- Email: [email protected]
Leptoquark pair production in γγ scattering: threshold resummation
Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
Leptoquark pair production in gg scattering: threshold resummation Johannes Blu. mleina,*, Alexander Kryukovb a
DESY, Deutsches Elektronen-Synchrontron, Platanenallee 6, D-15735 Zeuthen, Germany b Institute of Nuclear Physics, Moscow State University, RU-119899 Moscow, Russia
Abstract The possibilities to pair-produce leptoquarks in photon–photon collisions are discussed. QCD threshold corrections lead to a strong enhancement of the production cross-section. Suitably long-lived leptoquarks (GF t100 MeV) may form Leptoquarkonium states. r 2001 Elsevier Science B.V. All rights reserved. PACS: 12.60.i; 12.38.Cy Keywords: Leptoquark; Resummation; Leptoquarkonium
Leptoquarks are hypothetical particles which combine quantum numbers of the fundamental fermions of the Standard Model and emerge as bosonic (scalar and vector) states in various extensions of the Standard Model such as unified theories and sub-structure models. In most of the scenarios, the mass spectrum of these states is not predicted. In a series of models, however, one expects states in the range of several hundred GeV to a few TeV. These particles can be searched at the next generation colliders as LHC and future eþ e linear colliders. Currently, the following mass ranges are excluded by experiment for scalar leptoquarks: 1st generation leptoquarks: M > 242 GeV ½1 ðCDF þ D0Þ; 2nd generation leptoquarks: M > 202 GeV ½2 ðCDFÞ; 2nd generation leptoquarks: M > 200 GeV ½3 ðD0Þ; 3rd generation leptoquarks: M > 99 GeV ½4 ðCDFÞ; 3rd generation leptoquarks: M > 94 GeV ½5 ðD0Þ at 95% CL irrespective of the size of the fermion–leptoquark couplings, which is limited to very small values [6,7] for most of the leptoquark species. Somewhat higher bounds are derived for vector leptoquark states, depending on the size of their anomalous couplings to the gluon [8]. Unlike the fermionic couplings the couplings of the leptoquarks to the gauge bosons of the Standard Model are known, cf. Ref. [9], and are of the size of the standard gauge couplings of the fermions. Due to the dimunitive size of the fermionic *Corresponding author. E-mail address: [email protected] (J. Blu. mlein).
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 1 9 1 - 3
244
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
couplings, leptoquark pair production processes at high energy colliders1 allow to perform a widely model independent search for these states. At eþ e high energy linear colliders the largest production cross-sections are obtained for eþ e annihilation. This process was studied in detail in Refs. [9,14] both for scalar and vector leptoquark states. The QED radiative corrections to scalar leptoquark pair production were calculated at Oða logðs=m2e Þ; Oðða logðs=m2e ÞÞ2 Þ both for initial and final state radiations as well as the Oðas Þ QCD correction and the correction due to beamstrahlung in Ref. [16]. These corrections are large in the threshold range. The enhancement due to the QCD corrections, being dominated by the Coulomb singularity p1=b at low velocities is nearly balanced by the losses due to the QED corrections and Beamstrahlung despite the small size of the QED coupling constant. The second order QED corrections are still of the size of Oð10%Þ of the Born cross-section and therefore have to be taken into account. A second important production channel is photon–photon pair production of leptoquarks. Here, we consider the case that the photon beams are produced from the electron and positron beams, respectively, by laser beam Compton back-scattering. The photon energy spectrum [17] is described by 1 1 4z 4z2 Fg ðzÞ ¼ 1zþ þ ð1Þ NðxÞ 1 z xð1 zÞ x2 ð1 zÞ2 NðxÞ ¼
16 þ 32x þ 18x2 þ x3 x2 4x 8 þ logð1 þ xÞ x2 2xð1 þ xÞ2
ð2Þ
where pffiffiffi z denotes the longitudinal momentum fraction of the photons after beam conversion and x ¼ 2 ð 2 þ 1Þ; with zpx=ð1 þ xÞ: Alternatively, one may consider leptoquark pair production by photon– photon scattering by preparing the initial state through Weizs.acker–Williams emission from the eþ e beams. These contributions are, however, much smaller than those due to Compton conversion, cf. Refs. [13,8]. The photon–photon cross-section reads Z zmax Z zmax sFF% ðsÞ ¼ dz1 dz2 Fg ðz1 ÞFg ðz2 Þs# FF% ðz1 z2 sÞyðz1 z2 s 4MF2 Þ: ð3Þ 0
0
For scalar leptoquarks, the direct contribution to the sub-system cross-section is given by 1 þ b pa2 4 2 4 : Q 2ð2 b Þb ð1 b Þ log s# FS F% S ðsÞ ¼ 1 b s F
ð4Þ
