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=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@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=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@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=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@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=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@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=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@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 ( q 2 + k 2 ) 2 − 4 m 4 q 4 − 2 k q q 2 + k 2 + 2 m 2 q 2 + 2 η k 2 q 2 ] ln ⁡ [ η + η 2 − 1 ( A − η ) / ( A 0 − η ) η − η 2 − 1 ( A − η ) / ( A 0 − η ) ] − 2 k 2 q 2 η 2 − 1 ( A − η ) ( A 0 − η ) × [ 1 + ( 1 + 2 m 2 / q 2 ) ( 1 + 2 m 2 / k 2 ) η 2 − ( η 2 − 1 ) ( A − η ) / ( A 0 − η ) ] } , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@F3AB@

where k≡k2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdUgaRjabggMi6oaakaaabaGaem4AaS2aaWbaaSqabeaacqaIYaGmaaaabeaaaaa@30AD@, q≡q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdghaXjabggMi6oaakaaabaGaemyCae3aaWbaaSqabeaacqaIYaGmaaaabeaaaaa@30C5@, and the following notations are introduced:

A = q 2 + k 2 − 4 m 2 2 q k , A 0 = q 2 + k 2 2 q k . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemyqaeKaeyypa0tcfa4aaSaaaeaacqWGXbqCdaahaaqabeaacqaIYaGmaaGaey4kaSIaem4AaS2aaWbaaeqabaGaeGOmaidaaiabgkHiTiabisda0iabd2gaTnaaCaaabeqaaiabikdaYaaaaeaacqaIYaGmcqWGXbqCcqWGRbWAaaGccqGGSaalaeaacqWGbbqqdaWgaaWcbaGaeGimaadabeaakiabg2da9KqbaoaalaaabaGaemyCae3aaWbaaeqabaGaeGOmaidaaiabgUcaRiabdUgaRnaaCaaabeqaaiabikdaYaaaaeaacqaIYaGmcqWGXbqCcqWGRbWAaaGaeiOla4caaaaa@4A2D@

This formula has been derived by calculating QCD diagrams in the first order in the strong coupling constant (see our comments after Eq. (3)). In the massless case (m = 0), we immediately come to the function E(k²/q²) (10), while by neglecting small corrections O(m²/q²), one can derive (after variables are properly changed) the explicit form of the function ΔE_Q(k²/m²) = E_Q(q², k², m²) - E(q², k²) (19). In our case (q² = W², m = m_t) we will estimate the integrals in Eqs. (26), (27) numerically (for details, see Section 4).

Now let us calculate another quantity in Eq. (25), n_t, which describes the hadronic multiplicity of the t-quark decay products. The top weakly decays into W⁺ boson and b-quark. In its turn, the W⁺ boson decays into a quark-antiquark pair. Remember that we are interested in hadronic decays of the W boson. The quark-antiquark system results in massive jets which fragment into hadrons (see Fig. 3).

The gluon jets can be also emitted by the on-shell t-quark before its weak decay (the first diagram in Fig. 4) or by off-shell bottom quark (the second diagram in Fig. 4). At the end of these emissions, the on-shell b-quark weakly decays into hadrons whose average multiplicity is equal to n_b. Since W-boson is a colorless particle, the diagrams in Fig. 4 do not interfere with those presented in Fig. 3.

Thus, the multiplicity n_t is a sum of three terms:

n_t = n_W + n_tb + n_b.

The quantity n_bis experimentally measurable one 4. The first term in Eq. (29), n_W, is the hadron multiplicity of the W boson decay products. The second term, n_tb, is the hadron multiplicity in the gluon jets emitted by the on-shell top quark before its weak decay as well as by the bottom quark after the top decay.

