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Sizes and couplings of composite leptons, quarks and W bosons
Volume 144B, number 1,2
PHYSICS LETTERS
23 August 1984
SIZES AND COUPLINGS OF COMPOSITE LEPTONS, QUARKS AND W BOSONS F.M. RENARD
Ddpartement de PhysiqueMath~matique l, USTL, 1:-34060Montpellier Cedex, France Received 19 April 1984
We assume that leptons and quarks are much smaller objects than W bosons (A~,q ~- 3.6 TeV; Aw =Mw). This explains the weakness of W couplings to leptons and quarks (as opposed to strong couplings required by radiative Z° decays), the strength of sinZ0w and the present absence of direct lepton and quark substructure effects. We also define a relativistic description of W substructure which should replace the usual non-relativistic wave function and allow better estimates of several processes.
The anomalously large rate o f radiative Z decays [1 ] renewed the interest in composite models for leptons, quarks and W bosons. Several explanations have been proposed for these decays [2]. Most o f them are based on the assumption that weak interactions are residual effects due to the underlying substructure [3]. It was shown [4] before W, Z discovery that such a composite picture can lead to a phenomenology of electroweak interactions in many aspects very similar to the standard one but also [5] that a priori rare W boson decays could be enhanced by substructure effects. The analogy with hadronic interactions allows to expect a large effective coupling to appear in one place or another (ZZT, ZST, Z ~ * , ...) and for example to lead to the enhancement required by Z -+ ££7 events. However there are puzzling questions and deficiencies in this picture. In the present paper we address and answer the following questions: Why are the couplings o f W bosons to leptons and quarks much weaker than usual hadronic couplings (required for enhancing Z ~ ~£7 decays)? Why is the value o f sin20 w (due to 7 - W 3 mixing) so large? Why is there no direct signal o f lepton and quark substructure (no form factor, no contact term, no 1 Physique Math~matique et Th~orique, Equipe de Recherche Associ6e au CNRS. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
g - 2 effects .... )? How could one be confident in non-relativistic descriptions o f W bosons as sub-constituent bound states? Our explanations are based on the assumption that leptons and quarks are much smaller objects than W bosons. We start by showing how one can replace the nonrelativistic approximation (and the weak binding limit) by a fully relativistic description allowing the use o f Feynman diagram techniques. We define a W - h - h ' vertex ~ (1 + 75) ~ W(k, k') whose correspondence with the 3S 1 bound state wave function ¢ ( k ) in momentum space is given by [6] ~(k) .~ [(2n)3Mw ] -1/2
×f
dk 0
4m 2
w(k,k')(k z _m2)(k,2
-m2) ,
(1)
m being the effective mass o f the subconstituents h and h' o f the W bosons. For illustration we take the simple form I~(k, k') = gw A 4 / ( k2 - A 2 ) ( k'2 - A 2 ) •
(2)
As a first application we consider the 7 - W 3 junction Ml2eueU represented in fig. 1. A straightforward calculation o f the Feynman diagram using the vertex function l¢(k,k') gives (for m <~ A w ~ M w ) 119
Volume 144B, number 1,2
PHYSICS LETTERS h
')
~c~ Fig. 1. "r-Wa junction.
X = -(gW/247r2)e(Q)V~H,
(3)
where (Q) and n H are the average constituent charge and hypercolor number. Eq. (3) replaces the usual non-relativistic expression [7] X = [~b~O)/x/~]
e(Q)'~H/Mw3/2 ,
(4)
with ~ 0 ) = (2rr)-3/2fd3@(k). The normalization of gw [equivalent to that of 4~(k) through (1)] can be estimated from fd3k l¢(k)12/47r = 1 and assuming an extension Mff 1 . The result Igw[ ~ 1 2 r r 2 v ~ ,
(5)
roughly agrees with eq. (3) taking for example (Q) = 1/X/~,n H = 3 and the value [ 8 - 1 1 ] X-~ sin0 w -~0.5. We then consider the couplings o f W bosons to other bosonic or fermionic bound states, From the pictures of fig. 2a and 2b and the hadronic analogy we would expect a rather large dimensionless coupling constant f. A comparison with gorrrr -~ 6 and g~rNN ~ 13 suggests that f c o u l d easily be of order 10. In fact this is precisely the order of magnitude required by the interpretations [2,3] of Z -+ £~/events based on high mass intermediate states like S, Z or 2*. So we adopt this magnitude for the couplings o f W bosons to other extended objects (type 2a). In early times of composite models based on low compositeness scales it was found natural to expect [7,12] that the couplingg of W to leptons or quarks be also of orderf. In turn asg2/8M2w = G/x/~this led to expect large M w masses. Later additional constraints (like asymptotic symmetry [8], unitarity [9], current algebra [10] or W-dominance [ 11]) enforced g a n d M w to be closer to the standard values but raised the question: why is g much smaller than f. We t
___[
w -
....
