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Measuring static quark properties at LEP
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
M E A S U R I N G S T A T I C Q U A R K P R O P E R T I E S AT L E P ~" A. D J O U A D I a n d M. S P I R A Instttut fur Theorettsche Phystk, R WTH Aachen, D-51 O0 Aachen, FRG
Received 1 June 1989
The measurement of the hadromc Z branching ratios and jet angular distributions can set stringent limits on static quark properties, radn and anomalous (Z) magnetic moments For heavy quarks, in particular the third generation, e+e - anmhdatlon is the only method to measure these quantRles. Bounds on radn of the order of 10-16 cm down to 10-17 cm can be expected from LEP experiments. The radn and magnehc moments can be interrelated in phenomenologlcal models for quark substructures
1. Praeliminaria Theoretical speculations on an intrinsic finite size o f quarks, a non-zero radius and an a n o m a l o u s magnetic m o m e n t , have been p u r s u e d for a long time [ 1 ]. The proliferation o f quarks and leptons revived the idea o f quark and lepton substructures - following the old philosophy to derive complex structures for simple building blocks. However, no d y n a m i c a l m o d e l has emerged so far in which the very small size o f preonlc b o u n d states could be reconciled with nearly zero-mass, a p r o b l e m related to Helsenberg's uncertainty principle. Stringent b o u n d s on the size o f electrons a n d m u o n s have been set by m e a s u r e m e n t s o f the a n o m a l o u s magnetic m o m e n t s which agree on a level o f u n p r e c e d e n t e d accuracy with the Q E D predictions. A n y genuine nonpointlike structure o f these particles is restricted to energy scales above a n d length scales below the following values [ 2 - 4 ]: Ae>~76GeV,
Re~<0.6×10-15cm,
A,>~720GeV,
R,~0.7×10-16cm.
These b o u n d s are d e r i v e d from J ( g - 2 ) e/~ = 6 X 1 0 - ~~/ 17 × 10- 9 with a quadratic mass d e p e n d e n c e ( m l / A ) 2 as suggested by chiral symmetry. Preon interchange m o d e l s induce contact terms in Bhabha scattering amplitudes, defining a particle size R = x / / ~ f r A ~t i where geff is the novel effective strong coupling constant on the preon level [ 5 ]. P E T R A a n d P E P experiments (see ref. [ 6 ] for a c o m p i l a t i o n ) set limits o f A~t>~½-2TeV
and
Re~<10-15-10-16cm
on these quantities. To estimate the radius, the strong coupling has been chosen = 1. The b o u n d s d e p e n d on the specific current × current form o f the contact interactions. The analysis o f q u a r k - q u a r k scattering in the C E R N pp colhder gave limits [ 7 ] o f A~t >_-½ TeV
and
Rq ~< 1 0 - l~ c m
for light-quark radn, d e r i v e d from b o u n d s on the contact interactions. Bounds on finite-size properties o f leptons a n d quarks o f the first two generations cannot readily be extrapolated to the t h i r d generation. In fact, a variety o f compositeness m o d e l s have been investigated in which the t h i r d generation plays a special role (see e.g. ref. [ 8 ] ). The large top quark mass m a y seem indicative for possibly distinctive properties o f this generation. This idea has spurred analyses o f the x magnetic m o m e n t in lower Supported m part by the West German Bundesmlmstermm fur Forschung und Technologle. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
443
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
energy e+e - collisions [ 9 ]. LEP, however, is a unique facility to study also the static properties of b quarks, besides x and all the other quarks and leptons. The large number of these quarks in Z decays - 1.5 × 106 b13 pairs in 107 Z decays offers an excellent opportunity to look into the structure of these particles. In this note we shall describe the phenomenological basis for the analysis of static quark (and lepton) properties in the e+e - energy range around the Z ~. We will give rough estimates of the limits on quark radii and anomalous (Z) magnetic moments to be expected from LEP. Finally, a model calculation will be presented in which radii and anomalous magnetic moments are linked to finite-size effects caused by diquark exchanges at the quark-Z vertex.