In order to estimate the multiplicity of the decay products of the top (formula (47)), one has to know the energy dependence of the hadron multiplicity in the light quark event. The latter is defined by Eq. (7), where the function n^g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbd6gaUzaajaWaaSbaaSqaaiabdEgaNbqabaaaaa@2DEF@ is related with gluon jet multiplicity n_g(k²) (6). We have fitted the data on Nll¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGSbaBcuWGSbaBgaqeaaqabaaaaa@2F22@(W) by using the following QCD-motivated expression for n_g(k²):

n g ( k 2 ) = a + b exp ⁡ [ c ln ⁡ ( k 2 / Q 0 2 ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGUbGBdaWgaaWcbaGaem4zaCgabeaakiabcIcaOiabdUgaRnaaCaaaleqabaGaeGOmaidaaOGaeiykaKIaeyypa0JaemyyaeMaey4kaSIaemOyaiMagiyzauMaeiiEaGNaeiiCaa3aamWaaeaacqWGJbWydaGcaaqaaiGbcYgaSjabc6gaUjabcIcaOiabdUgaRnaaCaaaleqabaGaeGOmaidaaOGaei4la8Iaemyuae1aa0baaSqaaiabicdaWaqaaiabikdaYaaakiabcMcaPaWcbeaaaOGaay5waiaaw2faaiabcYcaSaaa@49A4@

where c = 1.63, and k² is the invariant mass of the jet. We have got the following values of the parameters:

a = 3.89,   b = 0.01,   Q₀ = 0.87 GeV.

The result of our fit is presented in Fig. 8 in comparison with the data. Note that χ²/d.o.f. becomes twice smaller if the experimental point at W = 58 GeV (open circle in Fig. 8), which lies much lower than neighboring points, is crossed out from the fit. In such a case, the values of the parameters are practically the same as in (51) with χ²/d.o.f. = 0.90.

For our numerical estimates we shall use m_W = 80.40 ± 0.03 GeV 11 and the recent value of the top mass 10:

m_t = 170.9 ± 1.8 GeV.

As for the bottom quark, its pole mass is quoted in 11 to be m_b = 4.7 – 5.0 GeV. We will take the average value

m_b = 4.85 ± 0.15 GeV.

By using our fit, we obtain Nll¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWGSbaBcuWGSbaBgaqeaaqabaaaaa@2F22@(m_W) = 19.09. Then we get (see Eqs. (40), (49), (41)):

n_W = 19.34 ± 0.10,

n_tb = 16.14 ± 0.24.

The error in Eq. (54) is defined by that of the multiplicity difference δ_cl (49), while that in Eq. (55) comes from uncertainties of the quark masses m_t (52) and m_b (53).

Our result (54) is in a nice agreement with the experimental values from Ref. 8,

n_W = 19.3 ± 0.3 ± 0.3,

and Ref. 9,

n_W = 19.44 ± 0.13 ± 0.12.

Now let us calculate the associated hadron multiplicity in tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@ event (48) at fixed energy W = 500 GeV. To do this, we need to estimate the multiplicity N_t(W, m_t) by using formulae (26) and (27):

N_t(W = 500 GeV) = 4.61 ± 0.11.

The errors in (58) come from top quark mass errors. It follows From Eqs. (61), (58) that Ntt¯(e+e−→tt¯→hadrons) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWG0baDcuWG0baDgaqeaaqabaGccqGGOaakcqWGLbqzdaahaaWcbeqaaiabgUcaRaaakiabdwgaLnaaCaaaleqabaGaeyOeI0caaOGaeyOKH4QaemiDaqNafmiDaqNbaebacqGHsgIRcqqGObaAcqqGHbqycqqGKbazcqqGYbGCcqqGVbWBcqqGUbGBcqqGZbWCcqGGPaqkaaa@4644@ = 86.67 ± 0.55.

In order to estimate possible theoretical uncertainties, we have repeated our calculations taking the different form of the average multiplicity of the gluon jet (compare with the QCD-based expression (50)):

n ^ g ( k 2 ) = A + B ln ⁡ 2 k 2 Q 0 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacuWGUbGBgaqcamaaBaaaleaacqWGNbWzaeqaaOGaeiikaGIaem4AaS2aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGH9aqpcqWGbbqqcqGHRaWkcqWGcbGqcyGGSbaBcqGGUbGBdaahaaWcbeqaaiabikdaYaaajuaGdaWcaaqaaiabdUgaRnaaCaaabeqaaiabikdaYaaaaeaacqWGrbqudaqhaaqaaiabicdaWaqaaiabikdaYaaaaaGccqGGUaGlaaa@40A3@

It appeared that the data on the average multiplicity in light quark events can be fitted well by using this expression (with A = 4.21, B = 0.012 and Q₀ = 0.93 GeV). In particular, we have obtained the following average values for the hadronic multiplicities: n_W = 19.52, n_tb = 16.43, N_t = 4.59. Thus, theoretical uncertainties can be estimated to be 0.47 and 0.96 for n_t and Ntt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWG0baDcuWG0baDgaqeaaqabaaaaa@2F42@, respectively.

