- Research article
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# Hadron multiplicity in e^{+}e^{-}events induced by top quark pairs at the ILC energy

*PMC Physics A*
**volume 2**, Article number: 3 (2008)

## Abstract

The average charged hadron multiplicity in the *e*^{+}*e*^{-} events with the primary $t\overline{t}$-pair at the collision energy 500 GeV, as well as the average multiplicity of charged hadrons from the top quark are calculated in QCD to be 86.7 ± 1.11 and 41.0 ± 0.54, respectively.

**PACS Codes**: 14.65.Ha, 13.66.Bc, 12.38.Bx

## 1 Introduction

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 ${N}_{t\overline{t}}$ in the *e*^{+}*e*^{-} events with $t\overline{t}$ pair at the collision energy of the ILC with the following prediction:

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 $c\overline{c}$ ($b\overline{b}$) events. The hadron multiplicities in *e*^{+}*e*^{-} events associated with the $t\overline{t}$-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.

## 2 Hadron multiplicity in e^{+}e^{-} annihilation associated with $\text{c}\overline{\text{c}}$ or $\text{b}\overline{\text{b}}$-pair production

Hadron multiplicity in $q\overline{q}$ event, ${N}_{q\overline{q}}$(*W*), can be represented in the following general form [1, 2]:

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), 2*n*_{
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 (2*n*_{
c
}= 5.2, 2*n*_{
b
}= 11.1 [4], and 2*n*_{
l
}= 2.4 [5]).

The quantity *n*_{
g
}(*k*^{2}) in (3) is the mean multiplicity of the gluon jet with a virtuality *k*^{2}, for which we will take a QCD-based parametric form, with parameters fit to data, while *E*_{
q
}(*k*^{2}/*W*^{2}) 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*(*k*^{2}/*W*^{2}) is an inclusive spectrum of the gluon jets is seen, e.g., from the fact that the average number of jets ∫*dk*^{2}/*k*^{2} *E*_{
q
}(*k*^{2}/*W*^{2}) ≠ 1.

*Q*_{0} 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

and

as well as notation

where *C*_{
F
}= (${N}_{c}^{2}$ - 1)/2*N*_{
c
}, and *N*_{
c
}= 3 is a number of colors. Then Eq. (3) can be represented as

In particular, ${N}_{l\overline{l}}$(*Y*) means the multiplicity of hadrons in light quark events, while ${N}_{Q\overline{Q}}$(*Y*) denotes the multiplicity of hadrons in a process when a pair of the heavy quarks is produced.

The physical meaning of the function

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 $q\overline{q}$ event except for multiplicity of the decay products of the primary quarks at the final stage of hadronization (the terms 2*n*_{
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

where Li_{2}(*z*) is the Euler dilogarithm. The function *E*(*η*) is presented in Fig. 1. It has the asymptotics

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
}(*k*^{2}) since

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

At *W* ≫ *m*_{
Q
}, one can neglect small power-like corrections O(*m*^{2}/*W*^{2}). In such a case, the quantity *δ*_{
Ql
}is defined by [1]

*δ*_{
Ql
}= 2(*n*_{
Q
}- *n*_{
l
}) - Δ*N*_{
Q
}(*Y*_{
Q
}),

where the notation

as well as variables

and

are introduced. The lower limit of integration in Eq. (15), -ln(*W*^{2}/${m}_{Q}^{2}$), 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 Δ*E*_{
Q
}was derived in our paper [1] (see Fig. 2):

where

The function Δ*E*_{
Q
}(*y*) decreases at *y* → -∞ (*ρ* → ∞) as

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

Thus, we get the relation between average multiplicities of hadrons in $Q\overline{Q}$ and $l\overline{l}$ events:

with the multiplicity difference Δ*N*_{
Q
}defined by Eq. (15).

Our calculations [1] of the multiplicity differences *δ*_{
Ql
}= ${N}_{l\overline{l}}$ - ${N}_{Q\overline{Q}}$ (*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].

## 3 Hadron multiplicity in e^{+}e^{-} annihilation associated with $\text{t}\overline{\text{t}}$-pair production

The goal of this paper is to calculate ${N}_{t\overline{t}}^{h}$, the average multiplicity of hadrons produced in *e*^{+}*e*^{-} events with the primary $t\overline{t}$ 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*^{-} → ${t}^{\ast}{\overline{t}}^{\ast}$ → *X* is factorized as follows:

where ${t}^{\ast}({\overline{t}}^{\ast})$ 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 $\overline{t}$-quarks. Note that the off-shell *t* and $\overline{t}$-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*^{-} → $q{\overline{q}}^{\prime}q{\overline{q}}^{\prime}$ and *e*^{+}*e*^{-}→ *W*^{+}*W*^{-} → $q{\overline{q}}^{\prime}l{\overline{\nu}}_{l}$ events. No evidence for final state interactions was found by measuring the difference $\u3008{n}_{4q}^{h}\u3009-2\u3008{n}_{2ql\overline{\nu}}^{h}\u3009$[8, 9]. 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 $\overline{t}$-quarks.

