Int J Theor Phys (2017) 56:1429–1439 DOI 10.1007/s10773-017-3283-0
Diquark Fragmentation Contribution in b Production T. Osati1 · M. Movlanaei1
Received: 5 July 2016 / Accepted: 7 January 2017 / Published online: 28 January 2017 © Springer Science+Business Media New York 2017
Abstract In the framework of the quark-diquark model of baryons, b can be considered as b constituent quark an ud constituent diquark. In this study, we investigate the effect ud scalar diquark fragmentation into b , therefor we calculate frgmentation functions of b quark and ud diquark into b baryon through the use of perturbative QCD. In the next stage, throuth the use of the obtained fragmentation functions, we calculate the total fragmentation probabilities and average fragmentation parameters for b −→ b and ud −→ b . Finally, the inclusive cross section of b baryon in electron-positron annihilation in ALEPH experiment is calculated with regard to ud diquark fragmentation contribution. Keywords Fragmentation function · Diquark · Cross section
1 Introduction The study of heavy hadrons have an important role in the development and acceptance of quantum chromodynamics(QCD) theory as the basic theory of strong interactions. Among heavy hadrons, heavy baryons drew highly attention, in particular baryons containing b or c quark. In the fragmentation process, a parton hit, and finally a hadron is produced by the transfer of momentum. This process has been studied extensively, and analyzed in different reactions such as annihilations of the electron-positron, electron-proton, protonantiproton and proton-proton collisions throuth the use of various models. The use of the
T. Osati
[email protected] 1
Department of Physics, Razi University, Kermanshah, Iran
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fragmentation function is one of the best ways to describe this process. The fragmentation process is described by the universal fragmentation function D(z), that gives the probability for splitting of the parton into the produced hadron plus other partons. It has been shown that the fragmentation functions for the heavy quarkonia can be calculated using the perturbative QCD [1]. The investigation of the processes with the heavy quarks is based on the factorization hypothesis [2]. The heavy hadron production amplitude may be presented as the product of the partonic part, which can be calculated using the perturbative QCD, and non-perturbative factor, which describes the free quarks into the final hadron transition. In the framework of the non-relativistic quark model, it is calculated by potential models [3–6]. Historically, the first theoretical study of the hadron production of heavy quarks was done by Bjorkan [7]. He used the simple quark model to describe the production and decay of a heavy quark to hadron in e+ e− −→ qq system. It was followed by Suzuki, he could inter spin into the heavy quark fragmentation phenomenon [8]. In 1964 Gell-Mann was the first in his writtings about quarks, raised the possibility diquarks [9]. Lichtenberg and Tassie also introduced diquarks as a combination of two quark to describe baryons in 1967 [10]. Diquark in its ground state has a positive parity, and may be an vector (spin1) or a scalar (spin 0) [11]. Diquark not only included the small momentum transfer, but also effective at large momentum transfer [12–14]. In large momentum transfer constitute quark decomposed into the current quarks, gluons and quark-antiquark pairs, in the same way, a diquark are also decomposed into the currents diquarks, gluons and diquark-antidiquark pairs. A.P.Martynenko and V.A.Saleev considered the heavy quark fragmentation into double heavy baryons in the quark-diquark model that they predicted the production rates and the shape of the energy spectra for cc and bc baryons in the region of the Z 0 at the LEP collider too [15] .The fragmentation functions for singly heavy baryons in a quarkdiquark model have been studied by Anatoly Adamov and Gary R.Goldstein [16], and the fragmentation function heavy baryons with spin-1/2 and spin-3/2 have been calculated in quark-diquark model by them. The b polarization in Z 0 boson decays has been studied [17], which predicted the longitudinal asymmetry for prompt b produced at the Z 0 resonance and estimate the spin-1/2 and spin-3/2 beauty baryon production rate. The framework based on the quasi-potential approach and relativistic quark model a new covariant expression for the heavy quark fragmentation amplitude to fragment into the pseudoscalar and vector S-wave heavy mesons is obtained [18]. The ud diquark fragmentation contribution has not been considered in [15–18], so in this work we want to consider the ud diquark fragmentation contribution into b production for the following reasons. First, recently the idea of diquark attracted renewed interest especially after pentaquarks results at the [19–24]. The second reason for considering diquark fragmentation is to reach higher fragmentation probabilities for heavy baryons production [25]. Finally, the scalar diquark fragmentation ud −→ c was done, and the contribution of the ud scalar diquark fragmentation was remarkable [26]. Our main purpose of study is to calculate the inclusive cross section production of b in the electron-positron annihilation considering the cotribution of scalar ud diquak fragmentation. Although there are various fragmentation channels of b production in the hadronization step, the contributions of two of these channels are only of the order of αs2 including: the b quark fragmentation and the ud diquark fragmentation into b . For this purpose, we devote Section 2.1 to calculate the fragmentation functions b quark into b (b −→ b ) in the quark-diquark model. In Section 2.2, the ud scalar diquark fragmentation function into b (ud −→ b ) is calculated. Besides, it is in Section 3, the inclusive cross section production b baryon is calculated in LEP by the fragmentation functions b −→ b and ud −→ b as well as the total hadronic cross section. Finally, Section 4 was devoted to finding and conclusion.
