The goal of this paper is to calculate , the average multiplicity of hadrons produced in e+e- events with the primary 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- → → X is factorized as follows:
(24)
where 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 -quarks. Note that the off-shell t and -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- → and e+e-→ W+W- → events. No evidence for final state interactions was found by measuring the difference [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 -quarks.
According to Eq. (24), the associative multiplicity in -event is given by the formula:
(25)
where
(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 looks like
(27)
where , , and the following notations are introduced:
(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(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
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 ( and pairs) or into () 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:
(30)
Now let us introduce the notation (not to confuse with ΔN
Q
from above):
(31)
Then the first term in the r.h.s. of Eq. (29) is given in terms of the function ΔN
cl
by the formula:
(32)
where and
(33)
The function (Y) in (32) is the hadronic multiplicity in light quark events. Thus, we need to find an expression for ΔN
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):
(34)
with the dimensionless function ΔE
Ql
:
(35)
Here ρ = exp(-y). The quantity J(ρ) was defined above (20). The function ΔE
Ql
(y) is shown in Fig. 5.
Since
(36)
the integral (34) converges rapidly at the lower limit. Asymptotics of ΔE
Ql
(y) at large y is the following:
(37)
We derive from Eqs. (22), (37) that ΔE
Ql
(y) = 0.5 ΔE
Q
(y) at large y. Numerical calculations show that 2ΔE
Ql
is very close to ΔE
Q
at all y (see Fig. 6). Thus, we can put for our further numerical estimates:
(38)
This relation means that
(39)
Correspondingly, we obtain:
(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:
(41)
where
(42)
and
(43)
with k2 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 , as well as on mass ratios and . 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 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 -pair. It is of the form:
(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 -primary pair, except for the decay products of the top(antitop) (26). The quantity (m
W
) is the mean hadron multiplicity in the light quark event taken at the energy E = m
W
. The hadron multiplicity ntb 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 [ΔN
cl
(m
c
) + n
l
- n
c
] is the difference of multiplicities in the processes with the primary - and -pairs. As for the hadron multiplicity resulting from the decay of the on-shell top (anti)quark, it is given by
(45)
The expressions for , ΔN
cl
are given by Eqs. (7), (34), respectively. Let us stress that and n
q
(q = b, c, l) are extracted from the data, and ΔN
cl
can be related with the measurable quantities (see formulae (38)–(40) in the end of subsection 3.1):
(46)
Then we obtain:
(47)
and
(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].