According to Eq. (24), the associative multiplicity in $t\overline{t}$-event is given by the formula:

where

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 $\sqrt{{k}^{2}}$ looks like

where $k\equiv \sqrt{{k}^{2}}$, $q\equiv \sqrt{{q}^{2}}$, and the following notations are introduced:

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*(*k*^{2}/*q*^{2}) (10), while by neglecting small corrections O(*m*^{2}/*q*^{2}), one can derive (after variables are properly changed) the explicit form of the function Δ*E*_{
Q
}(*k*^{2}/*m*^{2}) = *E*_{
Q
}(*q*^{2}, *k*^{2}, *m*^{2}) - *E*(*q*^{2}, *k*^{2}) (19). In our case (*q*^{2} = *W*^{2}, *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*_{
b
}is 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.

### 3.1 Multiplicity of W boson decay products

The *W*^{+} boson can decay either into two light quarks ($u\overline{d}$ and $u\overline{s}$ pairs) or into $c\overline{d}$ ($c\overline{s}$) pair. The former case is treated analogously to the light quark event in *e*^{+}*e*^{-} annihilation taken at the collision energy *W* = *m*_{
W
}. Here we will study the latter case.

Let *N*_{
Ql
}(*W*) be hadronic multiplicity associated with the production of *one* heavy quark (antiquark) of the type *Q* and *one* light antiquark (quark) of the type *l*:

Now let us introduce the notation (not to confuse with Δ*N*_{
Q
}from above):

Then the first term in the r.h.s. of Eq. (29) is given in terms of the function Δ*N*_{
cl
}by the formula:

where ${Y}_{c}=\mathrm{ln}({m}_{c}^{2}/{Q}_{0}^{2})$ and

The function ${N}_{l\overline{l}}$(*Y*) in (32) is the hadronic multiplicity in light quark events. Thus, we need to find an expression for Δ*N*_{
cl
}. Note that the formulae (15), (19) from Section 2 correspond to the case when *a pair* of heavy or *pair* of light quarks is produced. Now we have to study the case when hadrons are produced in association with *a single* heavy quark (namely, *c*-quark) and one light quark.

Our QCD calculations result in the following representation for the multiplicity difference (see Appendix for details):

with the dimensionless function Δ*E*_{
Ql
}:

Here *ρ* = exp(-*y*). The quantity *J*(*ρ*) was defined above (20). The function Δ*E*_{
Ql
}(*y*) is shown in Fig. 5.

Since

the integral (34) converges rapidly at the lower limit. Asymptotics of Δ*E*_{
Ql
}(*y*) at large *y* is the following:

We derive from Eqs. (22), (37) that Δ*E*_{
Ql
}(*y*) = 0.5 Δ*E*_{
Q
}(*y*) at large *y*. Numerical calculations show that 2Δ*E*_{
Ql
}is very close to Δ*E*_{
Q
}at all *y* (see Fig. 6). Thus, we can put for our further numerical estimates:

This relation means that

Correspondingly, we obtain:

### 3.2 Multiplicity of top and bottom decay products

As was already said above, the on-shell top quark can emit jets before it weakly decays into *W*^{+}*b*. After the weak decay of the top, the off-shell *b*-quark "throws off" its virtuality by emitting massive gluon jets. The fragmentation of these massive gluon jets into hadrons results in the average hadron multiplicity *n*_{
tb
}.

To calculate the multiplicity *n*_{
tb
}, one has to derive the inclusive spectrum of the gluon jets, emitted by the top and bottom quarks. Let us denote it as *E*_{
tb
}. Then the multiplicity *n*_{
tb
}will be given by the formula:

where

and

with *k*^{2} being the gluon jet invariant mass, (*m*_{
t
}- *m*_{
W
}- *m*_{
b
})^{2} its upper bound.

In the lowest order in the strong coupling constant, the quantity *E*_{
tb
}(*y*) is given by two diagrams in Fig. 4. It is presented by an integral which depends on the ratio ${k}^{2}/{m}_{t}^{2}$, as well as on mass ratios ${m}_{W}^{2}/{m}_{t}^{2}$ and ${m}_{b}^{2}/{m}_{t}^{2}$. This integral cannot be calculated analytically, but can be estimated numerically. The function *E*_{
tb
}(*y*) is presented in Fig. 7. It is worth to note that in the Feynman gauge the dominating contribution to *E*_{
tb
}(*y*) comes from the *interference* of two diagrams shown in Fig. 4.

