Experiments at LEP and SLAC revealed, besides other important results, quite interesting feature of the hadron multiple production dependent on the mass of the "primary" (anti)quarks which launch the process of the QCD evolution. It appeared that differences between the light and heavy quark-induced multiplicities become energy-independent. QCD calculations describe the phenomenon quite well.

Certainly, LEP could not give the information on the events induced by the top quarks. Recent discussions of the ILC project give us occasion to provide QCD predictions concerning the hadron multiple production in the events with primary t-quarks.

We manage to calculate the average hadron multiplicity Ntt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWG0baDcuWG0baDgaqeaaqabaaaaa@2F42@ in the e⁺e^- events with tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@ pair at the collision energy of the ILC with the following prediction:

N t t ¯ ( W = 500   GeV ) = 86.67 ± 0.55. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemiDaqNafmiDaqNbaebaaeqaaOGaeiikaGIaem4vaCLaeyypa0JaeGynauJaeGimaaJaeGimaaJaaGPaVlabbEeahjabbwgaLjabbAfawjabcMcaPiabg2da9iabiIda4iabiAda2iabc6caUiabiAda2iabiEda3iabgglaXkabicdaWiabc6caUiabiwda1iabiwda1iabc6caUaaa@46FE@

We also theoretically calculated the average hadronic multiplicity from the top quark:

n_t = 41.03 ± 0.27.

Both values correspond to the average value of the top mass m_t = 170.9. Everywhere below, it is assumed that we deal with average multiplicities of charged hadrons.

The paper is organized as follows. In order to make our calculations of the hadron multiplicity in top quark events more easy for understanding, we consider first the multiple hadron production in cc¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdogaJjqbdogaJzaaraaaaa@2DAD@ (bb¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkgaIjqbdkgaIzaaraaaaa@2DA9@) events. The hadron multiplicities in e⁺e^- events associated with the tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@-pair production are calculated in Section 3 in the framework of perturbative QCD. In Section 4 the numerical estimations and our main results are presented.

Hadron multiplicity in qq¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE5@ event, Nqq¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGXbqCcuWGXbqCgaqeaaqabaaaaa@2F36@(W), can be represented in the following general form 12:

N q q ¯ ( W ) = 2 n q + C F ∫ Q 0 2 W 2 d k 2 k 2 α s ( k 2 ) π n g ( k 2 ) E q ( k 2 W 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyCaeNafmyCaeNbaebaaeqaaOGaeiikaGIaem4vaCLaeiykaKIaeyypa0JaeGOmaiJaemOBa42aaSbaaSqaaiabdghaXbqabaGccqGHRaWkcqWGdbWqdaWgaaWcbaGaemOrayeabeaakmaapehabaqcfa4aaSaaaeaacqWGKbazcqWGRbWAdaahaaqabeaacqaIYaGmaaaabaGaem4AaS2aaWbaaeqabaGaeGOmaidaaaaadaWcaaqaaiabeg7aHnaaBaaabaGaem4CamhabeaacqGGOaakcqWGRbWAdaahaaqabeaacqaIYaGmaaGaeiykaKcabaGaeqiWdahaaOGaemOBa42aaSbaaSqaaiabdEgaNbqabaGccqGGOaakcqWGRbWAdaahaaWcbeqaaiabikdaYaaakiabcMcaPiabdweafnaaBaaaleaacqWGXbqCaeqaaOWaaeWaaKqbagaadaWcaaqaaiabdUgaRnaaCaaabeqaaiabikdaYaaaaeaacqWGxbWvdaahaaqabeaacqaIYaGmaaaaaaGccaGLOaGaayzkaaaaleaacqWGrbqudaqhaaadbaGaeGimaadabaGaeGOmaidaaaWcbaGaem4vaC1aaWbaaWqabeaacqaIYaGmaaaaniabgUIiYdGccqGGSaalaaa@637F@

where q means a type of quarks produced in the process of e⁺e^- annihilation into hadrons at the collision energy W. In what follows, the notation q = Q (heavy quark) will mean charm or beauty quark, while the notation q = l (light quark) will correspond to a massless case (when a pair of u, d or s-quarks is produced, whose masses are assumed to be equal to zero). The top quark production (q = t) will be studied in Sections 3 and 4.