-
B' (cO
.....
t' Cb)
T Co)
Fig. 2. Couplings ofW bosons and photon to other bound states, (a) extended bosons, (b) and (c) leptons and quarks.
120
23 August 1984
now explicitly show how this can be a consequence of a different size for W bosons and for lepton or quarks. It is intuitively obvious from fig. 2b that the effective couplings depend upon the overlap of W and leptons (or quarks) extensions. Contrarily to the case of fig. 2a where the overlap is of order 1, in fig. 2b we have the overlap
f d 3 k ~](k, k')I~(k, k')l 2 ,
(6)
for which a non-relativistic approximation with constant wave functions gives 6rr2(R~/Rw)3. Let us make a relativistic calculation using an effective lepton vertex function [ 13]
[~b~(k, k')i 2 = ere/[k ' 2 - A2] 2 .
(7)
The normalization is obtained by the computation of the electromagnetic Dirac form factor (fig. 2c) and F 1(0) = 1 ; this gives gre = 32rr2A 2 .
(8)
A similar calculation of the diagram of fig. 2b then gives [by identification to the usual W££' vertex (g/ 2~u/2)fi~(1 + @ ) ¢ u~,]:
g = ½X,/2fgw(Aw/AQ) 2 ~- 8~2x/~f(Aw/Are) 2 ,
(9)
using eq. (5). This clearly shows the dependence o f g upon the extension ratio (Aw/Are)2. I f f is of order 10 the actual value g -~ 0.65 (i.e, e/sin 0 w in the standard model) requires (Aw/Are) 2-'" 5 × 10 - 4 , i.e. A~ -~ 45A w which means Are -~ 3.6 TeV i f A w ~ M w. This small lepton (or quark) extension a posteriori justifies the use of a perturbation calculation based on the diagram o f fig. 2b. In turn we can now write the expression of sin20 w coming from 7 - W 3 mixing [4,8] using eqs. (4) and (9): sin20 w =
eX/g = (~/rr)((QN/~H/2X/~f) (Are/Aw) 2. (10)
It makes very explicit the origin of the enhancement with respect to the electromagnetic vacuum polarization factor c~hr. The quantity ((Q)~v/-ff-HHH/2\/~f)X ( A J A w ) 2 is of order 100. The large value of sin20w (20.23) is due to the large ratio o f W boson to lepton and quark extensions. Using duality arguments it was already argued [ 14] that this large value of sin20w could be due to a large level spacing between
Volume 144B, number 1,2
PHYSICS LETTERS
the ground W state and excited W(') states. Although the lines o f thought are different these results may be consiste nt. The re may be two effective scales [ 15 ] in this problem. One (A~) can be the basic compositeness scale. The other one (A w -~Mw) can just be an effective scale due to the actual mass of the ground state. We could also think that the main difference between leptons (or quarks) and W bosons is due to the presence o f scalar subconstituents with high masses. This may have something to do with supersymmetry breaking. It is striking that with our calculations A~ appears to lie just above the present values o f the lower limit on lepton and quark substructure coming from the absence o f residual contact terms in e+e- collisions [16]. The absence o f a g - 2 effect requires a few remarks. Hadronic analogy with 0NN and ~oNN couplings raised the question: why does a large Pauli (aU~'q~,)term not appear in W couplings to leptons or quarks. This Pauli term could through W-dominance propagate to photon couplings and give a g - 2 contribution o f order m~/A~. The answer uses both chirality and the high value o f A~. First in a true chiral world with left-handed or right-handed W - h - h ' vertices no direct contribution to Pauli couplings can appear in diagram o f fig. 2b. Chiral symmetry breaking terms which could be at the origin o f the lepton and quark masses may appear [13,17] at order (m~/A~) 2. With our value o f A~ they are not yet observable [ 18]. We now come back to the radiative W -+ ~ ' 7 decays. The diagrams o f fig. 3 can be computed in the non-relativistic or in the relativistic formalism using the l~(k, k') and ~ ( k , k') vertex functions defined above. The result can be written I'(W -+ ~ ' 7 ) ~- (a/4n)
(Q)2F(W-+~'),
(1 I)
where (Q) is some combination of the subconstituent charges which depends on the helicity properties of the h h ' + £~' transition. Obviously the ( ~ ' ) final state is not necessarily a vector. The factorization o f F ( W ~ ~ ' ) in eq. (11) is only done for convenient rate evaluation. We notice that it is only i n t h e non-
t
W Fig. 3. Composite diagrams for W~ ~'"/decays.
