Research article Hadron multiplicity in e + e - events induced by top quark pairs at the ILC energy Alexandre V Kisselev* and Vladimir A Petrov

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1 Research article Hadron multiplicity in e + e - events induced by top uark pairs at the ILC energy Alexandre V Kisselev* and Vladimir A Petrov Open Access Address: Institute for High Energy Physics, 48 Protvino, Russia Alexandre V Kisselev* - alexandre.kisselev@ihep.ru; Vladimir A Petrov - vladimir.petrov@ihep.ru * Corresponding author Published: 4 April 008 PMC Physics A 008, :3 doi:0.86/ Received: 5 December 007 Accepted: 4 April 008 This article is available from: Kisselev and Petrov This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( licenses/by/.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The average charged hadron multiplicity in the e + e - events with the primary tt -pair at the collision energy 500 GeV, as well as the average multiplicity of charged hadrons from the top uark are calculated in QCD to be 86.7 ±. and 4.0 ± 0.54, respectively. PACS Codes: 4.65.Ha, 3.66.Bc,.38.Bx Introduction Experiments at LEP and SLAC revealed, besides other important results, uite interesting feature of the hadron multiple production dependent on the mass of the "primary" (anti)uarks which launch the process of the QCD evolution. It appeared that differences between the light and heavy uark-induced multiplicities become energy-independent. QCD calculations describe the phenomenon uite well. Certainly, LEP could not give the information on the events induced by the top uarks. Recent discussions of the ILC project give us occasion to provide QCD predictions concerning the hadron multiple production in the events with primary t-uarks. e manage to calculate the average hadron multiplicity in the e + e - events with tt pair at the collision energy of the ILC with the following prediction: Ntt ( = 500GeV ) = ± () N tt Page of 3

2 PMC Physics A 008, :3 e also theoretically calculated the average hadronic multiplicity from the top uark: n t = 4.03 ± 0.7. () Both values correspond to the average value of the top mass m t = 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 uark events more easy for understanding, we consider first the multiple hadron production in cc ( bb ) events. The hadron multiplicities in e + e - events associated with the tt -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 e + e - annihilation associated with cc or bb -pair production Hadron multiplicity in event, (), can be represented in the following general form [,]: N N ( ) = n + C dk α s( k ) k n ( k ) E, k π F g Q 0 (3) where means a type of uarks produced in the process of e + e - annihilation into hadrons at the collision energy. In what follows, the notation = Q (heavy uark) will mean charm or beauty uark, while the notation = l (light uark) will correspond to a massless case (when a pair of u, d or s-uarks is produced, whose masses are assumed to be eual to zero). The top uark production ( = t) will be studied in Sections 3 and 4. The first term in the r.h.s. of E. (3), n, is the multiplicity of primary (anti)uark of the type (i.e. the multiplicity from the leading hadron which contains this (anti)uark). It is taken from an analysis of the data (n c = 5., n b =. [4], and n l =.4 [5]). The uantity n g (k ) in (3) is the mean multiplicity of the gluon jet with a virtuality k, for which we will take a QCD-based parametric form, with parameters fit to data, while E (k / ) is the inclusive spectrum of the gluon jet emitted by primary uarks. It was explained in detail in Ref. [] that one should not consider this mechanism of hadron production via gluon jets as due to "a single cascading gluon". That uantity E(k / ) is an inclusive spectrum of the gluon jets is seen, e.g., from the fact that the average number of jets dk /k E (k / ). Page of 3