The first term in the r.h.s. of Eq. (3), 2n_q, is the multiplicity of primary (anti)quark of the type q (i.e. the multiplicity from the leading hadron which contains this (anti)quark). It is taken from an analysis of the data (2n_c = 5.2, 2n_b = 11.1 4, and 2n_l = 2.4 5).

The quantity n_g(k²) in (3) is the mean multiplicity of the gluon jet with a virtuality k², for which we will take a QCD-based parametric form, with parameters fit to data, while E_q(k²/W²) is the inclusive spectrum of the gluon jet emitted by primary quarks. It was explained in detail in Ref. 1 that one should not consider this mechanism of hadron production via gluon jets as due to "a single cascading gluon". That quantity E(k²/W²) is an inclusive spectrum of the gluon jets is seen, e.g., from the fact that the average number of jets ∫dk²/k² E_q(k²/W²) ≠ 1.

Q₀ is a phenomenological parameter denoting the scale at which "precon-finement" of the off-shell partons occurs (as explained in Ref. 6).

Let us introduce variables

η = ln ⁡ W 2 k 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqaH3oaAcqGH9aqpcyGGSbaBcqGGUbGBjuaGdaWcaaqaaiabdEfaxnaaCaaabeqaaiabikdaYaaaaeaacqWGRbWAdaahaaqabeaacqaIYaGmaaaaaaaa@352A@

and

Y = ln ⁡ W 2 Q 0 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGzbqwcqGH9aqpcyGGSbaBcqGGUbGBjuaGdaWcaaqaaiabdEfaxnaaCaaabeqaaiabikdaYaaaaeaacqWGrbqudaqhaaqaaiabicdaWaqaaiabikdaYaaaaaGaeiilaWcaaa@3653@

as well as notation

n ^ g = C F α s ( k 2 ) π n g ( k 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacuWGUbGBgaqcamaaBaaaleaacqWGNbWzaeqaaOGaeyypa0tcfa4aaSaaaeaacqWGdbWqdaWgaaqaaiabdAeagbqabaGaeqySde2aaSbaaeaacqWGZbWCaeqaaiabcIcaOiabdUgaRnaaCaaabeqaaiabikdaYaaacqGGPaqkaeaacqaHapaCaaGccqWGUbGBdaWgaaWcbaGaem4zaCgabeaakiabcIcaOiabdUgaRnaaCaaaleqabaGaeGOmaidaaOGaeiykaKIaeiilaWcaaa@4268@

where C_F = (Nc2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaDaaaleaacqWGJbWyaeaacqaIYaGmaaaaaa@2E8A@ - 1)/2N_c, and N_c = 3 is a number of colors. Then Eq. (3) can be represented as

N q q ¯ ( Y ) = 2 n q + ∫ 0 Y d η n ^ g ( Y − η ) E q ( η ) ≡ 2 n q + N q ( Y ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyCaeNafmyCaeNbaebaaeqaaOGaeiikaGIaemywaKLaeiykaKIaeyypa0JaeGOmaiJaemOBa42aaSbaaSqaaiabdghaXbqabaGccqGHRaWkdaWdXbqaaiabdsgaKjabeE7aOjqbd6gaUzaajaWaaSbaaSqaaiabdEgaNbqabaGccqGGOaakcqWGzbqwcqGHsislcqaH3oaAcqGGPaqkcqWGfbqrdaWgaaWcbaGaemyCaehabeaakiabcIcaOiabeE7aOjabcMcaPiabggMi6kabikdaYiabd6gaUnaaBaaaleaacqWGXbqCaeqaaOGaey4kaSIaemOta40aaSbaaSqaaiabdghaXbqabaGccqGGOaakcqWGzbqwcqGGPaqkaSqaaiabicdaWaqaaiabdMfazbqdcqGHRiI8aOGaeiOla4caaa@5AD9@

In particular, Nll¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGSbaBcuWGSbaBgaqeaaqabaaaaa@2F22@(Y) means the multiplicity of hadrons in light quark events, while NQQ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGrbqucuWGrbqugaqeaaqabaaaaa@2EB6@(Y) denotes the multiplicity of hadrons in a process when a pair of the heavy quarks is produced.