23 August 1984
relativistic and weak binding approximation that one gets (Q) ~ (Qh - Qh') for a spin zero ( ~ ' ) state. In this special case the W-+ radiative .decay is forbidden in the haplon model (Qh = Qh' - 5 1) as observed by Baur et al. [3]. In general the charge combination is non zero but it can nevertheless be quantitatively different in W ± and in Z 0 cases. However the ratio (a/4rr) (Q)2 ~ 10-3 is much lower than the experimental [1] one (3/14). This confirms the need for enhancements for example coming from fermionic (~*) and/ or bosonic (S, V) intermediate states with strong couplings of hadronic type [2,3]. In conclusion we have shown that it is possible in composite models to understand why W couplings to leptons or quarks are much weaker than those expected from hadronic analogy and required by the interpretations of Z + ~ 7 events. The reason is a small ratio (of the order o f Aw/A Q -~ 1/45) o f lepton or quark extension to W extension. We expect A~ to be of the order o f 3.6 TeV. This also clearly shows why sin20 w due to 3'-W 3 mixing and enhanced by (A~/Aw)2 is so large. Such a value of A~ explains why no direct substructure effect ofleptons and quarks has been observed up to now (no form factor, no residual contact term, no g - 2 effects). On the opposite the low value o f the effective W scale (A w -~Mw) allows for composite effects to appear in processes involving W bosons and their extended partners (WWZ, WWS, ...). Instead o f the non-relativistic approximation (and weak binding limit) we define and use a relativistic W - h - h ' vertex which allows various types of calculations and gives more confidence in the evaluation [5] o f Z -+ 37, Z ~ Z3', W ~ WT, Z or W 3'ff, ... and any other perturbative process.
References [1] G. Arnison et al., Phys. Lett. 126B (1983) 398; P. Bagnaia et al., Phys. Lett. 129B (1983) 130. [2] For a review see F.M. Renard, talk given at XIX Rencontre de Moriond, ed. Tran Thanh Van, to be published (1984); preprint PM/84/5. [3] G. Gounaris, R. K6gerler and D. Schildknecht, Phys. Lett. 137B (1984) 261; U. Baur, M. Fritzsch and H. Faissner, Phys. Lett. 135B (1984) 313; F.M. Renard, Phys. Lett. 139B (1984) 449; R.D. Peccei, Phys. Lett. 136B (1984) 121. [4] J.J. Sakurai, in: Electroweak interactions, ed. H. Mitter, Acta Phys. Austr. Suppl. 24 (1982); 121
Volume 144B, number 1,2
[5 ] [6] [7] [8] [9] [10] [11]
122
PHYSICS LETTERS
R. K6gerler, in: Electroweak interactions at high energies, Proc. 1982 DESY Workshop, ed. R. K/Sgefler and D. Schildknecht; D. Schildknecht, in: Proc. Europhysics Study Conf. on Electroweak effects at high energies, ed. H. Newman; H. Fritzsch, Lecture at Int. School on Subnuclear physics, Erice, MPI-PAE/PTh 76/83 (1983). F.M. Renard, Nucl. Phys. B196 (1982) 83; Phys. Lett. 116B (1982) 269; 126B (1983) 59; preprint PM/84/4. M. Gourdin, M. Le BeUac, F.M. Renard and J. Tran Thanh Van, Nuovo Cimento 37 (1965) 524. H. Fritzsch and G. Mandelbaum, Phys. Lett. 102B (1981) 319. P. Hung and J.l. Sakurai, Nucl. Phys. B143 (1978) 81. P. Chen and F.M. Renard, Phys. Rev. D28 (1983) 1758. H. Fritzsch, R. KSgerler and D. Schildknecht, Phys. Lett. 114B (1982) 157. R. K6gerler and D. Schildknecht, CERN preprint TH. 3231 (1982).