### 3.3 Associated multiplicity of hadrons in $\text{t}\overline{\text{t}}$events

The formulae of the previous subsections enable us to derive the average multiplicity of the charged hadrons in *e*^{+}*e*^{-} annihilation at the collision energy *W* associated with the production of the $t\overline{t}$-pair. It is of the form:

Let us remind to the reader the meaning of all quantities in Eq. (44). The function *N*_{
t
}(*W*, *m*_{
t
}) describes the average number of hadrons produced in association with the $t\overline{t}$-primary pair, *except for* the decay products of the top(antitop) (26). The quantity ${N}_{l\overline{l}}$(*m*_{
W
}) is the mean hadron multiplicity in the light quark event taken at the energy *E* = *m*_{
W
}. The hadron multiplicity *n*_{tb} comes from the emission by *t* and *b* quarks (41). The quantity *n*_{
l
}is the mean multiplicity of hadrons produced in the decay of the on-shell primary quark *q* (*q* = *l*, *c*, *b*). Finally, the combination [Δ*N*_{
cl
}(*m*_{
c
}) + *n*_{
l
}- *n*_{
c
}] is the difference of multiplicities in the processes with the primary $l\overline{l}$- and $c\overline{l}$-pairs. As for the hadron multiplicity resulting from the decay of the on-shell top (anti)quark, it is given by

The expressions for ${N}_{l\overline{l}}$, Δ*N*_{
cl
}are given by Eqs. (7), (34), respectively. Let us stress that ${N}_{l\overline{l}}$ and *n*_{
q
}(*q* = *b*, *c*, *l*) are extracted from the data, and Δ*N*_{
cl
}can be related with the measurable quantities (see formulae (38)–(40) in the end of subsection 3.1):

Then we obtain:

and

where *n*_{
tb
}is defined above (41). In what follows, we will use the value

*δ*_{
cl
}= 1.03 ± 0.34

from Ref. [7].

## 4 Numerical estimates of hadron multiplicities

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 ${\widehat{n}}_{g}$ is related with gluon jet multiplicity *n*_{
g
}(*k*^{2}) (6). We have fitted the data on ${N}_{l\overline{l}}$(*W*) by using the following QCD-motivated expression for *n*_{
g
}(*k*^{2}):

where *c* = 1.63, and *k*^{2} is the invariant mass of the jet. We have got the following values of the parameters:

*a* = 3.89, *b* = 0.01, *Q*_{0} = 0.87 GeV.

The result of our fit is presented in Fig. 8 in comparison with the data. Note that *χ*^{2}*/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 *χ*^{2}/*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 ${N}_{l\overline{l}}$(*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 $t\overline{t}$ 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 ${N}_{t\overline{t}}({e}^{+}{e}^{-}\to t\overline{t}\to \text{hadrons})$ = 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)):

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 *Q*_{0} = 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 ${N}_{t\overline{t}}$, 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:

As a result, we obtain the average hadron multiplicity in *e*^{+}*e*^{-} annihilation with the primary $t\overline{t}$-pair:

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

Finally, we predict that

All estimations (63)–(65) correspond to the collision energy *W* = 500 GeV. We can mention the estimation of the hadron multiplicity from Ref. [12], ${N}_{t\overline{t}}({e}^{+}{e}^{-}\to t\overline{t}\to b\overline{b}{W}^{+}{W}^{-}+\text{hadrons})\approx 29$, which was obtained for *W* = 390 GeV and *m*_{
t
}= 175 GeV. For the same values of *W* and *m*_{
t
}, our formulae give ${N}_{t\overline{t}}({e}^{+}{e}^{-}\to t\overline{t}\to b\overline{b}{W}^{+}{W}^{-}+\text{hadrons})=34.3$.

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.

## Appendix A

Here we present some formulae for the case when the *W* boson decays into hadrons via production of $c\overline{s}$ (or $c\overline{d}$) 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

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 *k*^{2} 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*^{μ}Π_{
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*^{μν}*D*_{1} + *k*^{μ}*k*^{ν}*D*_{2} + ...,

depends on Lorentz-invariant variables *k*^{2}, *qk*, *lk*. Moreover, it is a polynomial of the second order in variable *lk*. One can use the following useful relation:

where

It is naturally to integrate the function *A*(*k*^{2}, *qk*, *lk*) first in variable *lk*, whose lower and upper limits are

by using the following formulae:

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

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):

The dimensionless quantity *n*_{
g
}(*k*^{2}) in (A.8) is the multiplicity of hadrons in the gluon jet whose virtuality is *k*^{2}, while the normalization *H* is given by

The analytic expressions for the functions *D*_{1}(*k*^{2}, *qk*), *D*_{2}(*k*^{2}, *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).

## Appendix B

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

where *m*_{
W
}is the mass of the *W* boson, Γ_{
W
}its full width. The mean multiplicity is given by the formula

where *n*_{
h
}(*n*_{
h
}= *n*_{
W
}or *n*_{
tb
}, see the main text) depends on the *W* boson virtuality *q*^{2}. Here

and the normalization *N* is

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,

we do obtain that the mean multiplicity is equal to *n*_{
h
}(${m}_{W}^{2}$).

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.

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## Acknowledgements

We are thankful to the referee for his comments and critical remarks that helped us to improve the presentation of some our results.

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Kisselev, A.V., Petrov, V.A. Hadron multiplicity in e^{+}e^{-}events induced by top quark pairs at the ILC energy.
*PMC Phys A* **2, **3 (2008). https://doi.org/10.1186/1754-0410-2-3

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### Keywords

- Heavy Quark
- Decay Product
- Bottom Quark
- Charged Hadron
- Average Multiplicity