The physical meaning of the function

N q ( Y ) = ∫ 0 Y d η n ^ g ( Y − η ) E q ( η ) ≡ ∫ 0 Y d η ′ n ^ g ( η ′ ) E q ( Y − η ′ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyCaehabeaakiabcIcaOiabdMfazjabcMcaPiabg2da9maapehabaGaemizaqMaeq4TdGMafmOBa4MbaKaadaWgaaWcbaGaem4zaCgabeaakiabcIcaOiabdMfazjabgkHiTiabeE7aOjabcMcaPiabdweafnaaBaaaleaacqWGXbqCaeqaaOGaeiikaGIaeq4TdGMaeiykaKIaeyyyIO7aa8qCaeaacqWGKbazcuaH3oaAgaqbaiqbd6gaUzaajaWaaSbaaSqaaiabdEgaNbqabaGccqGGOaakcuaH3oaAgaqbaiabcMcaPiabdweafnaaBaaaleaacqWGXbqCaeqaaOGaeiikaGIaemywaKLaeyOeI0Iafq4TdGMbauaacqGGPaqkaSqaaiabicdaWaqaaiabdMfazbqdcqGHRiI8aaWcbaGaeGimaadabaGaemywaKfaniabgUIiYdaaaa@5F4E@

is the following. It describes the average number of hadrons produced in virtual gluon jets emitted by the primary quark and antiquark of the type q. In other words, it is the multiplicity in qq¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaraaaaa@2DE5@ event except for multiplicity of the decay products of the primary quarks at the final stage of hadronization (the terms 2n_q in (7)).

For the massless case, the function E ≡ E_l was calculated in our paper 1. In terms of variable

σ = exp(-η),

it looks as

E [ η ( σ ) ] = ( 1 + 2 σ + 2 σ 2 ) ln ⁡ 1 σ − 3 + 7 σ 2 ( 1 − σ ) − σ ( 1 + σ ) ( ln ⁡ 1 σ ) 2 + 4 σ ( 1 + σ ) [ π 2 12 + ln ⁡ σ ln ⁡ ( 1 + σ ) + Li 2 ( − σ ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaGaemyrauKaei4waSLaeq4TdGMaeiikaGIaeq4WdmNaeiykaKIaeiyxa0Laeyypa0JaeiikaGIaeGymaeJaey4kaSIaeGOmaiJaeq4WdmNaey4kaSIaeGOmaiJaeq4Wdm3aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcyGGSbaBcqGGUbGBjuaGdaWcaaqaaiabigdaXaqaaiabeo8aZbaakiabgkHiTKqbaoaalaaabaGaeG4mamJaey4kaSIaeG4naCJaeq4WdmhabaGaeGOmaidaaOGaeiikaGIaeGymaeJaeyOeI0Iaeq4WdmNaeiykaKIaeyOeI0Iaeq4WdmNaeiikaGIaeGymaeJaey4kaSIaeq4WdmNaeiykaKYaaeWaaeaacyGGSbaBcqGGUbGBjuaGdaWcaaqaaiabigdaXaqaaiabeo8aZbaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaGcbaGaey4kaSIaeGinaqJaeq4WdmNaeiikaGIaeGymaeJaey4kaSIaeq4WdmNaeiykaKYaamWaaeaajuaGdaWcaaqaaiabec8aWnaaCaaabeqaaiabikdaYaaaaeaacqaIXaqmcqaIYaGmaaGccqGHRaWkcyGGSbaBcqGGUbGBcqaHdpWCcyGGSbaBcqGGUbGBcqGGOaakcqaIXaqmcqGHRaWkcqaHdpWCcqGGPaqkcqGHRaWkcqqGmbatcqqGPbqAdaWgaaWcbaGaeGOmaidabeaakiabcIcaOiabgkHiTiabeo8aZjabcMcaPaGaay5waiaaw2faaiabcYcaSaaaaaa@89DC@

where Li₂(z) is the Euler dilogarithm. The function E(η) is presented in Fig. 1. It has the asymptotics