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2 The Calculation of b Fragmentation Function It is established that in the hadronization stage, there are two dominant processes in b fragmentation production, first the heavy b quark fragmentation and the scalar ud diquark fragmentation into b is second. Therefore we calculate the fragmentation functions for b −→ b and ud −→ b fragmentation processes using pQCD. To cast b −→ b fragmentation into brief expressions, the heavy b quark at first emit a gluon and then this gluon creat a scalar ud diquark pair. It is important to note that the energy at least twice the scalar ud diquark mass(mud ). Finally the scalar ud diquark will combine with the heavy b quark and b baryon will be formed. The feynman diagram in the lowest order perturbative is given for b −→ b process in Fig. 1. About ud −→ b fragmentation, one can say that the scalar ud diquark also emit a gluon and this gluon creat a bb¯ pair at the next stage and b baryon will ultimately be created resulting from the combination of the scalar ud diquark with the heavy b quark. It would be appropriate to remind that b will have 12 - spin, considering the scalar diquark and quark have zero and 12 - spin, respectively. As illustrated in Fig. 2, the feynman diagarm in the lowest order perturbative QCD is given for ud −→ b (E+P ) fragmentation. It is remarked that fragmentation parameter z is defined as z = (E+P)QB (E+P )
EB EB (or z = (E+P)DB ) which decrease into z = E (or z = E ) in the framework of infinite Q D momentum. Here EB is the energy of baryon, and EQ and ED are the initial heavy b quark and diquark, respectively.
2.1 b −→ b Fragmentation Function According to feynman digram Fig. 1, using the current vertices of qgq (quark-gluon-quark) and dgd(diquark-gluon-diquak), the invariant amplitude transition of heavy b quark into b is as follows [26–29]:
1
1
4π αs2 (2mb )αs2 (2mud )fB CF Fs (q 2 ) μ [q u(p )γμ ν(p)]. AB = 2 q d0 2mb 2p0 p0 k0 k0
Fig. 1 The Feynman diagram illustrating the lowest order perturbative production of a baryon B (b ) in a b quark fragmentation
(1)
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Fig. 2 The lowest order Feynman diagram showing ud scalar diquark (D) fragmentation into b baryon in the quark-diquark model frame
Here q μ = (k + k )μ and d0 = [p0 + k0 + k0 − p0 ] , morever Fs (q 2 ) is the scalar diquark form factor. In an appropriate form, we will have [30]: Fs (q 2 ) =
qs2
qs2 − q2
(2)
Since qs is the pole of form factor, the values of it are above 1GeV. To obtain b −→ b fragmentation function in the initial scale μ0 = mb + 2mud , we must perform the phase space integration over the amplitude square |AB |2 , so we will have the following form [29]: 1 | AB |2 δ 3 (p + k + k − p )d 3 pd 3 kd 3 k . (3) Db→b (z, μ0 ) = 2 Here with the definition a = 8CF2 fB2 αs (2mb )αs (2mud )b4 , a 3 m2B
mb mB ,
b =
mud mB ,
α =
qs mb ,
β =
√
mB ,
and const =
it also defines the functions of D1 and D2 as follows:
D1 = ((−1 + z)5 z3 (−1 + a)4 α 4 ((−1+ a)ab2 +2zab2 (1−2a +a 2 −b2 )−2z3 (−1+a)a(−a 3 −a 4 +a 5 +(1 − 2b2 )β 2 + a(−2 + b2 )β 2 + a 2 (1 − 2b2 + β 2 )) + z4 (−1 + a)(a 2 + β 2 )(2a 4 − β 2 +4aβ 2 + a 2 (2 − 3b2 − 5β 2 ) + 2a 3 (−2 + b2 + β 2 )) + z2 (−6a 5 + 2a 6 + b2 β 2 − a(1 + 3b2 )β 2 +a 3 (−2 + 6b2 − 3β 2 ) + a 4 (6 − 5b2 + β 2 ) + a 2 (2b4 + 3β 2 + b2 (−1 + 2β 2 ))))).