3 PMC Physics A 008, :3 Q 0 is a phenomenological parameter denoting the scale at which "precon-finement" of the offshell partons occurs (as explained in Ref. [6]). Let us introduce variables η = ln k (4) and Y = ln, Q 0 (5) as well as notation nˆ g CFα s( k ) = ng( k ), (6) π N c where C F = ( - )/N c, and N c = 3 is a number of colors. Then E. (3) can be represented as N ll Y N ( Y) = n + dηn ( Y η) E ( η ) n + N ( Y). (7) g 0 N QQ In particular, (Y) means the multiplicity of hadrons in light uark events, while (Y) denotes the multiplicity of hadrons in a process when a pair of the heavy uarks is produced. The physical meaning of the function Y N( Y) = d ng( Y ) E( ) d ng( ) E( Y η η η η η η ) 0 Y 0 (8) is the following. It describes the average number of hadrons produced in virtual gluon jets emitted by the primary uark and antiuark of the type. In other words, it is the multiplicity in event except for multiplicity of the decay products of the primary uarks at the final stage of hadronization (the terms n in (7)). For the massless case, the function E E l was calculated in our paper []. In terms of variable Page 3 of 3

4 PMC Physics A 008, :3 σ = exp(-η), (9) it looks as 3 σ E[ ησ ( )] = ( + σ+ σ )ln + 7 ( σ) σ( + σ) ln σ σ π + 4σ( + σ) + lnσ ln( + σ) + Li ( σ), where Li (z) is the Euler dilogarithm. The function E(η) is presented in Fig.. It has the asymptotics (0) E( η) η. η = 3 () The derivative of E(η) is positive, and E(η)/ η = 0 at η = 0. As a result, the associated multiplicity N () (8) is a monotonic increasing function of the energy for any positive function n g (k ) since Nl( Y) = Y Y 0 EY ( η) d η ng( η ). () Y Now consider the multiplicity difference in events with the light and heavy flavors (Q = c or b): δ Ql N QQ N ll =. (3) E y y The Figure function E(y) The function E(y). Page 4 of 3

5 PMC Physics A 008, :3 At Ŭ m Q, one can neglect small power-like corrections O(m / ). In such a case, the uantity δ Ql is defined by [] δ Ql = (n Q - n l ) - ΔN Q (Y Q ), (4) where the notation as well as variables Y Q ΔN Q( YQ) = N l N Q = dyn g( YQ y) ΔEQ( y), (5) m Q y = ln k (6) and are introduced. The lower limit of integration in E. (5), -ln( / the fast convergence of the integral at negative y. m Q YQ = ln Q 0 (7) m Q ), is taken - because of Let us use another dimensionless variable ρ = exp(-y). (8) The explicit form of ΔE Q was derived in our paper [] (see Fig. ): ΔE Q 7 [( y ρ)] = + ρ ρ ln ρ ρ 7 + ρ( 7ρ 0) J( ρ) + 0 J( ρ) 4. ρ (9) Page 5 of 3

6 PMC Physics A 008, :3 E Q y y The Figure function ΔE Q (y) The function ΔE Q (y). where ρ ρ+ ρ 4 ln, ρ > 4, ρ 4 J( ρ) =, ρ = 4, ρ 4 ρ arctan, ρ < 4. 4 ρ ρ The function ΔE Q (y) decreases at y - (ρ ) as (0) y ΔEQ() y e, y 3 () and has the following asymptotics at y (ρ 0): ΔEQ() y y. y () Thus, we get the relation between average multiplicities of hadrons in QQ and ll events: N ( ) = N ( ) ΔN ( m ) + ( n n ), (3) QQ ll Q Q Q l with the multiplicity difference ΔN Q defined by E. (5). Our calculations [] of the multiplicity differences δ Ql = - (Q = b, c) with the use of formula (3) appeared to be in a good agreement with the data. Recently we have reconsidered the QCD upper limit on uantity δ bl [3] which appeared to be very close to all present experimen- N ll N QQ Page 6 of 3