2. Facta Assuming CP invariance, the Z-quark vertex is determined by three independent form factors
~
l"GuMz"~1/21"
--Z-
=t~)
lf2
)
(1)
tflTs'+2-~mq au'k~'+fAT"y''
q
which in the pointlike limit reduce to the familiar coupling constants of the standard model. Introducing the quark radii R1,A and the anomalous quark magnetic moment ~2 t¢ for couplings to the Z, and assuming both to be small, the form factors can be expanded in the following way A =Vq( l +- sl R l s2)
, f2=Vqa: , f a = a q ( l +_ ~IR A2s ) ,
(2)
with
vf=+_l-4ersinZOw,
af=_+l,
f=e,q
and
_+lforup/down
in electroweak standard model. From this general form we derive the cross section for e +e---,Z~q~l
da dcosO-
G 2 (v~ +a2) 32rc
M~s [ ( s - M 2 ) 2 + (SFz/Mz) 2 ~ ~( l +cos20)fl[f~ + fl2ffA--2f~f2 + f2]
(1-fl 2 + 3 (1- c o s 2O)fl\----~f~
(1-2fl2)fA +½(s/4m2q_4fl2)f~)+3cosO2fl2(flfA_fafA)]
'
(3)
fl= (1 -4m2/s)~/2 being the quark velocity in the final state for a non-zero quark mass mq. The cross section (3) is supposed to be compared to experimental data after QED, genuine electroweak and QCD corrections, as defined in the standard model, are substracted. Data purified in this way may then be used to set limits on the radii and the anomalous magnetic moments. (It has been tacitly assumed that the electron can be treated pointlike on the level of accuracy that can be envisaged for measurements of quark substructures. This assumption would not be valid if the vertex ( 1 ) were affected more by possible internal structures of the Z boson and less by that of the fermions; in this case we could not allude to the g - 2 measurements as an argument for deriving ( 3 ), and electron-Z form factors would have to be introduced in addition. ) Three independent measurements determine the cross section (3). As such we may choose the total cross section (equivalently the partial Z ~ q(:l decay width, eventually normalized to Z ~ IX+ Ix- to achieve higher precision), the forward-backward asymmetry of the q jet, and the Otq parameter defined by dN/dcos0oc 1 + aqCOS20+ .... Expressed in terms of the form factors, the various observables can be written as ~ Form factors for electrons and muons have been discussed m ref [ 10] ~2 x need not c o m o d e with g - 2 for photon couplings
444
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
GuM3z F ( Z ~ q d l ) = 8x/~Tr [½fl(3--fl2)f~ + fl3f2A--fl(3--2fl2)Af2 + l f l ( M z2/ 4 r n 2q + 2 - 4fl 2)f~2]
Aq~
aq
3 2Veae 2fl2(fllfA--f2fA) Ve +ae2 lfl(3--/32)f2 +/33f2A--fl(3--2f12)f~f2+ ~l f l ( M z2/ 4 m q 2 + 2-- 4f12)f~
-- _ _ 4 2
= f12
f2 + f 2 - - 4 f l f2 + ( 4-- M27 / 4mZq )f~2 (2-- fl2)f 2 +fl2fzA --4(1 -- fl2)f~ f2 + (1 -4f12+M2z/4m2q)f~2 "
(4)
If the quark mass can be neglected and just the leading terms are kept, the observables simplify considerably. The derivations from the Born terms G,M3z 3 2Veae 2Vqaq F~(Z-,qCl)= 8x/~rc (vZq+a~) , A q = 4 VZe+ a ~ V2q+a2q can be cast into form F(Z--,qCi) =FB(Z--,qdl) (1 _+IM~z/~2-g),
Aq =A~(1 _+-~M~zAR2 - AK),
aq = 1 - - 4 g ,
(5)
where the average radii/~2, etc., are given by R- 2 = V"2q R l2+ a q" 2R A ,2
~2 AR 2 = ( V q - a~2 q ( R j2 - R ~ )
g=K.Sq:[1--(M2/8m2q)tC] ,
A x = x [ a q-2+ v q-2 ( M z / 82 m q ) X: ]
,
2 2 g = x . b 2 [ 1 + (Mz/8mq)lC]
(6)
with vq(aq)~2 ~2 =v2(a2)/(vq2 +aq2). Note that the parameter Ogqis not affected by the quark radn in this approximation but is related directly to the anomalous magnetic moments. The same expressions hold for lepton final states, in particular the x, after eliminating the color factor Nc = 3 and altering the electroweak charges appropriately. Anticipating experimental accuracies up to + 4% for heavy quarks and _+ 1% for all quarks, quark radii can be probed down to R~
10-16-10-17
cm
and anomalous (Z) magnetic moments down to K~ 10-2-10 -3 ' Thls promises not only improvements of one or two orders of magnitude over existing limits (see e.g. refs. [ 9,1 1-1 3 ] ) but an exciting premiere for the third generation b quarks. Similar estimates hold for the z lepton.
3. Resolving the vertices. A phenomenological model There is no theory at the present time that reconciles the small lepton/quark masses with the tiny radn corresponding to the very large composlteness scales in energy. Nevertheless, one might formulate phenomenologlcal models in which static properties of the known quarks and leptons are traced back to the exchange of novel particles which are suggested in composite models. A nearly infinite variety of such models can be thought of, but we concentrate on just one typical example m which an isoscalar, charge - ~, color triplet, spin-zero dlquark D couples to two quarks with strengths 2 L,R ( f o r left/right handed components) in an SU (3) × SU (2) × U ( 1 ) lnvarlant form LD = ("].L/~bL +'~Rt~bR)D+h.c.