(4)
D2 = (2((−1 + za)2 b2 + z2 (−1 + a 2 )b2 )2 ((−1 + z)a 2 + b2 + z2 β 2 − a(−1 + z + zb2 + z2 β 2 ))2 ((−1 + za)2 b2 + z(−1 + a)(−(−1 + z)α 2 + z(−1 + a)β 2 ))2 ).
(5)
Thus for b −→ b fragmentation function we will have: Db→b (z, μ0 ) = const
D1 . D2
(6)
Note that mb is the b quark mass, and mud is the scalar ud diquark mass and mB is also mass of b baryon. Here,b quark fragmentation into b baryon is calculated in the framework of the quark-diquark model of baryon.
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2.2 The Scalar Diquark Fragmentation Function of ud into b We would like to make a comment on the scalar diquark fragmentation function of ud −→ b . Note that among different fragmentation channels to produce b in the hadronization stage, the vector or scalar ud diquark fragmentation contribution is from the order of magnitude αs2 . This is our main aim to consider the scalar ud diquark contribution in b fragmentation production. Besides, the scalar bc diquark fragmentation contribution into the heavy bcc and bbc baryons and the scalar ud diquark into c has already been studied [25, 26]. The authers have calculated bc → bcc and bc → bbc fragmentation functions qs2 . In the same approach with [25, q2 qs2 diquark form factor as Fs (q 2 ) = q 2 −q 2. s
with regard to the scalar diquark form factor as Fs (q 2 ) =
26], in the present work considering the scalar ud According to feynman diagram, in Fig. 2, and using the similar way applied in Section 2.1, ud −→ b fragmentation function in the initial scale μ0 = 2mb + mud is calculated using pQCD so. All the results as follows. Dud−→b (z, μ0 ) =
(4fB CF π 2 )2 a 4 αs (2mud )αs (2mb ) F (z) , G(z) 2m2B b3
(7)
that F (z) and G(z) are given by:
F (z) = (−1+z)4 z4 (−1+a)((1−2z)a 2 +za 3 −(−1+z)2 b2−z2 β 2 +a(−1+z−zb2 +z2 (b2 +β 2 )))2 γd4 . (8)
G(z) = 2a 2 (1 + 2z(−1 + a) + z2 (1 − 2a + a 2 + β 2 ))2 ((1 − 2z)a 2 + za 3 + (−1 + z)b2 − z2 β 2 +a(−1 + z + z2 β 2 ))2 (a(1 + 2z(−1 + a) + z2 (1 − 2a + a 2 + β 2 )) + (−1 + z)zγd2 )2 . (9)
Here γd =
qs mB .
3 b Inclusive Cross Section Production in the Pole of Z 0 As we know the hadronic total inclusive cross section production about the pole of Z 0 in electron- positron annihilation has already been measured in DELPHI, ALEPH, OPAL, L3, expriments [31]. It is important to know, that b is one of the produced hadrons in this process. It is intersting to note, that in the hadronization stage, b can be created either throgh u,d,s,c,b quarks fragmentation or by their anti, and either through ud, us, uc, ub, ds, dc, db, sc, sb,cb diquarks fragmentation or by the anti of these diquarks. As explained in
section, only b quark and ud diquark fragmentation contribution are from the order of αs2 . Therefore considering the ud diquark fragmentation contribution, b inclusive cross section production in the pole of Z 0 in the same approach with [26] can be obtained as follow: σ b = σtot
[
eb2 Db−→b e2 Dud−→b + ud ]dz. 2 q eq d ed2
(10)
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Table 1 The total fragmentation probabilities and the average fragmentation parameter in b → b and ud → b in different values of qs qs (GeV )
p(b → b )
p(ud → b )
1.2
3.48 × 10−4
0.709
3.15 × 10−5
0.529
1.4
6.62 × 10−3
0.736
5.89 × 10−5
0.529
1.42
1.33 × 10−2
0.741
6.24 × 10−5
0.529
1.43
2.25 × 10−2
0.745
6.42 × 10−5
0.529
1.444
7.62 × 10−2
0.750
6.69 × 10−5
0.529
1.45
0.225
0.753
6.8 × 10−5
0.529
Here σtot is the hadronic total cross section production [31] and eq , ed are quark and diquark electrical charge, respectivly [32]. The total fragmentation probabilities(F.P) and average fragmentation parameter(< z >) defined as follows: 1 D(z, μ0 )dz. (11) F.P = 0 1
< z >=
(12)
zD(z, μ0 )dz. 0
4 Discussion and Conclusion The fragmentation functions ud −→ b and b −→ b are calculated as the similar way with [26]. These functions given in (6) and (7). They have calculated in initial scale μ0 , which are equal to the sum of the masses particles in the finall state. The input parameters in these functions are taken the following valuse: mD = 0.6GeV [30], mb = 4.69GeV [33], mB = 5.6197GeV [33], CF = 11 12 [29], fB = 0.25GeV [29] and according to approparite momentum flow, we obtained αs (2mb ) = 0.18, and αs (2mud ) = 0.31. We also