7 PMC Physics A 008, :3 tal 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 tt -pair production h The goal of this paper is to calculate, the average multiplicity of hadrons produced in e + e - N tt events with the primary tt pair. e 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. X is factor- e will assume that the suare of the matrix element of the process e + e - ized as follows: t t + + Me ( e t t hadrons) = Me ( e t t tt + hadrons) Mt ( hadrons) Mt ( hadrons), (4) where t ( t ) denotes the virtual top uark (antiuark). The factorization of the matrix element (4) means that there is no significant space-time overlap in the decay products of the on-shell t and t -uarks. Note that the off-shell t and t - uarks fragment into hadrons through the emission of the gluon jets in a coherent way (the first term in the r.h.s of E. (4)). The QCD non-singlet evolution of the primary virtual t-uark is very slow because the difference of virtualities in logarithmic scale is very small down to the top uark mass. In other words, the virtual t-uark becomes "real" after just a few gluon radiations. h h 4 lν The effect of possible color reconnection was investigated by comparing hadronic multiplicities in e + e and e + e l ν l events. No evidence for final state interactions was found by measuring the difference n n [8,9]. From the space-time point of view bosons and t-uarks behave in a similar way, ie. the latter manage to cover the distance Δ l ~ /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 -uarks. According to E. (4), the associative multiplicity in tt -event is given by the formula: tt t t t t h N (, m ) = N (, m ) + n, (5) Page 7 of 3

8 PMC Physics A 008, :3 where N (, m ) = C ( ) dk α s( k ) n ( k ) E (, k, m ). k π t t F g t t Q m t 0 Here and in what follows we will assume that the collision energy is a typical ILC energy, = 500 GeV, for definiteness. In such a case, contrary to E. (8), power corrections O(m t /) should be taken into account. The explicit form of the inclusive distribution of the gluon jets with (6) the invariant mass k looks like A ( k ) 4 m 4 k k m Et(, k, m ) = k d k η η 4 + k η k η+ η A η A0 η ln ( )/( ) η η ( A η)/( A0 η) η ( A η) ( A0 η) ( + m / )( + m / k ) + η, ( η )( A η)/( A 0 η) (7) where k k,, and the following notations are introduced: A + k 4 m + k =, A0 =. (8) k k This formula has been derived by calculating QCD diagrams in the first order in the strong coupling constant (see our comments after E. (3)). In the massless case (m = 0), we immediately come to the function E(k / ) (0), while by neglecting small corrections O(m / ), one can derive (after variables are properly changed) the explicit form of the function ΔE Q (k /m ) = E Q (, k, m ) - E(, k ) (9). In our case ( =, m = m t ) we will estimate the integrals in Es. (6), (7) numerically (for details, see Section 4). Now let us calculate another uantity in E. (5), n t, which describes the hadronic multiplicity of the t-uark decay products. The top weakly decays into + boson and b-uark. In its turn, the + boson decays into a uark-antiuark pair. Remember that we are interested in hadronic Page 8 of 3

9 PMC Physics A 008, :3 b b t k + t k order Figure The emission in 3the strong of the coupling gluon jets constant (spiral lines) by the uark pair resulting from the decay of the + -boson in the first The emission of the gluon jets (spiral lines) by the uark pair resulting from the decay of the + -boson in the first order in the strong coupling constant. The + boson is produced in the weak decay of the top. decays of the boson. The uark-antiuark system results in massive jets which fragment into hadrons (see Fig. 3). The gluon jets can be also emitted by the on-shell t-uark before its weak decay (the first diagram in Fig. 4) or by off-shell bottom uark (the second diagram in Fig. 4). At the end of these emissions, the on-shell b-uark weakly decays into hadrons whose average multiplicity is eual to n b. Since -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 + n tb + n b. (9) The uantity n b is experimentally measurable one [4]. The first term in E. (9), n, is the hadron multiplicity of the boson decay products. The second term, n tb, is the hadron multi- t t b k + t b b k The strong Figure emission coupling 4 of constant the gluon jets by the on-shell top uark and by off-shell bottom uark in the first order in the The emission of the gluon jets by the on-shell top uark and by off-shell bottom uark in the first order in the strong coupling constant. The emissions take place before and after the weak decay of the top, respectively. Offshell uarks are denoted as t* and b*. Page 9 of 3