Taking into account the phenomenological value of n_b 7,

n_b = 5.55 ± 0.09,

we obtain from (47):

n_t(t → hadrons) = 41.03 ± 0.54.

In the case when the W boson decays into leptons, we predict:

n t ( t → l ν ¯ l + hadrons ) = 21.69 ± 0.53. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGUbGBdaWgaaWcbaGaemiDaqhabeaakiabcIcaOiabdsha0jabgkziUkabdYgaSjqbe27aUzaaraWaaSbaaSqaaiabdYgaSbqabaGccqGHRaWkcqqGObaAcqqGHbqycqqGKbazcqqGYbGCcqqGVbWBcqqGUbGBcqqGZbWCcqGGPaqkcqGH9aqpcqaIYaGmcqaIXaqmcqGGUaGlcqaI2aGncqaI5aqocqGHXcqScqaIWaamcqGGUaGlcqaI1aqncqaIZaWmcqGGUaGlaaa@4E01@

As a result, we obtain the average hadron multiplicity in e⁺e^- annihilation with the primary tt¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsha0jqbdsha0zaaraaaaa@2DF1@-pair:

N t t ¯ ( e + e − → t t ¯ → hadrons ) = 86.67 ± 1.11. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemiDaqNafmiDaqNbaebaaeqaaOGaeiikaGIaemyzau2aaWbaaSqabeaacqGHRaWkaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTaaakiabgkziUkabdsha0jqbdsha0zaaraGaeyOKH4QaeeiAaGMaeeyyaeMaeeizaqMaeeOCaiNaee4Ba8MaeeOBa4Maee4CamNaeiykaKIaeyypa0JaeGioaGJaeGOnayJaeiOla4IaeGOnayJaeG4naCJaeyySaeRaeGymaeJaeiOla4IaeGymaeJaeGymaeJaeiOla4caaa@5201@

In the case when both W bosons decay into leptons, the mean multiplicity of the hadrons is expected to be

N t t ¯ ( e + e − → t t ¯ → W + W − + hadrons ) = 47.99 ± 0.59. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemiDaqNafmiDaqNbaebaaeqaaOGaeiikaGIaemyzau2aaWbaaSqabeaacqGHRaWkaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTaaakiabgkziUkabdsha0jqbdsha0zaaraGaeyOKH4Qaem4vaC1aaWbaaSqabeaacqGHRaWkaaGccqWGxbWvdaahaaWcbeqaaiabgkHiTaaakiabgUcaRiabbIgaOjabbggaHjabbsgaKjabbkhaYjabb+gaVjabb6gaUjabbohaZjabcMcaPiabg2da9iabisda0iabiEda3iabc6caUiabiMda5iabiMda5iabgglaXkabicdaWiabc6caUiabiwda1iabiMda5iabc6caUaaa@57A8@

Finally, we predict that

N t t ¯ ( e + e − → t t ¯ → b b ¯ W + W − + hadrons ) = 36.89 ± 0.56. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemiDaqNafmiDaqNbaebaaeqaaOGaeiikaGIaemyzau2aaWbaaSqabeaacqGHRaWkaaGccqWGLbqzdaahaaWcbeqaaiabgkHiTaaakiabgkziUkabdsha0jqbdsha0zaaraGaeyOKH4QaemOyaiMafmOyaiMbaebacqWGxbWvdaahaaWcbeqaaiabgUcaRaaakiabdEfaxnaaCaaaleqabaGaeyOeI0caaOGaey4kaSIaeeiAaGMaeeyyaeMaeeizaqMaeeOCaiNaee4Ba8MaeeOBa4Maee4CamNaeiykaKIaeyypa0JaeG4mamJaeGOnayJaeiOla4IaeGioaGJaeGyoaKJaeyySaeRaeGimaaJaeiOla4IaeGynauJaeGOnayJaeiOla4caaa@5A4E@