The corresponding relations for vector leptoquarks were derived in Ref. [13] and are somewhat lengthy due to the emergence of anomalous couplings. % ; direct–resolved and resolved–resolved terms are present, Besides the direct contributions gg-FF % Þ ¼ sdir þ sres;dir þ sres : sðg þ g-FF ð5Þ The latter ones are hadronic contributions and were calculated for both scalar and vector leptoquarks in Refs. [12,13,8] including two anomalous couplings for vector leptoquarks. The numerical analysis shows that these terms contribute significantly only far away from the threshold, i.e. typically for larger values of the cms velocities bX0:8 of the leptoquarks [8]. In the search region (threshold) the photo-pair production cross-sections behave as pQ4F and may vary by a factor of 625 between the production cross-sections for jQF j ¼ 13 and for jQF j ¼ 53 states, which have the same cross-section at a hadron collider. For vector leptoquark, the cross-sections are strongly sensitive to the anomalous couplings kg and lg [13,8]. In the threshold region, QCD corrections to the photon–photon process, similar to the in eþ e annihilation [16], are very important. These corrections can only be calculated for the case of scalar 1
This applies to high energy pp [8,10,11], ep [12,8,11], ge [8], gg [13,8] and eþ e [9,13–15] collisions.
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
245
leptoquark pair production, since for vector leptoquarks the effective Lagrangian does not correspond to a renormalizable theory. We will therefore limit the consideration here to the case of scalar leptoquarks. Threshold resummations of the universal terms have been considered in the literature before, cf. Refs. [18– 20]. For a final state of a pair of scalar leptoquarks we follow Ref. [21]. The Born cross-section dsB obtains the correction factor KS ðEÞ=KSðBÞ ðEÞ; ds ¼ dsB
KS ðEÞ KSB ðEÞ
ð6Þ
K ðBÞ ðEÞ ¼
MF pffiffiffiffiffiffiffiffiffiffiffi Mf E 4p
ð7Þ
KS ðEÞ ¼
with E ¼
8 > > M2 < k F
þ
4p > M > : F
þ
X N 2k1 kþ 2k21 arctan þ MF k MF2 n4 n¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 2 2 2 > GF k1 n þ kþ n E þ GF þ k1 =MF > = ½E þ k21 =ðMF n2 Þ2 þ G2F
> > ;
ð8Þ
pffiffi s 2MF : MS and GS denote the mass and width of the scalar leptoquark, respectively and
k1 ¼ 23as ðMF ÞMF
Fig. 1. Ratio pffiffiof the leptoquark pair production cross-section including the threshold resummation and the Born cross-section as a function of s: Left full line: M ¼ 200 GeV; dashed line: M ¼ 250 GeV; dotted line: M ¼ 300 GeV; dash–dotted line: M ¼ 350 GeV; right full line: M ¼ 400 GeV:
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
246
pffiffi % FÞ as a function of s: Each pair of equally drawn lines corresponds to the Fig. 2. Leptoquark pair production cross-section sðgg-F same leptoquark mass. Upper line: sB þ sresum ; lower line: sB ; Upper full lines: M ¼ 200 GeV; dashed lines: M ¼ 250 GeV; dotted lines: M ¼ 300 GeV; dash–dotted lines: M ¼ 350 GeV; lower full lines: M ¼ 400 GeV:
k7
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MF 2 2 E þ GF 7E : ¼ 2
Here, the strong coupling constant as ¼ as ðMS2 Þ was considered to be fixed. Fig. 1 shows the ratio of the integrated cross-section with threshold resummation and the Born crosssection as assuming a decay width of GS ¼ 1 GeV:2 Threshold enhancements of a factor 5–8 can be obtained. If the width of the leptoquarks turns out to be t100 MeV Leptoquarkonia can be formed in ggcollisions, cf. Ref. [16].3 The b-behavior at the threshold favors the gg-process (spb) in comparison to the eþ e process (spb3 ). In Fig. 2, shows the total cross-sections with and without threshold resummations. To summarize, photon pair production of scalar and vector leptoquarks at future eþ e linear colliders were studied. The cross-sections vary by the leptoquark charge as jQF j4 ; which may mean a variation up to a factor of 625. This production process offers a background free window to study the anomalous couplings kA and lA of potential vector leptoquark states to the photon. The threshold QCD enhancement of the photon–photon process is of O(5–8) for typical choices of the parameters. Leptoquarkonia can be formed in the photon–photon process iff their width is GF t100 MeV: This work is supported in part by EU contract FMRX-CT98-0194(DG 12-MIHT). 2 The ‘idealized’ decay widths are by far smaller due to the small fermionic couplings. However, the real leptoquark width should be larger since these particles are supposed to fragment into hadrons like the quarks. A theoretically safe value of the width is hard to obtain due to these non-perturbative effects. 3 This possibility was later discussed in Refs. [22,23] too.