10 PMC Physics A 008, :3 plicity in the gluon jets emitted by the on-shell top uark before its weak decay as well as by the bottom uark after the top decay. 3. Multiplicity of boson decay products The + boson can decay either into two light uarks ( ud and us pairs) or into cd ( cs ) pair. The former case is treated analogously to the light uark event in e + e - annihilation taken at the collision energy = m. Here we will study the latter case. Let N Ql () be hadronic multiplicity associated with the production of one heavy uark (antiuark) of the type Q and one light antiuark (uark) of the type l: N ( ) = ( n + n ) + N ( ). (30) Ql Q l Ql Now let us introduce the notation (not to confuse with ΔN Q from above): ΔNQl = Nl N ˆ Ql. (3) Then the first term in the r.h.s. of E. (9) is given in terms of the function ΔN cl by the formula: n = Nll ( Y) + [ Ncl( Yc) + nc nl], Δ (3) where Y = ln( m / Q ) c c 0 and The function N ll m Y = ln. Q 0 (33) (Y) in (3) is the hadronic multiplicity in light uark events. Thus, we need to find an expression for ΔN cl. Note that the formulae (5), (9) from Section correspond to the case when a pair of heavy or pair of light uarks is produced. Now we have to study the case when hadrons are produced in association with a single heavy uark (namely, c-uark) and one light uark. Our QCD calculations result in the following representation for the multiplicity difference (see Appendix for details): Y Q ΔN Ql( YQ) = dyn g( YQ y) ΔEQl( y), (34) Page 0 of 3

11 PMC Physics A 008, :3 E Ql y y Figure The function 5 ΔE Ql (y) The function ΔE Ql (y). with the dimensionless function ΔE Ql : ΔE Ql [( y ρ)] = [ + ρ( ρ )]ln + ( + ρ) 4 3 ρ J( ρ + ρ( 3ρ 8) J( ρ) + 6 ). ρ 4 Here ρ = exp(-y). The uantity J(ρ) was defined above (0). The function ΔE Ql (y) is shown in Fig. 5. (35) Since y ΔEQl() y 3 e, y (36) the integral (34) converges rapidly at the lower limit. Asymptotics of ΔE Ql (y) at large y is the following: ΔEQl() y y. y (37) e derive from Es. (), (37) that ΔE Ql (y) = 0.5 ΔE Q (y) at large y. Numerical calculations show that ΔE Ql is very close to ΔE Q at all y (see Fig. 6). Thus, we can put for our further numerical estimates: ΔN cl = ΔNc. (38) Page of 3

12 PMC Physics A 008, :3 E Q y, E Ql y y The Figure function 6 ΔE Q (y) (solid line) vs. function ΔE Ql (y) (dashed line) The function ΔE Q (y) (solid line) vs. function ΔE Ql (y) (dashed line). This relation means that NQl = [ Nll + NQQ] = Nll + δ Ql. (39) Correspondingly, we obtain: n = Nll ( m) + cl. 4 δ (40) 3. Multiplicity of top and bottom decay products As was already said above, the on-shell top uark can emit jets before it weakly decays into + b. After the weak decay of the top, the off-shell b-uark "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 uarks. Let us denote it as E tb. Then the multiplicity n tb will be given by the formula: Y tb ntb = dynˆ g( Ytb y) Etb( y), 0 (4) where y mt m = ln ( mb ), k (4) Page of 3

13 PMC Physics A 008, :3 and Y tb = m t m m ln ( b ), Q 0 (43) with k being the gluon jet invariant mass, (m t - m - m b ) its upper bound. In the lowest order in the strong coupling constant, the uantity E tb (y) is given by two diagrams in Fig. 4. It is presented by an integral which depends on the ratio k / m t, as well as on t b t mass ratios m / m and m / m. 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 Associated multiplicity of hadrons in tt 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 associated with the production of the tt -pair. It is of the form: tt t t t ll tb b h N (, m ) = N (, m ) + [ N ( m ) + n + n ] + [ ΔN ( m ) + n n ]. cl c c l (44) Let us remind to the reader the meaning of all uantities in E. (44). The function N t (, m t ) describes the average number of hadrons produced in association with the tt -primary pair, except for the decay products of the top(antitop) (6). The uantity N ll (m ) is the mean hadron multiplicity in the light uark event taken at the energy E = m. The hadron multiplicity n tb comes from the emission by t and b uarks (4). The uantity n l is the mean multiplicity of hadrons produced in the decay of the on-shell primary uark ( = 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 ll - and cl -pairs. As for the hadron multiplicity resulting from the decay of the on-shell top (anti)uark, it is given by nt = n + ntb + nb = Nll ( m) + ntb + nb + [ Ncl( mc) + nc nl]. Δ (45) Page 3 of 3