23 August 1984
[12] L. Abbott and E. Farhi, Nucl. Phys. B189 (1981) 547; Phys. Lett. 101B (1981) 69. [13] S.J. Brodsky and S.D. Drell, Phys. Rev. D22 (1980) 2236. [14] P. Chen and J.J. SakuraJ, Phys. Lett. 100B (1982) 181 ; S. Narison, Phys. Lett. 122B (1983) 171 ; G. Gounaris, R. K6gerler and D. Schildknecht, Phys. Lett. 133B (1983) 118. [15] N.S. Craigie, preprint IC/83/183; ICTP Summer Workshop, Trieste (1983). [ 16 ] E J . Eichten, K.D. Lane and M.E. Peskin, Phys. Rev. Lett. 50 (1983) 811; R. Riickl, talk given at XIX Rencontre de Moriond, ed. Tran Thanh Van, to be published (1984). [17] M.E. Peskin, Proc. 1981 Int. Symp. on Lepton and photon interactions, Bonn, ed. W. Pfeil. [18] L. Lyons, Prog. Part. Nucl. Phys. 10 (1983) 227; Oxford preprint 2/84.
PHYSICS LETTERS
23 August 1984
SIZES AND COUPLINGS OF COMPOSITE LEPTONS, QUARKS AND W BOSONS F.M. RENARD
Ddpartement de PhysiqueMath~matique l, USTL, 1:-34060Montpellier Cedex, France Received 19 April 1984
We assume that leptons and quarks are much smaller objects than W bosons (A~,q ~- 3.6 TeV; Aw =Mw). This explains the weakness of W couplings to leptons and quarks (as opposed to strong couplings required by radiative Z° decays), the strength of sinZ0w and the present absence of direct lepton and quark substructure effects. We also define a relativistic description of W substructure which should replace the usual non-relativistic wave function and allow better estimates of several processes.
The anomalously large rate o f radiative Z decays [1 ] renewed the interest in composite models for leptons, quarks and W bosons. Several explanations have been proposed for these decays [2]. Most o f them are based on the assumption that weak interactions are residual effects due to the underlying substructure [3]. It was shown [4] before W, Z discovery that such a composite picture can lead to a phenomenology of electroweak interactions in many aspects very similar to the standard one but also [5] that a priori rare W boson decays could be enhanced by substructure effects. The analogy with hadronic interactions allows to expect a large effective coupling to appear in one place or another (ZZT, ZST, Z ~ * , ...) and for example to lead to the enhancement required by Z -+ ££7 events. However there are puzzling questions and deficiencies in this picture. In the present paper we address and answer the following questions: Why are the couplings o f W bosons to leptons and quarks much weaker than usual hadronic couplings (required for enhancing Z ~ ~£7 decays)? Why is the value o f sin20 w (due to 7 - W 3 mixing) so large? Why is there no direct signal o f lepton and quark substructure (no form factor, no contact term, no 1 Physique Math~matique et Th~orique, Equipe de Recherche Associ6e au CNRS. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
g - 2 effects .... )? How could one be confident in non-relativistic descriptions o f W bosons as sub-constituent bound states? Our explanations are based on the assumption that leptons and quarks are much smaller objects than W bosons. We start by showing how one can replace the nonrelativistic approximation (and the weak binding limit) by a fully relativistic description allowing the use o f Feynman diagram techniques. We define a W - h - h ' vertex ~ (1 + 75) ~ W(k, k') whose correspondence with the 3S 1 bound state wave function ¢ ( k ) in momentum space is given by [6] ~(k) .~ [(2n)3Mw ] -1/2
×f
dk 0
4m 2
w(k,k')(k z _m2)(k,2
-m2) ,
(1)
m being the effective mass o f the subconstituents h and h' o f the W bosons. For illustration we take the simple form I~(k, k') = gw A 4 / ( k2 - A 2 ) ( k'2 - A 2 ) •
(2)
As a first application we consider the 7 - W 3 junction Ml2eueU represented in fig. 1. A straightforward calculation o f the Feynman diagram using the vertex function l¢(k,k') gives (for m <~ A w ~ M w ) 119
Volume 144B, number 1,2
PHYSICS LETTERS h
')
~c~ Fig. 1. "r-Wa junction.