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
247
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16]
[17] [18] [19] [20] [21] [22] [23]
CDF and D0 Collaboration, FERMILAB-PUB-98=312-E. F. Abe, et al., CDF Collaboration, Phys. Rev. Lett. 78 (1997) 2906. B. Abbott, et al., D0 Collaboration, FERMILAB-PUB-99=314-E and hep-ex=9910049. F. Abe, et al., CDF Collaboration, Phys. Rev. Lett. 81 (1998) 4806. B. Abbott, et al., D0 Collaboration, Phys. Rev. Lett. 81 (1998) 38. A.F. Zarnecki, hep-ph=0006196. S. Davidson, D. Bailey, B.A. Campbell, Z. Phys. C 61 (1994) 613. J. Blu. mlein, E. Boos, A. Kryukov, Z. Phys. C 76 (1997) 137. J. Blu. mlein, R. Ru. ckl, Phys. Lett. B 304 (1993) 337. J. Blu. mlein, E. Boos, A. Kryukov, DESY 97-067, hep-ph=9811271. J. Blu. mlein, Z. Phys. C 74 (1997) 605. J. Blu. mlein, E. Boos, A. Kryukov, Mod. Phys. Lett. A 9 (1994) 3007. J. Blu. mlein, E. Boos, Nucl. Phys. Proc. B 37 (Suppl.) (1994) 181. J. Blu. mlein, E. Boos, A. Kryukov, Phys. Lett. B 392 (1997) 105. J. Blu. mlein, P. Frampton, IFP-475-UNC and in: Proceedings of the second Internnational Workshop on Physics with Linear eþ e Colliders, Waikoloa, HI, 26–30 April 1993, World Scientific, Singapore, 1993, p. 926; F. Cuypers, P. Frampton, R. Ru. ckl, Phys. Lett. B 390 (1997) 221. J. Blu. mlein, DESY 93–132, Proceedings of the second International Workshop on Physics and Experiments with Linear eþ e Colliders, Waikoloa, HI, 26–30 April 1999, p. 542; J. Blu. mlein, DESY 93–153, Proceedings of the Linear Collider Workshop 1993, p. 419. I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Nucl. Instr. and Meth. 205 (1983) 47. V.S. Fadin, V.A. Khoze, JETP Lett. 46 (1987) 525; V.S. Fadin, V.A. Khoze, Sov. J. Nucl. Phys. 48 (1988) 309. I.I. Bigi, V.S. Fadin, V.A. Khoze, Nucl. Phys. B 377 (1992) 461. M.J. Strassler, M.E. Peskin, Phys. Rev. D 43 (1991) 1500. I.I. Bigi, F. Gabbiani, UND-HEP-91-BIG08. V.V. Kiselev, Phys. Atom. Nucl. 62 (1999) 300; V.V. Kiselev, Phys. Rev. D 58 (1998) 054008. D. Browser-Chao, T.D. Imbo, B.A. King, E.C. Martell, Phys. Lett. B 432 (1998) 167.