E ( η ) | η → ∞ = η − 3 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaabcaqaaiabdweafjabcIcaOiabeE7aOjabcMcaPaGaayjcSdWaaSbaaSqaaiabeE7aOjabgkziUkabg6HiLcqabaGccqGH9aqpcqaH3oaAcqGHsisljuaGdaWcaaqaaiabiodaZaqaaiabikdaYaaakiabc6caUaaa@3CAE@

The derivative of E(η) is positive, and ∂E(η)/∂η = 0 at η = 0. As a result, the associated multiplicity N_q(W) (8) is a monotonic increasing function of the energy W for any positive function n_g(k²) since

∂ N l ( Y ) ∂ Y = ∫ 0 Y d η n ^ g ( η ) ∂ E ( Y − η ) ∂ Y . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabgkGi2kabd6eaonaaBaaabaGaemiBaWgabeaacqGGOaakcqWGzbqwcqGGPaqkaeaacqGHciITcqWGzbqwaaGccqGH9aqpdaWdXaqaaiabdsgaKjabeE7aOjqbd6gaUzaajaWaaSbaaSqaaiabdEgaNbqabaGccqGGOaakcqaH3oaAcqGGPaqkjuaGdaWcaaqaaiabgkGi2kabdweafjabcIcaOiabdMfazjabgkHiTiabeE7aOjabcMcaPaqaaiabgkGi2kabdMfazbaakiabc6caUaWcbaGaeGimaadabaGaemywaKfaniabgUIiYdaaaa@4F7A@

Now consider the multiplicity difference in events with the light and heavy flavors (Q = c or b):

δ Q l = N Q Q ¯ − N l l ¯ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqaH0oazdaWgaaWcbaGaemyuaeLaemiBaWgabeaakiabg2da9iabd6eaonaaBaaaleaacqWGrbqucuWGrbqugaqeaaqabaGccqGHsislcqWGobGtdaWgaaWcbaGaemiBaWMafmiBaWMbaebaaeqaaOGaeiOla4caaa@3992@

At W ≫ m_Q, one can neglect small power-like corrections O(m²/W²). In such a case, the quantity δ_Ql is defined by 1

δ_Ql = 2(n_Q - n_l) - ΔN_Q(Y_Q),

where the notation

Δ N Q ( Y Q ) = N l − N Q = ∫ − ∞ Y Q d y n ^ g ( Y Q − y ) Δ E Q ( y ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@569A@

as well as variables

y = ln ⁡ m Q 2 k 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWG5bqEcqGH9aqpcyGGSbaBcqGGUbGBjuaGdaWcaaqaaiabd2gaTnaaDaaabaGaemyuaefabaGaeGOmaidaaaqaaiabdUgaRnaaCaaabeqaaiabikdaYaaaaaaaaa@3650@

and

Y Q = ln ⁡ m Q 2 Q 0 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGzbqwdaWgaaWcbaGaemyuaefabeaakiabg2da9iGbcYgaSjabc6gaULqbaoaalaaabaGaemyBa02aa0baaeaacqWGrbquaeaacqaIYaGmaaaabaGaemyuae1aa0baaeaacqaIWaamaeaacqaIYaGmaaaaaaaa@382B@

are introduced. The lower limit of integration in Eq. (15), -ln(W²/mQ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGrbquaeaacqaIYaGmaaaaaa@2EA4@), is taken -∞ because of the fast convergence of the integral at negative y.

Let us use another dimensionless variable

ρ = exp(-y).

