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:

\begin{array}{lll}|M({e}^{+}{e}^{-}\to {t}^{\ast}{\overline{t}}^{\ast}\to \text{hadrons}){|}^{2}\hfill & =\hfill & |M({e}^{+}{e}^{-}\to {t}^{\ast}{\overline{t}}^{\ast}\to t\overline{t}+\text{hadrons}){|}^{2}\hfill \\ \times \hfill & |M(t\to \text{hadrons}){|}^{2}\hfill \\ \times \hfill & |M(\overline{t}\to \text{hadrons}){|}^{2},\hfill \end{array}

(24)

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:

{N}_{t\overline{t}}^{h}(W,{m}_{t})={N}_{t}(W,{m}_{t})+2{n}_{t},

(25)

where

{N}_{t}(W,{m}_{t})={C}_{F}{\displaystyle \underset{{Q}_{0}^{2}}{\overset{{(W-2{m}_{t})}^{2}}{\int}}\frac{d{k}^{2}}{{k}^{2}}\frac{{\alpha}_{s}({k}^{2})}{\pi}{n}_{g}({k}^{2}){E}_{t}({W}^{2},{k}^{2},{m}_{t}^{2})}.

(26)

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

\begin{array}{lll}{E}_{t}({q}^{2},{k}^{2},{m}^{2})\hfill & =\hfill & \left(-k\frac{\partial}{\partial k}\right){\displaystyle \underset{1}{\overset{A}{\int}}d\eta}\{[\frac{1}{\eta}\frac{{({q}^{2}+{k}^{2})}^{2}-4{m}^{4}}{{q}^{4}}-2\frac{k}{q}\frac{{q}^{2}+{k}^{2}+2{m}^{2}}{{q}^{2}}\hfill \\ +\hfill & 2\eta \frac{{k}^{2}}{{q}^{2}}]\mathrm{ln}\left[\frac{\eta +\sqrt{{\eta}^{2}-1}\sqrt{(A-\eta )/({A}_{0}-\eta )}}{\eta -\sqrt{{\eta}^{2}-1}\sqrt{(A-\eta )/({A}_{0}-\eta )}}\right]\hfill \\ -\hfill & 2\frac{{k}^{2}}{{q}^{2}}\sqrt{{\eta}^{2}-1}\frac{\sqrt{(A-\eta )}}{\sqrt{({A}_{0}-\eta )}}\hfill \\ \times \hfill & \left[1+\frac{(1+2{m}^{2}/{q}^{2})(1+2{m}^{2}/{k}^{2})}{{\eta}^{2}-({\eta}^{2}-1)(A-\eta )/({A}_{0}-\eta )}\right]\},\hfill \end{array}

(27)

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

\begin{array}{cc}A=\frac{{q}^{2}+{k}^{2}-4{m}^{2}}{2qk},& {A}_{0}=\frac{{q}^{2}+{k}^{2}}{2qk}.\end{array}

(28)

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*:

{N}_{Ql}(W)=({n}_{Q}+{n}_{l})+{\widehat{N}}_{Ql}(W).

(30)

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

\Delta {N}_{Ql}={N}_{l}-{\widehat{N}}_{Ql}.

(31)

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:

{n}_{W}={N}_{l\overline{l}}({Y}_{W})+\frac{1}{2}[-\Delta {N}_{cl}({Y}_{c})+{n}_{c}-{n}_{l}],

(32)

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

{Y}_{W}=\mathrm{ln}\frac{{m}_{W}^{2}}{{Q}_{0}^{2}}.

(33)

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):

\Delta {N}_{Ql}({Y}_{Q})={\displaystyle \underset{-\infty}{\overset{{Y}_{Q}}{\int}}dy{\widehat{n}}_{g}({Y}_{Q}-y)\Delta {E}_{Ql}(y)},

(34)

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

\begin{array}{ccc}\Delta {E}_{Ql}[y(\rho )]& =& \frac{1}{4}[2+\rho (3\rho -2)]\mathrm{ln}\frac{1}{\rho}+\frac{1}{4}(5+6\rho )\\ +& \frac{1}{2}\rho (3\rho -8)J(\rho )+6\frac{1-J(\rho )}{\rho -4}.\end{array}

(35)

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

Since

{\Delta {E}_{Ql}(y)|}_{y\to -\infty}\simeq \frac{3}{2}{e}^{-\left|y\right|},

(36)

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

{\Delta {E}_{Ql}(y)|}_{y\to \infty}\simeq \frac{1}{2}\left(y-\frac{1}{2}\right).

(37)

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:

\Delta {N}_{cl}=\frac{1}{2}\Delta {N}_{c}.

(38)

This relation means that

{N}_{Q\overline{l}}=\frac{1}{2}[{N}_{l\overline{l}}+{N}_{Q\overline{Q}}]={N}_{l\overline{l}}+\frac{1}{2}{\delta}_{Ql}.

(39)

Correspondingly, we obtain:

{n}_{W}={N}_{l\overline{l}}({m}_{W})+\frac{1}{4}{\delta}_{cl}.

(40)

### 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:

{n}_{tb}={\displaystyle \underset{0}{\overset{{Y}_{tb}}{\int}}dy{\widehat{n}}_{g}({Y}_{tb}-y){E}_{tb}(y)},

(41)

where

y=\mathrm{ln}\frac{{({m}_{t}-{m}_{W}-{m}_{b})}^{2}}{{k}^{2}},

(42)

and

{Y}_{tb}=\mathrm{ln}\frac{{({m}_{t}-{m}_{W}-{m}_{b})}^{2}}{{Q}_{0}^{2}},

(43)

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:

\begin{array}{ccc}{N}_{t\overline{t}}^{h}(W,{m}_{t})& =& {N}_{t}(W,{m}_{t})+2[{N}_{l\overline{l}}({m}_{W})+{n}_{tb}+{n}_{b}]\\ +& [-\Delta {N}_{cl}({m}_{c})+{n}_{c}-{n}_{l}].\end{array}

(44)

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

\begin{array}{ccc}{n}_{t}& =& {n}_{W}+{n}_{tb}+{n}_{b}\\ =& {N}_{l\overline{l}}({m}_{W})+{n}_{tb}+{n}_{b}+\frac{1}{2}[-\Delta {N}_{cl}({m}_{c})+{n}_{c}-{n}_{l}].\end{array}

(45)

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):

\Delta {N}_{cl}\simeq \frac{1}{2}\Delta {N}_{c}={n}_{c}-{n}_{l}-\frac{1}{2}{\delta}_{cl}.

(46)

Then we obtain:

{n}_{t}={N}_{l\overline{l}}({m}_{W})+\frac{1}{4}{\delta}_{cl}+{n}_{tb}+{n}_{b},

(47)

and

{N}_{t\overline{t}}^{h}(W,{m}_{t})={N}_{t}(W,{m}_{t})+2\left[{N}_{l\overline{l}}({m}_{W})+\frac{1}{4}{\delta}_{cl}+{n}_{tb}+{n}_{b}\right],

(48)

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

*δ*_{
cl
}= 1.03 ± 0.34

from Ref. [7].