All estimations (63)–(65) correspond to the collision energy W = 500 GeV. We can mention the estimation of the hadron multiplicity from Ref. 12, Ntt¯(e+e−→tt¯→bb¯W+W−+hadrons)≈29 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWG0baDcuWG0baDgaqeaaqabaGccqGGOaakcqWGLbqzdaahaaWcbeqaaiabgUcaRaaakiabdwgaLnaaCaaaleqabaGaeyOeI0caaOGaeyOKH4QaemiDaqNafmiDaqNbaebacqGHsgIRcqWGIbGycuWGIbGygaqeaiabdEfaxnaaCaaaleqabaGaey4kaScaaOGaem4vaC1aaWbaaSqabeaacqGHsislaaGccqGHRaWkcqqGObaAcqqGHbqycqqGKbazcqqGYbGCcqqGVbWBcqqGUbGBcqqGZbWCcqGGPaqkcqGHijYUcqaIYaGmcqaI5aqoaaa@5226@, which was obtained for W = 390 GeV and m_t = 175 GeV. For the same values of W and m_t, our formulae give Ntt¯(e+e−→tt¯→bb¯W+W−+hadrons)=34.3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd6eaonaaBaaaleaacqWG0baDcuWG0baDgaqeaaqabaGccqGGOaakcqWGLbqzdaahaaWcbeqaaiabgUcaRaaakiabdwgaLnaaCaaaleqabaGaeyOeI0caaOGaeyOKH4QaemiDaqNafmiDaqNbaebacqGHsgIRcqWGIbGycuWGIbGygaqeaiabdEfaxnaaCaaaleqabaGaey4kaScaaOGaem4vaC1aaWbaaSqabeaacqGHsislaaGccqGHRaWkcqqGObaAcqqGHbqycqqGKbazcqqGYbGCcqqGVbWBcqqGUbGBcqqGZbWCcqGGPaqkcqGH9aqpcqaIZaWmcqaI0aancqGGUaGlcqaIZaWmaaa@534B@.

The formulae (61)–(65) is our main result. We hope that the hadron multiplicities of the top decay products (Eqs. (61) and (62)) will be measured at the LHC.

Here we present some formulae for the case when the W boson decays into hadrons via production of cs¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdogaJjqbdohaZzaaraaaaa@2DCD@ (or cd¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdogaJjqbdsgaKzaaraaaaa@2DAF@) pair. Since the total width of the W boson, Γ_W, is much less than its mass, and the hadron multiplicity is a smooth function of energy, we will use zero width approximation and take the multiplicity at W = m_W. It can be shown that the account of the W boson width results in corrections which are numerically small (less than 1.6%, see Appendix B).

To calculate the inclusive spectrum of the gluon jets emitted by the decay products of the W-boson, we need to calculate two sub-diagrams of the diagram presented in Fig. 9.

The exterior part of the diagram in Fig. 9 describes the emission of the W boson by the top quark, with l being a 4-momentum of the t-quark, q is a 4-momentum of the W boson, and (l - q) is a 4-momentum of the b-quark. The corresponding expression for this part of the diagram (after convolution with the tensor parts of the W boson propagators) looks like

Π μ ν ( l , q ) = 4 { 4 l μ l ν − 2 ( l μ q ν + q μ l ν ) − g μ ν ( m t 2 + m b 2 − m W 2 ) − 2 m W 2 [ ( m t 2 − m b 2 ) l μ − m t 2 q μ ] q ν − 2 m W 2 q μ [ ( m t 2 − m b 2 ) l ν − m t 2 q ν ] + 1 m W 4 q μ q ν [ ( m t 2 − m b 2 ) 2 − q 2 ( m t 2 + m b 2 ) ] + 2 i ε μ ν ρ σ l ρ q σ } , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@E80D@

where m_t and m_b are masses of the top and beauty quark, respectively. In what follows, we will neglect power corrections of the type O(m_c/m_t) and O(m_b/m_t).