The explicit form of ΔE_Q was derived in our paper 1 (see Fig. 2):

Δ E Q [ y ( ρ ) ] = [ 1 + ρ ( 7 2 ρ − 3 ) ln ⁡ 1 ρ + ( 9 2 + 7 ρ ) + ρ ( 7 ρ − 20 ) J ( ρ ) + 20 1 − J ( ρ ) ρ − 4 ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6F87@

where

J ( ρ ) = { ρ ρ − 4 ln ⁡ ( ρ + ρ − 4 2 ) , ρ > 4 , 1 , ρ = 4 , ρ 4 − ρ arctan ⁡ ( 4 − ρ ρ ) , ρ < 4. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGkbGscqGGOaakcqaHbpGCcqGGPaqkcqGH9aqpdaGabaqaauaabaqadiaaaeaadaGcaaqcfayaamaalaaabaGaeqyWdihabaGaeqyWdiNaeyOeI0IaeGinaqdaaaWcbeaakiGbcYgaSjabc6gaUnaabmaajuaGbaWaaSaaaeaadaGcaaqaaiabeg8aYbqabaGaey4kaSYaaOaaaeaacqaHbpGCcqGHsislcqaI0aanaeqaaaqaaiabikdaYaaaaOGaayjkaiaawMcaaiabcYcaSaqaaiabeg8aYjabg6da+iabisda0iabcYcaSaqaaiabigdaXiabcYcaSaqaaiabeg8aYjabg2da9iabisda0iabcYcaSaqaamaakaaajuaGbaWaaSaaaeaacqaHbpGCaeaacqaI0aancqGHsislcqaHbpGCaaaaleqaaOGagiyyaeMaeiOCaiNaei4yamMaeiiDaqNaeiyyaeMaeiOBa42aaeWaaKqbagaadaWcaaqaamaakaaabaGaeGinaqJaeyOeI0IaeqyWdihabeaaaeaacqaHbpGCaaaakiaawIcacaGLPaaacqGGSaalaeaacqaHbpGCcqGH8aapcqaI0aancqGGUaGlaaaacaGL7baaaaa@6AE1@

The function ΔE_Q(y) decreases at y → -∞ (ρ → ∞) as

Δ E Q ( y ) | y → − ∞ ≃ 11 3 e − | y | , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaabcaqaaiabfs5aejabdweafnaaBaaaleaacqWGrbquaeqaaOGaeiikaGIaemyEaKNaeiykaKcacaGLiWoadaWgaaWcbaGaemyEaKNaeyOKH4QaeyOeI0IaeyOhIukabeaakiabloKi7KqbaoaalaaabaGaeGymaeJaeGymaedabaGaeG4mamdaaOGaemyzau2aaWbaaSqabeaacqGHsisldaabdaqaaiabdMha5bGaay5bSlaawIa7aaaakiabcYcaSaaa@4590@

and has the following asymptotics at y → ∞ (ρ → 0):

Δ E Q ( y ) | y → ∞ ≃ y − 1 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaabcaqaaiabfs5aejabdweafnaaBaaaleaacqWGrbquaeqaaOGaeiikaGIaemyEaKNaeiykaKcacaGLiWoadaWgaaWcbaGaemyEaKNaeyOKH4QaeyOhIukabeaakiabloKi7iabdMha5jabgkHiTKqbaoaalaaabaGaeGymaedabaGaeGOmaidaaiabc6caUaaa@3EFF@

Thus, we get the relation between average multiplicities of hadrons in QQ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdgfarjqbdgfarzaaraaaaa@2D65@ and ll¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdYgaSjqbdYgaSzaaraaaaa@2DD1@ events:

N Q Q ¯ ( W ) = N l l ¯ ( W ) − Δ N Q ( m Q ) + 2 ( n Q − n l ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyuaeLafmyuaeLbaebaaeqaaOGaeiikaGIaem4vaCLaeiykaKIaeyypa0JaemOta40aaSbaaSqaaiabdYgaSjqbdYgaSzaaraaabeaakiabcIcaOiabdEfaxjabcMcaPiabgkHiTiabfs5aejabd6eaonaaBaaaleaacqWGrbquaeqaaOGaeiikaGIaemyBa02aaSbaaSqaaiabdgfarbqabaGccqGGPaqkcqGHRaWkcqaIYaGmcqGGOaakcqWGUbGBdaWgaaWcbaGaemyuaefabeaakiabgkHiTiabd6gaUnaaBaaaleaacqWGSbaBaeqaaOGaeiykaKIaeiilaWcaaa@4D90@

with the multiplicity difference ΔN_Q defined by Eq. (15).

Our calculations 1 of the multiplicity differences δ_Ql = Nll¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGSbaBcuWGSbaBgaqeaaqabaaaaa@2F22@ - NQQ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGrbqucuWGrbqugaqeaaqabaaaaa@2EB6@ (Q = b, c) with the use of formula (23) appeared to be in a good agreement with the data. Recently we have reconsidered the QCD upper limit on quantity δ_bl 3 which appeared to be very close to all present experimental data on δ_bl. The data on N_ll as well as on δ_bl at different energies corrected for detector effects as well as for initial state radiation were recently cited in 7.

The goal of this paper is to calculate Ntt¯h MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaDaaaleaacqWG0baDcuWG0baDgaqeaaqaaiabdIgaObaaaaa@309C@, the average multiplicity of hadrons produced in e⁺e^- events with the primary tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@ pair. We consider the case when the top (antitop) decay mode is pure hadronic. As a byproduct, we will calculate n_t, the hadron multiplicity of the on-shell top decay products.

We will assume that the square of the matrix element of the process e⁺e^- → t∗t¯∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0naaCaaaleqabaGaey4fIOcaaOGafmiDaqNbaebadaahaaWcbeqaaiabgEHiQaaaaaa@3033@ → X is factorized as follows:

| M ( e + e − → t ∗ t ¯ ∗ → hadrons ) | 2 = | M ( e + e − → t ∗ t ¯ ∗ → t t ¯ + hadrons ) | 2 × | M ( t → hadrons ) | 2 × | M ( t ¯ → hadrons ) | 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeWadaaabaGaeiiFaWNaemyta0KaeiikaGIaemyzau2aaWbaaSqabeaacqGHRaWkaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTaaakiabgkziUkabdsha0naaCaaaleqabaGaey4fIOcaaOGafmiDaqNbaebadaahaaWcbeqaaiabgEHiQaaakiabgkziUkabbIgaOjabbggaHjabbsgaKjabbkhaYjabb+gaVjabb6gaUjabbohaZjabcMcaPiabcYha8naaCaaaleqabaGaeGOmaidaaaGcbaGaeyypa0dabaGaeiiFaWNaemyta0KaeiikaGIaemyzau2aaWbaaSqabeaacqGHRaWkaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTaaakiabgkziUkabdsha0naaCaaaleqabaGaey4fIOcaaOGafmiDaqNbaebadaahaaWcbeqaaiabgEHiQaaakiabgkziUkabdsha0jqbdsha0zaaraGaey4kaSIaeeiAaGMaeeyyaeMaeeizaqMaeeOCaiNaee4Ba8MaeeOBa4Maee4CamNaeiykaKIaeiiFaW3aaWbaaSqabeaacqaIYaGmaaaakeaaaeaacqGHxdaTaeaacqGG8baFcqWGnbqtcqGGOaakcqWG0baDcqGHsgIRcqqGObaAcqqGHbqycqqGKbazcqqGYbGCcqqGVbWBcqqGUbGBcqqGZbWCcqGGPaqkcqGG8baFdaahaaWcbeqaaiabikdaYaaaaOqaaaqaaiabgEna0cqaaiabcYha8jabd2eanjabcIcaOiqbdsha0zaaraGaeyOKH4QaeeiAaGMaeeyyaeMaeeizaqMaeeOCaiNaee4Ba8MaeeOBa4Maee4CamNaeiykaKIaeiiFaW3aaWbaaSqabeaacqaIYaGmaaGccqGGSaalaaaaaa@996A@

where t∗(t¯∗) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0naaCaaaleqabaGaey4fIOcaaOGaeiikaGIafmiDaqNbaebadaahaaWcbeqaaiabgEHiQaaakiabcMcaPaaa@31EF@ denotes the virtual top quark (antiquark).