14 PMC Physics A 008, :3 E tb y y Figure The function 7 E tb (y) The function E tb (y). The expressions for N ll, ΔN cl are given by Es. (7), (34), respectively. Let us stress that N ll and n ( = b, c, l) are extracted from the data, and ΔN cl can be related with the measurable uantities (see formulae (38) (40) in the end of subsection 3.): Then we obtain: ΔNcl ΔNc = nc nl δ cl. (46) and nt = Nll ( m) + cl + ntb + nb, 4 δ (47) Ntt h (, mt) = Nt(, mt) + Nll ( m ) + cl + n tb + n b, 4 δ (48) where n tb is defined above (4). In what follows, we will use the value δ cl =.03 ± 0.34 (49) 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 uark event. The latter is Page 4 of 3

15 PMC Physics A 008, :3 defined by E. (7), where the function ˆn g is related with gluon jet multiplicity n g (k ) (6). e N ll have fitted the data on () by using the following QCD-motivated expression for n g (k ): ng( k ) = a + bexp c ln( k / Q0 ), (50) where c =.63, and k is the invariant mass of the jet. e have got the following values of the parameters: a = 3.89, b = 0.0, Q 0 = 0.87 GeV. (5) The result of our fit is presented in Fig. 8 in comparison with the data. Note that χ /d.o.f. becomes twice smaller if the experimental point at = 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 (5) with χ /d.o.f. = For our numerical estimates we shall use m = ± 0.03 GeV [] and the recent value of the top mass [0]: m t = 70.9 ±.8 GeV. (5) As for the bottom uark, its pole mass is uoted in [] to be m b = GeV. e will take the average value The fit of the data on the hadron multiplicity in light uark events (solid line, see formula in the text) Figure 8 The fit of the data on the hadron multiplicity in light uark events (solid line, see formula in the text). The data are taken from Ref. [6]. Page 5 of 3

16 PMC Physics A 008, :3 m b = 4.85 ± 0.5 GeV. (53) N ll By using our fit, we obtain (m ) = Then we get (see Es. (40), (49), (4)): n = 9.34 ± 0.0, (54) n tb = 6.4 ± 0.4. (55) The error in E. (54) is defined by that of the multiplicity difference δ cl (49), while that in E. (55) comes from uncertainties of the uark masses m t (5) and m b (53). Our result (54) is in a nice agreement with the experimental values from Ref. [8], n = 9.3 ± 0.3 ± 0.3, (56) and Ref. [9], n = 9.44 ± 0.3 ± 0.. (57) Now let us calculate the associated hadron multiplicity in tt event (48) at fixed energy = 500 GeV. To do this, we need to estimate the multiplicity N t (, m t ) by using formulae (6) and (7): N t ( = 500 GeV) = 4.6 ± 0.. (58) The errors in (58) come from top uark mass errors. It follows From Es. (6), (58) that Ntt ( + e e tt hadrons ) = ± 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)): k nˆ g( k ) = A + Bln. Q 0 (59) Page 6 of 3