Leptoquark pair production in gg scattering: threshold resummation Johannes Blu. mleina,*, Alexander Kryukovb a
DESY, Deutsches Elektronen-Synchrontron, Platanenallee 6, D-15735 Zeuthen, Germany b Institute of Nuclear Physics, Moscow State University, RU-119899 Moscow, Russia
Abstract The possibilities to pair-produce leptoquarks in photon–photon collisions are discussed. QCD threshold corrections lead to a strong enhancement of the production cross-section. Suitably long-lived leptoquarks (GF t100 MeV) may form Leptoquarkonium states. r 2001 Elsevier Science B.V. All rights reserved. PACS: 12.60.i; 12.38.Cy Keywords: Leptoquark; Resummation; Leptoquarkonium
Leptoquarks are hypothetical particles which combine quantum numbers of the fundamental fermions of the Standard Model and emerge as bosonic (scalar and vector) states in various extensions of the Standard Model such as unified theories and sub-structure models. In most of the scenarios, the mass spectrum of these states is not predicted. In a series of models, however, one expects states in the range of several hundred GeV to a few TeV. These particles can be searched at the next generation colliders as LHC and future eþ e linear colliders. Currently, the following mass ranges are excluded by experiment for scalar leptoquarks: 1st generation leptoquarks: M > 242 GeV ½1 ðCDF þ D0Þ; 2nd generation leptoquarks: M > 202 GeV ½2 ðCDFÞ; 2nd generation leptoquarks: M > 200 GeV ½3 ðD0Þ; 3rd generation leptoquarks: M > 99 GeV ½4 ðCDFÞ; 3rd generation leptoquarks: M > 94 GeV ½5 ðD0Þ at 95% CL irrespective of the size of the fermion–leptoquark couplings, which is limited to very small values [6,7] for most of the leptoquark species. Somewhat higher bounds are derived for vector leptoquark states, depending on the size of their anomalous couplings to the gluon [8]. Unlike the fermionic couplings the couplings of the leptoquarks to the gauge bosons of the Standard Model are known, cf. Ref. [9], and are of the size of the standard gauge couplings of the fermions. Due to the dimunitive size of the fermionic *Corresponding author. E-mail address: [email protected] (J. Blu. mlein).
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 1 9 1 - 3
244
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
couplings, leptoquark pair production processes at high energy colliders1 allow to perform a widely model independent search for these states. At eþ e high energy linear colliders the largest production cross-sections are obtained for eþ e annihilation. This process was studied in detail in Refs. [9,14] both for scalar and vector leptoquark states. The QED radiative corrections to scalar leptoquark pair production were calculated at Oða logðs=m2e Þ; Oðða logðs=m2e ÞÞ2 Þ both for initial and final state radiations as well as the Oðas Þ QCD correction and the correction due to beamstrahlung in Ref. [16]. These corrections are large in the threshold range. The enhancement due to the QCD corrections, being dominated by the Coulomb singularity p1=b at low velocities is nearly balanced by the losses due to the QED corrections and Beamstrahlung despite the small size of the QED coupling constant. The second order QED corrections are still of the size of Oð10%Þ of the Born cross-section and therefore have to be taken into account. A second important production channel is photon–photon pair production of leptoquarks. Here, we consider the case that the photon beams are produced from the electron and positron beams, respectively, by laser beam Compton back-scattering. The photon energy spectrum [17] is described by 1 1 4z 4z2 Fg ðzÞ ¼ 1zþ þ ð1Þ NðxÞ 1 z xð1 zÞ x2 ð1 zÞ2 NðxÞ ¼
16 þ 32x þ 18x2 þ x3 x2 4x 8 þ logð1 þ xÞ x2 2xð1 þ xÞ2
ð2Þ
where pffiffiffi z denotes the longitudinal momentum fraction of the photons after beam conversion and x ¼ 2 ð 2 þ 1Þ; with zpx=ð1 þ xÞ: Alternatively, one may consider leptoquark pair production by photon– photon scattering by preparing the initial state through Weizs.acker–Williams emission from the eþ e beams. These contributions are, however, much smaller than those due to Compton conversion, cf. Refs. [13,8]. The photon–photon cross-section reads Z zmax Z zmax sFF% ðsÞ ¼ dz1 dz2 Fg ðz1 ÞFg ðz2 Þs# FF% ðz1 z2 sÞyðz1 z2 s 4MF2 Þ: ð3Þ 0
0
For scalar leptoquarks, the direct contribution to the sub-system cross-section is given by 1 þ b pa2 4 2 4 : Q 2ð2 b Þb ð1 b Þ log s# FS F% S ðsÞ ¼ 1 b s F
ð4Þ