X = -(gW/247r2)e(Q)V~H,
(3)
where (Q) and n H are the average constituent charge and hypercolor number. Eq. (3) replaces the usual non-relativistic expression [7] X = [~b~O)/x/~]
e(Q)'~H/Mw3/2 ,
(4)
with ~ 0 ) = (2rr)-3/2fd3@(k). The normalization of gw [equivalent to that of 4~(k) through (1)] can be estimated from fd3k l¢(k)12/47r = 1 and assuming an extension Mff 1 . The result Igw[ ~ 1 2 r r 2 v ~ ,
(5)
roughly agrees with eq. (3) taking for example (Q) = 1/X/~,n H = 3 and the value [ 8 - 1 1 ] X-~ sin0 w -~0.5. We then consider the couplings o f W bosons to other bosonic or fermionic bound states, From the pictures of fig. 2a and 2b and the hadronic analogy we would expect a rather large dimensionless coupling constant f. A comparison with gorrrr -~ 6 and g~rNN ~ 13 suggests that f c o u l d easily be of order 10. In fact this is precisely the order of magnitude required by the interpretations [2,3] of Z -+ £~/events based on high mass intermediate states like S, Z or 2*. So we adopt this magnitude for the couplings o f W bosons to other extended objects (type 2a). In early times of composite models based on low compositeness scales it was found natural to expect [7,12] that the couplingg of W to leptons or quarks be also of orderf. In turn asg2/8M2w = G/x/~this led to expect large M w masses. Later additional constraints (like asymptotic symmetry [8], unitarity [9], current algebra [10] or W-dominance [ 11]) enforced g a n d M w to be closer to the standard values but raised the question: why is g much smaller than f. We t
___[
w -
....
-
B' (cO
.....
t' Cb)
T Co)
Fig. 2. Couplings ofW bosons and photon to other bound states, (a) extended bosons, (b) and (c) leptons and quarks.
120
23 August 1984
now explicitly show how this can be a consequence of a different size for W bosons and for lepton or quarks. It is intuitively obvious from fig. 2b that the effective couplings depend upon the overlap of W and leptons (or quarks) extensions. Contrarily to the case of fig. 2a where the overlap is of order 1, in fig. 2b we have the overlap
f d 3 k ~](k, k')I~(k, k')l 2 ,
(6)
for which a non-relativistic approximation with constant wave functions gives 6rr2(R~/Rw)3. Let us make a relativistic calculation using an effective lepton vertex function [ 13]
[~b~(k, k')i 2 = ere/[k ' 2 - A2] 2 .
(7)
The normalization is obtained by the computation of the electromagnetic Dirac form factor (fig. 2c) and F 1(0) = 1 ; this gives gre = 32rr2A 2 .