17 PMC Physics A 008, :3 It appeared that the data on the average multiplicity in light uark events can be fitted well by using this expression (with A = 4., B = 0.0 and Q 0 = 0.93 GeV). In particular, we have obtained the following average values for the hadronic multiplicities: n = 9.5, n tb = 6.43, N t = Thus, theoretical uncertainties can be estimated to be 0.47 and 0.96 for n t and, respectively. N tt Taking into account the phenomenological value of n b [7], n b = 5.55 ± 0.09, (60) we obtain from (47): n t (t hadrons) = 4.03 ± (6) In the case when the boson decays into leptons, we predict: n ( t lν + hadrons) =. 69 ± (6) t l As a result, we obtain the average hadron multiplicity in e + e - annihilation with the primary tt - pair: Ntt ( + e e tt hadrons ) = ±.. (63) In the case when both bosons decay into leptons, the mean multiplicity of the hadrons is expected to be Ntt + e e + tt + hadrons ) = ± (64) Finally, we predict that Ntt ( + + e e tt bb + hadrons ) = ± (65) All estimations (63) (65) correspond to the collision energy = 500 GeV. e can mention the estimation of the hadron multiplicity from Ref. [], + + Ntt ( e e tt bb + hadrons) 9, which was obtained for = 390 GeV and m t = 75 GeV. For the same values of and m t, our formulae give + + Ntt ( e e tt bb + hadrons) = Page 7 of 3

18 PMC Physics A 008, :3 The formulae (6) (65) is our main result. e hope that the hadron multiplicities of the top decay products (Es. (6) and (6)) will be measured at the LHC. Appendix A Here we present some formulae for the case when the boson decays into hadrons via production of cs (or cd ) pair. Since the total width of the boson, Γ, 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 = m. It can be shown that the account of the boson width results in corrections which are numerically small (less than.6%, see Appendix B). To calculate the inclusive spectrum of the gluon jets emitted by the decay products of the - 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 boson by the top uark, with l being a 4-momentum of the t-uark, is a 4-momentum of the boson, and (l - ) is a 4-momentum of the b-uark. The corresponding expression for this part of the diagram (after convolution with the tensor parts of the boson propagators) looks like l k k μ μ ν ν l The the Figure generalized boson 9 (wavy diagram line) describing the inclusive spectrum of the gluon jets (curly line) with the virtuality k inside The generalized diagram describing the inclusive spectrum of the gluon jets (curly line) with the virtuality k inside the boson (wavy line). In its turn, the boson is a product of the weak decay of the top uark (solid line with 4- momentum l). Page 8 of 3

19 PMC Physics A 008, :3 k k The Figure inclusive 0 distribution of the massive gluon jet with the virtuality k The inclusive distribution of the massive gluon jet with the virtuality k. The wavy line is the boson, whose 4- momentum is. The thick uark line is a heavy uark, while the thin line is a light uark. The cut uark lines mean that these uarks are on-shell uarks. (, l) { l l ( l l ) g ( mt + mb m) Π μν = 4 4 μν μ ν + μν μν [( mt m m ) ] [( m m ) l m ] m b lμ mt μ ν μ t b ν t ν + μν[( mt mb ρ σ ) ( mt + mb)] + iε l, m 4 μνρσ (A.) where m t and m b are masses of the top and beauty uark, 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 produced by the boson. Let D μν be the expression corresponding to this k k The Figure same as in Fig. 0, but with the gluon jet emitted by the light uark The same as in Fig. 0, but with the gluon jet emitted by the light uark. Page 9 of 3

20 PMC Physics A 008, :3 k k The inside Figure interference the boson diagram which also contributes to the inclusive distribution of the gluon jets with the virtuality k The interference diagram which also contributes to the inclusive distribution of the gluon jets with the virtuality k inside the boson. The diagram is taken in the sum of the diagrams with the factor. diagram. In the first order in the strong coupling constant, D μν is represented by the sum of three QCD diagrams presented in Figs. 0, and. Since 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 + k μ k ν D +..., A = Π D = g Π D ( k, k) + k k Π D ( k, k), (A.) μν μν μν μν μ ν μν depends on Lorentz-invariant variables k, k, lk. Moreover, it is a polynomial of the second order in variable lk. One can use the following useful relation: d 4 k Ak klk 3 (,, ) = ( π) ( π) mt ( r) dk d( k) d( lk) A( k, k, lk), (A.3) where r m = m. (A.4) t It is naturally to integrate the function A(k, k, lk) first in variable lk, whose lower and upper limits are ( lk) ± = ( k)( ) ( ) ( ), r + r ± r k m k (A.5) by using the following formulae: Page 0 of 3