The inner part of the diagram in Fig. 9 describes the distribution of the massive gluon jet with the invariant mass k² produced by the W boson. Let D^μν be the expression corresponding to this diagram. In the first order in the strong coupling constant, D^μν is represented by the sum of three QCD diagrams presented in Figs. 10, 11 and 12.

Since q^μΠ_μν = 0, one needs to calculate only two tensor structures in D^μν, namely, g_μν and k_μ k_ν The convolution of the tensor Π_μν with the tensor D^μν = g^μνD₁ + k^μ k^νD₂ + ...,

A = Π μ ν D μ ν = g μ ν Π μ ν D 1 ( k 2 , q k ) + k μ k ν Π μ ν D 2 ( k 2 , q k ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6319@

depends on Lorentz-invariant variables k², qk, lk. Moreover, it is a polynomial of the second order in variable lk. One can use the following useful relation:

∫ d 4 k ( 2 π ) 3 A ( k 2 , q k , l k ) = 1 ( 2 π ) 2 m t 2 ( 1 − r ) ∫ ∫ ∫ d k 2 d ( q k ) d ( l k ) A ( k 2 , q k , l k ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@7686@

where

r = m W 2 m t 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGYbGCcqGH9aqpjuaGdaWcaaqaaiabd2gaTnaaDaaabaGaem4vaCfabaGaeGOmaidaaaqaaiabd2gaTnaaDaaabaGaemiDaqhabaGaeGOmaidaaaaakiabc6caUaaa@35EB@

It is naturally to integrate the function A(k², qk, lk) first in variable lk, whose lower and upper limits are

( l k ) ± = 1 2 r [ ( q k ) ( 1 + r ) ± ( 1 − r ) ( q k ) 2 − m W 2 k 2 ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGGOaakcqWGSbaBcqWGRbWAcqGGPaqkdaWgaaWcbaGaeyySaelabeaakiabg2da9KqbaoaalaaabaGaeGymaedabaGaeGOmaiJaemOCaihaaOWaamWaaeaacqGGOaakcqWGXbqCcqWGRbWAcqGGPaqkcqGGOaakcqaIXaqmcqGHRaWkcqWGYbGCcqGGPaqkcqGHXcqScqGGOaakcqaIXaqmcqGHsislcqWGYbGCcqGGPaqkdaGcaaqaaiabcIcaOiabdghaXjabdUgaRjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeyOeI0IaemyBa02aa0baaSqaaiabdEfaxbqaaiabikdaYaaakiabdUgaRnaaCaaaleqabaGaeGOmaidaaaqabaaakiaawUfacaGLDbaacqGGSaalaaa@55FF@

by using the following formulae:

∫ ( l k ) − ( l k ) + d ( l k ) = 1 r ( 1 − r ) ( q k ) 2 − m W 2 k 2 , ∫ ( l k ) − ( l k ) + ( l k ) d ( l k ) = 1 2 r 2 ( 1 − r 2 ) ( q k ) ( q k ) 2 − m W 2 k 2 , ∫ ( l k ) − ( l k ) + ( l k ) 2 d ( l k ) = 1 12 r 3 ( 1 − r ) [ 4 ( q k ) 2 ( 1 + r + r 2 ) − m W 2 k 2 ( 1 − r ) 2 ] × ( q k ) 2 − m W 2 k 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeabbaaaaeaadaWdXaqaaiabdsgaKjabcIcaOiabdYgaSjabdUgaRjabcMcaPaWcbaGaeiikaGIaemiBaWMaem4AaSMaeiykaKYaaSbaaWqaaiabgkHiTaqabaaaleaacqGGOaakcqWGSbaBcqWGRbWAcqGGPaqkdaWgaaadbaGaey4kaScabeaaa0Gaey4kIipakiabg2da9KqbaoaalaaabaGaeGymaedabaGaemOCaihaaOGaeiikaGIaeGymaeJaeyOeI0IaemOCaiNaeiykaKYaaOaaaeaacqGGOaakcqWGXbqCcqWGRbWAcqGGPaqkdaahaaWcbeqaaiabikdaYaaakiabgkHiTiabd2gaTnaaDaaaleaacqWGxbWvaeaacqaIYaGmaaGccqWGRbWAdaahaaWcbeqaaiabikdaYaaaaeqaaOGaeiilaWcabaWaa8qmaeaacqGGOaakcqWGSbaBcqWGRbWAcqGGPaqkcqWGKbazcqGGOaakcqWGSbaBcqWGRbWAcqGGPaqkaSqaaiabcIcaOiabdYgaSjabdUgaRjabcMcaPmaaBaaameaacqGHsislaeqaaaWcbaGaeiikaGIaemiBaWMaem4AaSMaeiykaKYaaSbaaWqaaiabgUcaRaqabaaaniabgUIiYdGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYiabdkhaYnaaCaaabeqaaiabikdaYaaaaaGccqGGOaakcqaIXaqmcqGHsislcqWGYbGCdaahaaWcbeqaaiabikdaYaaakiabcMcaPiabcIcaOiabdghaXjabdUgaRjabcMcaPmaakaaabaGaeiikaGIaemyCaeNaem4AaSMaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqGHsislcqWGTbqBdaqhaaWcbaGaem4vaCfabaGaeGOmaidaaOGaem4AaS2aaWbaaSqabeaacqaIYaGmaaaabeaakiabcYcaSaqaamaapedabaGaeiikaGIaemiBaWMaem4AaSMaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqWGKbazcqGGOaakcqWGSbaBcqWGRbWAcqGGPaqkaSqaaiabcIcaOiabdYgaSjabdUgaRjabcMcaPmaaBaaameaacqGHsislaeqaaaWcbaGaeiikaGIaemiBaWMaem4AaSMaeiykaKYaaSbaaWqaaiabgUcaRaqabaaaniabgUIiYdGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabigdaXiabikdaYiabdkhaYnaaCaaabeqaaiabiodaZaaaaaGccqGGOaakcqaIXaqmcqGHsislcqWGYbGCcqGGPaqkdaWadaqaaiabisda0iabcIcaOiabdghaXjabdUgaRjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeiikaGIaeGymaeJaey4kaSIaemOCaiNaey4kaSIaemOCai3aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGHsislcqWGTbqBdaqhaaWcbaGaem4vaCfabaGaeGOmaidaaOGaem4AaS2aaWbaaSqabeaacqaIYaGmaaGccqGGOaakcqaIXaqmcqGHsislcqWGYbGCcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaOGaay5waiaaw2faaaqaaiabgEna0oaakaaabaGaeiikaGIaemyCaeNaem4AaSMaeiykaKYaaWbaaSqabeaacqaIYaGmaaGccqGHsislcqWGTbqBdaqhaaWcbaGaem4vaCfabaGaeGOmaidaaOGaem4AaS2aaWbaaSqabeaacqaIYaGmaaaabeaakiabc6caUaaaaaa@DD2D@

Note that the parts of tensors Π_μν and D^μν antisymmetric in indices (as the last term in Eq. (A.1)) give no contribution after integration in lk.

Integration limits in variable qk looks like

( q k ) − = m W 2 k 2 , ( q k ) + = 1 2 ( m W 2 + k 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaGaeiikaGIaemyCaeNaem4AaSMaeiykaKYaaSbaaSqaaiabgkHiTaqabaGccqGH9aqpdaGcaaqaaiabd2gaTnaaDaaaleaacqWGxbWvaeaacqaIYaGmaaGccqWGRbWAdaahaaWcbeqaaiabikdaYaaaaeqaaOGaeiilaWcabaGaeiikaGIaemyCaeNaem4AaSMaeiykaKYaaSbaaSqaaiabgUcaRaqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabcIcaOiabd2gaTnaaDaaaleaacqWGxbWvaeaacqaIYaGmaaGccqGHRaWkcqWGRbWAdaahaaWcbeqaaiabikdaYaaakiabcMcaPiabc6caUaaaaaa@4B2C@

As a result, we obtain the hadron multiplicity associated with the charm quark in a hadronic decay of the W boson (with the multiplicity of primary quark decay products subtracted):