The corresponding relations for vector leptoquarks were derived in Ref. [13] and are somewhat lengthy due to the emergence of anomalous couplings. % ; direct–resolved and resolved–resolved terms are present, Besides the direct contributions gg-FF % Þ ¼ sdir þ sres;dir þ sres : sðg þ g-FF ð5Þ The latter ones are hadronic contributions and were calculated for both scalar and vector leptoquarks in Refs. [12,13,8] including two anomalous couplings for vector leptoquarks. The numerical analysis shows that these terms contribute significantly only far away from the threshold, i.e. typically for larger values of the cms velocities bX0:8 of the leptoquarks [8]. In the search region (threshold) the photo-pair production cross-sections behave as pQ4F and may vary by a factor of 625 between the production cross-sections for jQF j ¼ 13 and for jQF j ¼ 53 states, which have the same cross-section at a hadron collider. For vector leptoquark, the cross-sections are strongly sensitive to the anomalous couplings kg and lg [13,8]. In the threshold region, QCD corrections to the photon–photon process, similar to the in eþ e annihilation [16], are very important. These corrections can only be calculated for the case of scalar 1
This applies to high energy pp [8,10,11], ep [12,8,11], ge [8], gg [13,8] and eþ e [9,13–15] collisions.
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
245
leptoquark pair production, since for vector leptoquarks the effective Lagrangian does not correspond to a renormalizable theory. We will therefore limit the consideration here to the case of scalar leptoquarks. Threshold resummations of the universal terms have been considered in the literature before, cf. Refs. [18– 20]. For a final state of a pair of scalar leptoquarks we follow Ref. [21]. The Born cross-section dsB obtains the correction factor KS ðEÞ=KSðBÞ ðEÞ; ds ¼ dsB
KS ðEÞ KSB ðEÞ
ð6Þ
K ðBÞ ðEÞ ¼
MF pffiffiffiffiffiffiffiffiffiffiffi Mf E 4p
ð7Þ
KS ðEÞ ¼
with E ¼
8 > > M2 < k F
þ
4p > M > : F
þ
X N 2k1 kþ 2k21 arctan þ MF k MF2 n4 n¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 2 2 2 > GF k1 n þ kþ n E þ GF þ k1 =MF > = ½E þ k21 =ðMF n2 Þ2 þ G2F
> > ;
ð8Þ
pffiffi s 2MF : MS and GS denote the mass and width of the scalar leptoquark, respectively and
k1 ¼ 23as ðMF ÞMF
Fig. 1. Ratio pffiffiof the leptoquark pair production cross-section including the threshold resummation and the Born cross-section as a function of s: Left full line: M ¼ 200 GeV; dashed line: M ¼ 250 GeV; dotted line: M ¼ 300 GeV; dash–dotted line: M ¼ 350 GeV; right full line: M ¼ 400 GeV:
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
246
pffiffi % FÞ as a function of s: Each pair of equally drawn lines corresponds to the Fig. 2. Leptoquark pair production cross-section sðgg-F same leptoquark mass. Upper line: sB þ sresum ; lower line: sB ; Upper full lines: M ¼ 200 GeV; dashed lines: M ¼ 250 GeV; dotted lines: M ¼ 300 GeV; dash–dotted lines: M ¼ 350 GeV; lower full lines: M ¼ 400 GeV:
k7
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MF 2 2 E þ GF 7E : ¼ 2
Here, the strong coupling constant as ¼ as ðMS2 Þ was considered to be fixed. Fig. 1 shows the ratio of the integrated cross-section with threshold resummation and the Born crosssection as assuming a decay width of GS ¼ 1 GeV:2 Threshold enhancements of a factor 5–8 can be obtained. If the width of the leptoquarks turns out to be t100 MeV Leptoquarkonia can be formed in ggcollisions, cf. Ref. [16].3 The b-behavior at the threshold favors the gg-process (spb) in comparison to the eþ e process (spb3 ). In Fig. 2, shows the total cross-sections with and without threshold resummations. To summarize, photon pair production of scalar and vector leptoquarks at future eþ e linear colliders were studied. The cross-sections vary by the leptoquark charge as jQF j4 ; which may mean a variation up to a factor of 625. This production process offers a background free window to study the anomalous couplings kA and lA of potential vector leptoquark states to the photon. The threshold QCD enhancement of the photon–photon process is of O(5–8) for typical choices of the parameters. Leptoquarkonia can be formed in the photon–photon process iff their width is GF t100 MeV: This work is supported in part by EU contract FMRX-CT98-0194(DG 12-MIHT). 2 The ‘idealized’ decay widths are by far smaller due to the small fermionic couplings. However, the real leptoquark width should be larger since these particles are supposed to fragment into hadrons like the quarks. A theoretically safe value of the width is hard to obtain due to these non-perturbative effects. 3 This possibility was later discussed in Refs. [22,23] too.