21 PMC Physics A 008, :3 ( lk) + dlk ( ) = ( r) ( k) m k, ( lk) r ( lk) + ( lk) d( lk) = ( lk) ( r )( k) ( k) m k, r ( lk) + ( lk) d( lk) = 3 ( r) 4( k) ( + r + r ) m k ( r) ( lk) r ( k) m k. (A.6) Note that the parts of tensors Π μν and D μν antisymmetric in indices (as the last term in E. (A.)) give no contribution after integration in lk. Integration limits in variable k looks like ( k) = m k, ( k) + = ( m + k ). (A.7) As a result, we obtain the hadron multiplicity associated with the charm uark in a hadronic decay of the boson (with the multiplicity of primary uark decay products subtracted): Nˆ cl = ( π ) mt ( rh ) ( lk) ( lk) + m Q 0 dk ( k) + ng( k ) d( k) k dlkak ( ) (, klk, ). ( k) (A.8) The dimensionless uantity n g (k ) in (A.8) is the multiplicity of hadrons in the gluon jet whose virtuality is k, while the normalization H is given by H = m 4 t r + r 6π ( ) ( ). (A.9) The analytic expressions for the functions D (k, k), D (k, k) in (A.) 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 Es. (34), (35) in the main text). Page of 3

22 PMC Physics A 008, :3 Appendix B In this Appendix we will demonstrate that the account of the boson width results in only small corrections to the hadronic multiplicities. The denominator of the boson propagator (see the diagram in Fig. 9) is m + im Γ, (B.) where m is the mass of the boson, Γ its full width. The mean multiplicity is given by the formula n = h N ( m m ) t 0 b d m Γ F ( ) nh( ), ( m ) + m Γ (B.) where n h (n h = n or n tb, see the main text) depends on the boson virtuality. Here and the normalization N is F ( ) = ( m ) ( m + ), (B.3) t t N = 0 d mγ F ( ). (B.4) ( m ) + m Γ In both <n h > (B.) and N (B.3) the factor m Γ is introduced, while common constants are omitted. In the zero width limit, mγ Γ 0 πδ ( m ), ( m ) + m Γ (B.5) we do obtain that the mean multiplicity is eual to n h ( ). m The numerical calculations with the use of formulae (B.), (B.3) result in the following values: <n w > = 9.04, (B.6) <n tb > = (B.7) Page of 3

23 PMC Physics A 008, :3 Thus, the account of non-zero width of the boson slightly changes the average value of the multiplicities. Namely, n (54) has gone down by 0.3, while n tb (55) has gone up by 0.3, but their sum remains almost unchanged. Acknowledgements e are thankful to the referee for his comments and critical remarks that helped us to improve the presentation of some our results. References. Petrov VA, Kisselev AV: Z Phys C 995, 66:453.. Petrov VA, Kisselev AV: Nucl Phys B (Proc Suppl) 995, 39(-3): Kisselev AV, Petrov VA: Eur Phys J C 007, 50:. 4. Schumm BA, Dokshitzer YuL, Khoze VA, Koetke DS: Phys Rev Lett 99, 69: Chrin J, DELPHI Collaboration, et al.: Proceedings of the 7th International Conference on High Energy Physics: 0 7 July 994; Glasgow, UK : Bassetto A, Ciafaloni M, Marchesini G: Nucl Phys B 980, 63: Dokshitzer YuL, Fabbri F, Khoze VA, Ochs : Eur Phys J C 006, 45: Abbiendi G, OPAL Collaboration, et al.: Phys Lett B 999, 453: Abreu P, DELPHI Collaboration, et al.: Eur Phys J C 000, 8: Pleier M-A: Talk presented at the "XXVII Physics in Collision": 6 9 June 007; Annecy, France.. Review of Particle Physics (Particle Data Group): Nucl & Part Phys G 006, 33:.. Ballestrero A, Khoze VA, Maina E, Moretti S, Stirling J: Z Phys C 996, 7:7. Page 3 of 3