(8)
A similar calculation of the diagram of fig. 2b then gives [by identification to the usual W££' vertex (g/ 2~u/2)fi~(1 + @ ) ¢ u~,]:
g = ½X,/2fgw(Aw/AQ) 2 ~- 8~2x/~f(Aw/Are) 2 ,
(9)
using eq. (5). This clearly shows the dependence o f g upon the extension ratio (Aw/Are)2. I f f is of order 10 the actual value g -~ 0.65 (i.e, e/sin 0 w in the standard model) requires (Aw/Are) 2-'" 5 × 10 - 4 , i.e. A~ -~ 45A w which means Are -~ 3.6 TeV i f A w ~ M w. This small lepton (or quark) extension a posteriori justifies the use of a perturbation calculation based on the diagram o f fig. 2b. In turn we can now write the expression of sin20 w coming from 7 - W 3 mixing [4,8] using eqs. (4) and (9): sin20 w =
eX/g = (~/rr)((QN/~H/2X/~f) (Are/Aw) 2. (10)
It makes very explicit the origin of the enhancement with respect to the electromagnetic vacuum polarization factor c~hr. The quantity ((Q)~v/-ff-HHH/2\/~f)X ( A J A w ) 2 is of order 100. The large value of sin20w (20.23) is due to the large ratio o f W boson to lepton and quark extensions. Using duality arguments it was already argued [ 14] that this large value of sin20w could be due to a large level spacing between
Volume 144B, number 1,2
PHYSICS LETTERS
the ground W state and excited W(') states. Although the lines o f thought are different these results may be consiste nt. The re may be two effective scales [ 15 ] in this problem. One (A~) can be the basic compositeness scale. The other one (A w -~Mw) can just be an effective scale due to the actual mass of the ground state. We could also think that the main difference between leptons (or quarks) and W bosons is due to the presence o f scalar subconstituents with high masses. This may have something to do with supersymmetry breaking. It is striking that with our calculations A~ appears to lie just above the present values o f the lower limit on lepton and quark substructure coming from the absence o f residual contact terms in e+e- collisions [16]. The absence o f a g - 2 effect requires a few remarks. Hadronic analogy with 0NN and ~oNN couplings raised the question: why does a large Pauli (aU~'q~,)term not appear in W couplings to leptons or quarks. This Pauli term could through W-dominance propagate to photon couplings and give a g - 2 contribution o f order m~/A~. The answer uses both chirality and the high value o f A~. First in a true chiral world with left-handed or right-handed W - h - h ' vertices no direct contribution to Pauli couplings can appear in diagram o f fig. 2b. Chiral symmetry breaking terms which could be at the origin o f the lepton and quark masses may appear [13,17] at order (m~/A~) 2. With our value o f A~ they are not yet observable [ 18]. We now come back to the radiative W -+ ~ ' 7 decays. The diagrams o f fig. 3 can be computed in the non-relativistic or in the relativistic formalism using the l~(k, k') and ~ ( k , k') vertex functions defined above. The result can be written I'(W -+ ~ ' 7 ) ~- (a/4n)
(Q)2F(W-+~'),
(1 I)
where (Q) is some combination of the subconstituent charges which depends on the helicity properties of the h h ' + £~' transition. Obviously the ( ~ ' ) final state is not necessarily a vector. The factorization o f F ( W ~ ~ ' ) in eq. (11) is only done for convenient rate evaluation. We notice that it is only i n t h e non-
t
W Fig. 3. Composite diagrams for W~ ~'"/decays.
23 August 1984
relativistic and weak binding approximation that one gets (Q) ~ (Qh - Qh') for a spin zero ( ~ ' ) state. In this special case the W-+ radiative .decay is forbidden in the haplon model (Qh = Qh' - 5 1) as observed by Baur et al. [3]. In general the charge combination is non zero but it can nevertheless be quantitatively different in W ± and in Z 0 cases. However the ratio (a/4rr) (Q)2 ~ 10-3 is much lower than the experimental [1] one (3/14). This confirms the need for enhancements for example coming from fermionic (~*) and/ or bosonic (S, V) intermediate states with strong couplings of hadronic type [2,3]. In conclusion we have shown that it is possible in composite models to understand why W couplings to leptons or quarks are much weaker than those expected from hadronic analogy and required by the interpretations of Z + ~ 7 events. The reason is a small ratio (of the order o f Aw/A Q -~ 1/45) o f lepton or quark extension to W extension. We expect A~ to be of the order o f 3.6 TeV. This also clearly shows why sin20 w due to 3'-W 3 mixing and enhanced by (A~/Aw)2 is so large. Such a value of A~ explains why no direct substructure effect ofleptons and quarks has been observed up to now (no form factor, no residual contact term, no g - 2 effects). On the opposite the low value o f the effective W scale (A w -~Mw) allows for composite effects to appear in processes involving W bosons and their extended partners (WWZ, WWS, ...). Instead o f the non-relativistic approximation (and weak binding limit) we define and use a relativistic W - h - h ' vertex which allows various types of calculations and gives more confidence in the evaluation [5] o f Z -+ 37, Z ~ Z3', W ~ WT, Z or W 3'ff, ... and any other perturbative process.