J. Blu.mlein, A. Kryukov / Nuclear Instruments and Methods in Physics Research A 472 (2001) 243–247
247
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16]
[17] [18] [19] [20] [21] [22] [23]
CDF and D0 Collaboration, FERMILAB-PUB-98=312-E. F. Abe, et al., CDF Collaboration, Phys. Rev. Lett. 78 (1997) 2906. B. Abbott, et al., D0 Collaboration, FERMILAB-PUB-99=314-E and hep-ex=9910049. F. Abe, et al., CDF Collaboration, Phys. Rev. Lett. 81 (1998) 4806. B. Abbott, et al., D0 Collaboration, Phys. Rev. Lett. 81 (1998) 38. A.F. Zarnecki, hep-ph=0006196. S. Davidson, D. Bailey, B.A. Campbell, Z. Phys. C 61 (1994) 613. J. Blu. mlein, E. Boos, A. Kryukov, Z. Phys. C 76 (1997) 137. J. Blu. mlein, R. Ru. ckl, Phys. Lett. B 304 (1993) 337. J. Blu. mlein, E. Boos, A. Kryukov, DESY 97-067, hep-ph=9811271. J. Blu. mlein, Z. Phys. C 74 (1997) 605. J. Blu. mlein, E. Boos, A. Kryukov, Mod. Phys. Lett. A 9 (1994) 3007. J. Blu. mlein, E. Boos, Nucl. Phys. Proc. B 37 (Suppl.) (1994) 181. J. Blu. mlein, E. Boos, A. Kryukov, Phys. Lett. B 392 (1997) 105. J. Blu. mlein, P. Frampton, IFP-475-UNC and in: Proceedings of the second Internnational Workshop on Physics with Linear eþ e Colliders, Waikoloa, HI, 26–30 April 1993, World Scientific, Singapore, 1993, p. 926; F. Cuypers, P. Frampton, R. Ru. ckl, Phys. Lett. B 390 (1997) 221. J. Blu. mlein, DESY 93–132, Proceedings of the second International Workshop on Physics and Experiments with Linear eþ e Colliders, Waikoloa, HI, 26–30 April 1999, p. 542; J. Blu. mlein, DESY 93–153, Proceedings of the Linear Collider Workshop 1993, p. 419. I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Nucl. Instr. and Meth. 205 (1983) 47. V.S. Fadin, V.A. Khoze, JETP Lett. 46 (1987) 525; V.S. Fadin, V.A. Khoze, Sov. J. Nucl. Phys. 48 (1988) 309. I.I. Bigi, V.S. Fadin, V.A. Khoze, Nucl. Phys. B 377 (1992) 461. M.J. Strassler, M.E. Peskin, Phys. Rev. D 43 (1991) 1500. I.I. Bigi, F. Gabbiani, UND-HEP-91-BIG08. V.V. Kiselev, Phys. Atom. Nucl. 62 (1999) 300; V.V. Kiselev, Phys. Rev. D 58 (1998) 054008. D. Browser-Chao, T.D. Imbo, B.A. King, E.C. Martell, Phys. Lett. B 432 (1998) 167.