take < kT2 > = 1GeV , which is an optimum value for this quantity. One of the important input parameters is qs . The fragmentation functions are very sensitive to the values of qs . Therefore, the derermination of the values of qs is one of our challenges, and it is important to know that there is not a lot of informations about it. Using quark-diquark model of the nucleon and electromagnetic form factors of the nucleon, Kroll et al., obtained diquark form factors. The scalar diquark form factor is assumed to has simple pole and dipole forms, respecttively, with pole positions qs above 1GeV [34]. A. Adamov and G. R. Goldstien have Table 2 The total fragmentation probabilities and the average fragmentation parameter in ud → b in different values of qs qs (GeV )
p(ud → b )
2
2.56 × 10−4
0.529
3
1.45 × 10−3
0.529
4.5
9.62 × 10−3
0.529
5
1.67 × 10−2
0.529
8.3
0.944
0.533
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12
10
D(z) 10
2
8
6
4
2
0 0
0.2
0.4
z
0.6
0.8
1
Fig. 3 Fragmentation function with qs = 1.42GeV for the process b −→ b
predicted the total fragmentation probability 1 % and 0.52 %, respectively, in their investigations on a heavy flavor baryons in c −→ c and b −→ b direct fragmentation. They also obtained the total fragmentation probability for b −→ b fragmentation processes includ∗ ∗ ing decays of b and b and c −→ c including c and c decays, 5.8 % and 3.26 %, respectively [30], whereas in the ALEPH and OPAL expriments have been measured 7.6 ± 4.2 % and 5.6 ± 2.6 %, respectively [35, 36]. The value of qs has a justly effect over the valuse of the total fragmentation probabilities and cross section production that is shown in
16 14 12
D(z) 10
5
10 8 6 4 2 0 0
0.2
0.4
z
0.6
0.8
Fig. 4 Fragmentation function with qs = 1.42GeV for the process ud −→ b
1
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Table 3 Inclusive production cross section b in e+ e− -annahilation in pole of Z 0 with qs = 1.42GeV Ecm
σ (b −→ b ) × 102 (nb)
σ (ud −→ b ) × 102 (nb)
σ ((ud + b) −→ b ) × 102 (nb)
88.464
2.6524
0.00100458
2.6534
89.455
4.85383
0.00183836
4.85567
90.212
8.8397
0.00334798
8.84305
91.207
14.833
0.00561793
14.8387
91.238
14.8524
0.00562527
14.8581
91.952
12.2728
0.00464824
12.2774
92.952
7.07467
0.00267949
7.07735
93.701
4.94597
0.00187325
4.94784
our studies. Our investigations also show the values are given qs should be satisfied these following condition Simultaneously. 1-
The values of the total fragmentation probabilities should always be less than 1 in the fragmentation process ud −→ b and b −→ b . 2- The sum of the total fragmentation probabilities for production of b should be less than 1 in the fragmentation process ud −→ b and b −→ b . 3- The values of qs should be selected in the way ,that the fragmentation probabilities finite values. Considering the obtained results by A. Adamov et al., and the above mentioned conditions, our investigations show that in c −→ c fragmentation, qs may have the values between 1GeV till 1.43GeV, respectively, that they are twice the mass of the scalar ud diquark, while in ud −→ c fragmentation the maximum value of qs may be 3GeV, that it is twice c quark mass. We obtaine the optimum value of qs = 1.17GeV for c through set equal the resultant c −→ c total fragmentation probability of results with similar results obtained by A. Adamov et al. [30]. According to the above statement the contribution of c quark fragmentation and ud diquark fragmentation in the production of c are 0.50 % and 0.11 %, respectively. Besides, in c −→ c and ud −→ c fragmentation, qs may have the values between 1.2GeV − 1.43GeV . For this values, the total fragmentation probability of c −→ c is between 0.657 % − 45.5 % and the ud −→ c total fragmentation probability is obtained between 0.133 % − 0.297 %. In the ud −→ c fragTable 4 Inclusive production cross section b in e+ e− -annahilation in pole of Z 0 with qs = 1.45GeV Ecm
σ (b −→ b ) × 102 (nb)
σ (ud −→ b ) × 102 (nb)
σ ((ud + b) −→ b ) × 102 (nb)
88.464