N ^ c l = 1 ( 2 π ) 2 m t 2 ( 1 − r ) H ∫ Q 0 2 m W 2 d k 2 [ ∂ ∂ k 2 n g ( k 2 ) ] ∫ ( q k ) − ( q k ) + d ( q k ) × ∫ ( l k ) − ( l k ) + d ( l k ) A ( k 2 , q k , l k ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@9321@

The dimensionless quantity n_g(k²) in (A.8) is the multiplicity of hadrons in the gluon jet whose virtuality is k², while the normalization H is given by

H = 1 6 π 2 m t 4 ( 1 − r ) 2 ( 1 + 2 r ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGibascqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabiAda2iabec8aWnaaCaaabeqaaiabikdaYaaaaaGaemyBa02aa0baaeaacqWG0baDaeaacqaI0aanaaGaeiikaGIaeGymaeJaeyOeI0IaemOCaiNaeiykaKYaaWbaaeqabaGaeGOmaidaaiabcIcaOiabigdaXiabgUcaRiabikdaYiabdkhaYjabcMcaPiabc6caUaaa@4291@

The analytic expressions for the functions D₁(k², qk), D₂(k², qk) in (A.2) are rather complicated to be shown here. That is why we present only the final results of our QCD calculations based on the formulae of this Appendix (see Eqs. (34), (35) in the main text).

In this Appendix we will demonstrate that the account of the W boson width results in only small corrections to the hadronic multiplicities.

The denominator of the W boson propagator (see the diagram in Fig. 9) is

q 2 − m W 2 + i m W Γ W , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGXbqCdaahaaWcbeqaaiabikdaYaaakiabgkHiTiabd2gaTnaaDaaaleaacqWGxbWvaeaacqaIYaGmaaGccqGHRaWkcqWGPbqAcqWGTbqBdaWgaaWcbaGaem4vaCfabeaakiabfo5ahnaaBaaaleaacqWGxbWvaeqaaOGaeiilaWcaaa@3A5C@

where m_W is the mass of the W boson, Γ_W its full width. The mean multiplicity is given by the formula

〈 n h 〉 = 1 N ∫ 0 ( m t − m b ) 2 d q 2 m W Γ W ( q 2 − m W 2 ) 2 + m W 2 Γ W 2 F ( q 2 ) n h ( q 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6B30@

where n_h (n_h = n_W or n_tb, see the main text) depends on the W boson virtuality q². Here

F ( q 2 ) = ( m t 2 − q 2 ) 2 ( m t 2 + 2 q 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGgbGrcqGGOaakcqWGXbqCdaahaaWcbeqaaiabikdaYaaakiabcMcaPiabg2da9iabcIcaOiabd2gaTnaaDaaaleaacqWG0baDaeaacqaIYaGmaaGccqGHsislcqWGXbqCdaahaaWcbeqaaiabikdaYaaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeiikaGIaemyBa02aa0baaSqaaiabdsha0bqaaiabikdaYaaakiabgUcaRiabikdaYiabdghaXnaaCaaaleqabaGaeGOmaidaaOGaeiykaKIaeiilaWcaaa@4607@

and the normalization N is

N = ∫ 0 ∞ d q 2 m W Γ W ( q 2 − m W 2 ) 2 + m W 2 Γ W 2 F ( q 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@53B3@

In both <n_h> (B.2) and N (B.3) the factor m_W Γ_W is introduced, while common constants are omitted.

In the zero width limit,

m W Γ W ( q 2 − m W 2 ) 2 + m W 2 Γ W 2 → Γ W → 0 π δ ( q 2 − m W 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@56BE@

we do obtain that the mean multiplicity is equal to n_h(mW2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGxbWvaeaacqaIYaGmaaaaaa@2EB0@).

The numerical calculations with the use of formulae (B.2), (B.3) result in the following values:

<n_w> = 19.04,

<n_tb> = 16.37.

Thus, the account of non-zero width of the W boson slightly changes the average value of the multiplicities. Namely, n_W (54) has gone down by 0.3, while n_tb (55) has gone up by 0.23, but their sum remains almost unchanged.