observable. In this “angle tag” procedure, the jets are identi?ed with the partons in such a way that the sum of jet-parton angles are minimized.

25

10

6

κ

10

N(κ), hardest q-jet
5

Ng(κ)

Merc., angle tag soft tag CE, angle tag 20 soft tag 0.5Ngg

25

CE, angle tag soft tag 0.5Nqq

6

κ

10

Figure 12: Comparison of jet identi?cation methods. The cluster algorithms presented are designed to properly construct gluon jets in a large part of phase space. Assuming the gluon jet to be softest (symbols) restricts the available phase space to some extent, but the result is in very good agreement with more sophisticated gluon tagging methods (shaded areas).

Figure 13: Multiplicities in the hardest quark jet obtained by the Cone Exclusion method compared to half an e+ e? annihilation event. Results are plotted for several values of y. The range of results is given by a shaded region for the “angle tag”. A reliable analysis can be performed in the kinematical region where the gluon jet is predominantly the softest one.

Multiplicities in Gluon Jets

In Fig 11a, the multiplicity of the gluon jet obtained by di?erent algorithms is presented as a function of κ. For each κ, results for several values of y are plotted. The gluon jet was identi?ed with the “angle tag” method. The Angular Ordering algorithms (Cambridge and Mercedes) have similar behaviours. They perform well at low transverse momenta, but start to show a y-dependence at higher virtualities. This is especially so for the Mercedes algorithm. The OSC algorithms (CE and Boost) give results independent of y and in very h good agreement with 0.5Ngg . In Fig 11b, the multiplicity of the gluon jet for ?xed κ is presented as a function of gluon jet rapidity y. To allow the analysis to include large y, the “angle tag” method is used. With h h the Boost algorithm, Ng (κ) is independent of y and very close to the predicted 1 Ngg (κ). 2 The result from the Cambridge algorithm is independent of y in a large range, but somewhat 1 h larger than 2 Ngg . The steep rise of the Cambridge multiplicity at y ? 1.6 re?ects the di?erent treatment of the gluon jet when it is not the softest one. A conventional cluster algorithm – in this case the Durham algorithm – assigns particles from the region of coherent emission to the gluon jet. For large y this region becomes larger and the Durham multiplicity increases. In Fig 12 it is shown how the results using the very simple “soft tag” identi?cation of gluon jets is in very good agreement with more sophisticated methods.

26

44

60

a)
N(Lz,κ<κr) N(Lz,κ=κr)

b)

36 6

angle tag soft tag

40 6

angle tag soft tag

κr

10

κr

10

Figure 14: MC results of the multiplicity dependence on the jet resolution scale κr for e+ e? √ annihilation events with ?xed CMS energy s = MZ . The gluon jet is identi?ed both using the direction of the hardest gluon in the cascade and simply assuming the gluon jet to be softer than the quark jets. Solid lines are obtained using Eqs (25) and (26). a) The mean multiplicity in all events with no jet above κr b) The mean multiplicity in all events with the hardest gluon jet at κr . Multiplicities in Quark Jets In the Cone Exclusion algorithm, the OSC is used to de?ne one-scale dependent regions also for the quark and antiquark jet. The results for the hardest quark jet are presented in Fig 13. Again we note that jet mis-identi?cations in the “soft tag” method have little e?ect, and that this simple tagging procedure gives reliable results in a large kinematical region.

7.3

Two-Scale Dependence

The analysis required to compare data from ?xed energy experiments with model predictions of the virtuality scale dependence is simple and straightforward. In Fig 14 the mean multiplicity in events where no gluon jet is found using a resolution scale κr , and in events where the hardest gluon jet is found at κr are presented. Since the full multiplicity of a two-jet or three-jet event is independent of how the particles are distributed among the jets, any k⊥ -based algorithm may be used. There is a fair agreement between MC simulations using the Durham algorithm, and the expectations from Eqs (25) and (26). With the Mercedes and Boost algorithms, the multiplicities in quark jets are expected to h be 0.5Nqq (L + 2y, κ < κr ), where y and κr are the kinematical variables of the gluon jet. Thus the energy evolution of the two-scale dependence can be studied in a ?xed-energy experiment. The expected linear dependence in y (c.f. Eq (25)) is seen in Fig 15, especially for the OSC-based Boost algorithm.