44.748
0.00109417
44.7491
89.455
81.8881
0.0020023
81.8901
90.212
149.133
0.00364656
149.136
91.207
250.245
0.00611893
250.252
91.238
250.573
0.00612693
250.579
91.952
207.052
0.00506277
207.057
92.952
119.355
0.00291844
119.358
93.701
83.4424
0.00204031
83.4444
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16 14
(nb) 10
2
12 10 8 6 4 2 88
89
90
E
91 cm
(GeV)
92
93
94
Fig. 5 Inclusive cross section with qs = 1.42GeV for the process (b + ud) −→ b in ALEPH
mentation, qs can also increase until 3GeV, that in this case the contribution of the scalar ud diquark fragmentation in c production will give to 39 %, that in the phenomenologycal view, this is a correct value, because in this circumstances the contribution of c quark fragmentation into c is bigger than ud diquark fragmentation into c too. Therefore one 2.5
(nb)
2
1.5
1
.5
0 88
89
90
E
91 cm
(GeV)
92
93
94
Fig. 6 Inclusive cross section with qs = 1.45GeV for the process (b + ud) −→ b in ALEPH
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can concluded that the scalar ud diquark fragmentation in the production of c has the remarkable contribution [26]. Our investigations show that in b −→ b quark fragmentation, qs can have values between 1GeV − 1.45GeV that twice the mass of the scalar ud diquark. In the ud −→ b fragmentation process, qs can increase until 8.3GeV that it is approximaitly twice the mass of heavy b quark. In the phenomenologycal view, the contribution of the scalar ud diquark fragmentation in the production of b should be smaller than b quark fragmentation contribution. So, in ud −→ b fragmentation process, qs could not be more than 2GeV. The optimum value of qs = 1.42GeV , through set equal the total fragmentation probability resultant b of our calculation with the similar result obtained by A. Adamov et al. [30], that in this case the contribution of b quark fragmentation in the production of b is obtained 1.33 %, that have an excellent agreement with [30]. In Tables 1 and 2 the total fragmentation probabilities and average fragmentation parameter are given for b −→ b and ud −→ b fragmentation processes in the different values of qs . The contribution of the scalar ud diquark in the production of b is 0.006 % considering qs = 1.42GeV . Our investigations show that generally in b −→ b , qs can have the values from 1.42GeV – 1.45GeV. As we showed that the contribution of b quark fragmentation in the production of b is between 1.33 % − 22.5 %, that arrive at an good agreement with the obtained result in AELPH expriment [36]. For all qs there are between, 1.42GeV − 1.45GeV, the total fragmentation probabilities for ud −→ b are between 0.006 % − 0.007 %. If we also consider qs = 2GeV , in ud −→ b fragmentation process, the total fragmentation probability will be 0.256 %. These data show generally, the scalar ud diquark fragmentation has not the remarkable contribution in b production. In Figs. 3 and 4 the behavior of fragmentation functions of b −→ b and ud −→ b given according to z. Note that our investigations demonstrate that the value of qs show natural behavior in ud −→ c fragmentation process. As we declared in ud −→ c , the value of qs can increase until 3GeV, namely twice the mass of c quark, whereas in ud −→ b because of the limitation of phenomenological, qs can ultimately increase until 2GeV. The inclusive cross section production of b in ALEPH expriment is calculated in the fragmentation processes for ud −→ b and b −→ b using (10) for qs = 1.42GeV and qs = 1.45GeV , see Tables 3 and 4. The behavior of total cross section production of b according to Ecm has been shown in Figs. 5 and 6. The obtained data show that first of all, the contribution of ud diquark in the production of b is petty. secondly, the possiblity of b production in the pole of Z 0 , namely the centeral of mass energy 91.2GeV is maximum, and the dominant mechanism in b production, is the b quark fragmentation,see Tables 1 and 2. Acknowledgments I would like to take this opportunity to acknowledge Ms.Lila Ghaderi Vahed to this paper make corrections.
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