The factorization of the matrix element (24) means that there is no significant space-time overlap in the decay products of the on-shell t and t¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdsha0zaaraaaaa@2C80@-quarks. Note that the off-shell t and t¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdsha0zaaraaaaa@2C80@-quarks fragment into hadrons through the emission of the gluon jets in a coherent way (the first term in the r.h.s of Eq. (24)). The QCD non-singlet evolution of the primary virtual t-quark is very slow because the difference of virtualities in logarithmic scale is very small down to the top quark mass. In other words, the virtual t-quark becomes "real" after just a few gluon radiations.

The effect of possible color reconnection was investigated by comparing hadronic multiplicities in e⁺e^- → W⁺W^- → qq¯′qq¯′ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaryaafaGaemyCaeNafmyCaeNbaeHbauaaaaa@30E9@ and e⁺e^-→ W⁺W^- → qq¯′lν¯l MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdghaXjqbdghaXzaaryaafaGaemiBaWMafqyVd4MbaebadaWgaaWcbaGaemiBaWgabeaaaaa@32AE@ events. No evidence for final state interactions was found by measuring the difference 〈n4qh〉−2〈n2qlν¯h〉 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabgMYiHlabd6gaUnaaDaaaleaacqaI0aancqWGXbqCaeaacqWGObaAaaGccqGHQms8cqGHsislcqaIYaGmcqGHPms4cqWGUbGBdaqhaaWcbaGaeGOmaiJaemyCaeNaemiBaWMafqyVd4MbaebaaeaacqWGObaAaaGccqGHQms8aaa@41B5@89. From the space-time point of view W bosons and t-quarks behave in a similar way, ie. the latter manage to cover the distance Δ_l ~ 1/Γ_t, where Γ_t is the full width of the top. Since Γ_t ≃ Γ_w, we expect no interference effects in the decays of the on-shell t and t¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdsha0zaaraaaaa@2C80@-quarks.

According to Eq. (24), the associative multiplicity in tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@-event is given by the formula:

N t t ¯ h ( W , m t ) = N t ( W , m t ) + 2 n t , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaqhaaWcbaGaemiDaqNafmiDaqNbaebaaeaacqWGObaAaaGccqGGOaakcqWGxbWvcqGGSaalcqWGTbqBdaWgaaWcbaGaemiDaqhabeaakiabcMcaPiabg2da9iabd6eaonaaBaaaleaacqWG0baDaeqaaOGaeiikaGIaem4vaCLaeiilaWIaemyBa02aaSbaaSqaaiabdsha0bqabaGccqGGPaqkcqGHRaWkcqaIYaGmcqWGUbGBdaWgaaWcbaGaemiDaqhabeaakiabcYcaSaaa@473D@

where

N t ( W , m t ) = C F ∫ Q 0 2 ( W − 2 m t ) 2 d k 2 k 2 α s ( k 2 ) π n g ( k 2 ) E t ( W 2 , k 2 , m t 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6D21@

Here and in what follows we will assume that the collision energy is a typical ILC energy, W = 500 GeV, for definiteness. In such a case, contrary to Eq. (8), power corrections O(m_t/W) should be taken into account. The explicit form of the inclusive distribution of the gluon jets with the invariant mass k2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaakaaabaGaem4AaS2aaWbaaSqabeaacqaIYaGmaaaabeaaaaa@2D85@ looks like

E t ( q 2 , k 2 , m 2 ) = ( − k ∂ ∂ k ) ∫ 1 A d η { [ 1 η