27

30

Nq(Lz+2y, κ <7.11)

Mercedes Boost 0.5Nqq(Lz+2y, κ <7.11)

softest q-jet

hardest q-jet

10 -2 y -1 0 1 y 2

Figure 15: With the Mercedes and Boost algorithms, the energy scales of the quark jets are well de?ned and equals ln(s/Λ2 ) ± 2y, where the rapidity y of the gluon jet is de?ned by 1 y = 2 ln[(1 ? xq )/(1 ? xq )]. Thus it is possible to study the energy dependence of the two-scale dependent multiplicity. MLLA predicts this dependence to be linear, which also is the result for the Boost algorithm, while the Mercedes algorithm (based on AO) deviates somewhat at large |y|.

8

Summary

The ratio of the hadron multiplicity in quark and gluon jets is predicted to be 4/9 at very high energies. Corrections to this values calculated in the Modi?ed Leading Log Approximation (MLLA) cannot expalin the di?erence between experimental data and this asymptotic value. In this paper we estimate the contribution from recoil e?ects to the ratio Nq /Ng . Although formally of order 1/lns this contribution is quantitatively sizeable. Combined with the MLLA expression the result agrees well with MC simulations and with preliminary experimental data for jet energies equal to MZ /2 [10, 11]. We also discuss gluon jet de?nitions by which the scale evolution of gluon jets can be studied, using data from a ?xed energy e+ e? experiment. These jet de?nitions thus should enable an extended analysis which would be interesting to combine with the results of [11]. Another important point in this paper is that multiplicities in jets depend on two scales, the energy and the virtuality or maximal allowed transverse momentum. We derive expressions for the two-scale dependence and note that it can be examined with simple methods: Using a k⊥ -based cluster algorithm to construct exactly three jets, the multiplicities can be examined as a function of jet transverse momentum. We discuss the “One-Scale Criterion” (OSC), which states that a one-scale dependent jet will, in some Lorentz frame, correspond to one hemisphere of a two-parton event where the energy and transverse momentum scales coincide. The relevant scale for the jet is the energy of the corresponding hemisphere. Most of the commonly used cluster schemes of today concentrate on reproducing the jet energy and direction, and put less emphasis on the assignment of the

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soft particles to di?erent jets. One exception is the Cambridge algorithm [9], which based on Angular Ordering (AO) arguments constructs jets with one-scale dependent multiplicities. We present a set of algorithms designed to construct one-scale dependent jets. These are based either on AO or explicitly on the OSC. We examine the algorithms by analysing MC-generated events and comparing the obtained multiplicities in the jets with complete MC-simulated events at corresponding energies. Our study shows that all the presented algorithms perform well, but that the OSC methods are somewhat better than the AO ones for quantitative analyses of multiplicities in jets. With the OSC algorithms the gluon jet properties depend on only one scale. The treatment of quark jets di?er, however. The “Boost algorithm” is particularly suited for studies of the two-scale dependence of quark jets, while the “Cone Exclusion” algorithm is designed for a study of one-scale jets, where the virtuality coincides with the energy of the jet. We note that the algorithms can be used in a large kinematical region. We do not require the events to be Mercedes-like or Y-shaped. In events where one jet is signi?cantly softer than the others, this jet is predominantly the gluon jet. Performing the analysis on these events makes it less important to identify the gluon jet via quark taggings. Acknowledgments This work was supported in part by the EU Fourth Framework Programme ‘Training and Mobility of Researchers’, Network ‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’, contract FMRX-CT98-0194 (DG 12 - MIHT).

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