References [1] G. Arnison et al., Phys. Lett. 126B (1983) 398; P. Bagnaia et al., Phys. Lett. 129B (1983) 130. [2] For a review see F.M. Renard, talk given at XIX Rencontre de Moriond, ed. Tran Thanh Van, to be published (1984); preprint PM/84/5. [3] G. Gounaris, R. K6gerler and D. Schildknecht, Phys. Lett. 137B (1984) 261; U. Baur, M. Fritzsch and H. Faissner, Phys. Lett. 135B (1984) 313; F.M. Renard, Phys. Lett. 139B (1984) 449; R.D. Peccei, Phys. Lett. 136B (1984) 121. [4] J.J. Sakurai, in: Electroweak interactions, ed. H. Mitter, Acta Phys. Austr. Suppl. 24 (1982); 121
Volume 144B, number 1,2
[5 ] [6] [7] [8] [9] [10] [11]
122
PHYSICS LETTERS
R. K6gerler, in: Electroweak interactions at high energies, Proc. 1982 DESY Workshop, ed. R. K/Sgefler and D. Schildknecht; D. Schildknecht, in: Proc. Europhysics Study Conf. on Electroweak effects at high energies, ed. H. Newman; H. Fritzsch, Lecture at Int. School on Subnuclear physics, Erice, MPI-PAE/PTh 76/83 (1983). F.M. Renard, Nucl. Phys. B196 (1982) 83; Phys. Lett. 116B (1982) 269; 126B (1983) 59; preprint PM/84/4. M. Gourdin, M. Le BeUac, F.M. Renard and J. Tran Thanh Van, Nuovo Cimento 37 (1965) 524. H. Fritzsch and G. Mandelbaum, Phys. Lett. 102B (1981) 319. P. Hung and J.l. Sakurai, Nucl. Phys. B143 (1978) 81. P. Chen and F.M. Renard, Phys. Rev. D28 (1983) 1758. H. Fritzsch, R. KSgerler and D. Schildknecht, Phys. Lett. 114B (1982) 157. R. K6gerler and D. Schildknecht, CERN preprint TH. 3231 (1982).
23 August 1984
[12] L. Abbott and E. Farhi, Nucl. Phys. B189 (1981) 547; Phys. Lett. 101B (1981) 69. [13] S.J. Brodsky and S.D. Drell, Phys. Rev. D22 (1980) 2236. [14] P. Chen and J.J. SakuraJ, Phys. Lett. 100B (1982) 181 ; S. Narison, Phys. Lett. 122B (1983) 171 ; G. Gounaris, R. K6gerler and D. Schildknecht, Phys. Lett. 133B (1983) 118. [15] N.S. Craigie, preprint IC/83/183; ICTP Summer Workshop, Trieste (1983). [ 16 ] E J . Eichten, K.D. Lane and M.E. Peskin, Phys. Rev. Lett. 50 (1983) 811; R. Riickl, talk given at XIX Rencontre de Moriond, ed. Tran Thanh Van, to be published (1984). [17] M.E. Peskin, Proc. 1981 Int. Symp. on Lepton and photon interactions, Bonn, ed. W. Pfeil. [18] L. Lyons, Prog. Part. Nucl. Phys. 10 (1983) 227; Oxford preprint 2/84.