(7)
(the fields are antisymmetrlzed in the color indices). Similarly to the third generation, the interaction density may be defined for the first two generations. 445
Volume 228, number 3
PHYSICS LETTERS B
T
21 September 1989
_
--z--
+ -z-15
b Fig 1.
The two vertex diagrams shown in fig. 1 introduce finite-size effects and anomalous (Z) magnetic moments. Anticipating a mass range of several hundred GeV for mD >> mt, the radii and the magnetic moment are given by ~3 122+2212 R2=2n 4~r 1 2R+2L 2 2
R2-2n x=
4n
2eD Ut --3~-b x w + - v b ( L - 4 ) +
vt 1 2e_pD~xw+__~(L m 2 --3ab ab
1 2R2L mbmt ( 2 v ~ ut Xw+-(L-i) 4n m2 Vb
)
+
U~ b
((L-7
4)+ -a~
)
'
(L_7)
ab
122+22m~(eDxw+lVt--~at ~ ,
n
4n
m2
\V b
6
vb
(8)
/
where the abreviations m2 - L = l o g mt2
2 2 2 R --2 L (= 22+22 , Xw=Sln20w
have been used. (The signs in eq. (2) are to be identified with the sign of the sum between the bars in R21 and R 2. ) eD = -- ½is the diquark charge. Thus an experimental bound o f R ~<1 × 10 - 16cm would translate into a compositeness scale ofA = x/~R - 1>~1 TeV, corresponding to a diquark mass of rnD~>200x/~-/4n GeV. The predicted value for the (Z) anomalous magnetic moment depends less dramatically on the assumption whether the theory is chiral or not (2 R2L= 0 or 0) than for ( g - 2 ) e [2,1 5,1 6] because the mass ratio mJmt is not expected to deviate significantly from O ( 1 0 - ' ), in contrast to the ratio mc/ms or even mc/mt from dilepton and leptoquark exchanges. We have investigated variants of this model in which the effects of additional form factors at the diquark vertices are estimated and the internal boson-fermion loop masses are chosen close to each other. The overall picture, however, does not change significantly [ 14 ]. 4. Summa
It has been shown in this brief note that LEP is a unique facility to probe the static properties of quarks, in particular heavy quarks of the third generation. There is no other way to set limits on their radii and anomalous magnetic moments. Furthermore, for hght quarks and x the existing hmits can greatly be improved. (Note that the radius R used here is the physical particle radius; it is not plagued by the ambiguities introduced in the definition of a radms by the unknown coupling constants in contact terms. ) We had assumed from the outset that electrons could be considered pointlike for the experimental limits we are envisioning for x and quarks. This assumption has been inferred from the ( g - 2 ) e measurements. However, if experimental analyses would reveal deviations from the values F ( Z ~ e + e - ) etc. expected m the standard model, it would be a natural consequence m the frame of composite models to question the polntlike behawor of the Z boson ~tself. ~3 The expressions for arbitrary ratios m~/m2 have been reported m ref. [ 14]. 446
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
Acknowledgement W e a r e m d e b t e d t o P . M . Z e r w a s f o r h i s c o n t i n u o u s h e l p d u r i n g all t h e stages o f t h i s w o r k .
References [ 1 ] M Chanowltz and S D Drell, Phys Rev Lett 30 (1973) 807, Phys Rev D 9 (1974) 2078, G.B West, Phys Rev D 10 (1974) 329, G B West and P Zerwas, Phys Rev D 10 (1974) 2130. [2] S.J Brodsky and S D Drell, Phys Rev D 22 (1980) 2236 [3] M.E Peskln, Proc Intern Symp. on Lepton and photon lnterachons at high energies (Kyoto, 1985), eds M Konuma and K Takahashl [4] P Mery et al, Marsellle preprlnt CPT-89/P.2226 (1989) [ 5 ] E. Elchten, K D. Kane and M.E Peskm, Phys Rev Lett 50 (1983) 811. [6 ] K Haglwara and S. Komamlya, in High energy electron-positron physics, eds A All and P Sodlng (World Scientific, Singapore, 1988) [ 7 ] UA2 Collab., J A Appel et al., Phys Lett. B 160 (1985) 349, UA1 Collab, G Arnlson et al, Phys Lett. B 172 (1986) 641. [8 ] J C Patl and H Stremnltzer, Phys Rev Lett 56 (1986) 2152. [ 9 ] D J Sllverman and G L Shaw, Phys. Rev D 27 (1983) 1196 [ 10] J Hall and S F King, Nucl Phys B 287 (1987) 541. [ 11 ] S Failer and R J. Oakes, Phys Rev D 28 (1983) 2881 [ 12 ] T Kamae, Proc 24 Intern. Conf on High energy physics (Mumch, 1988), eds R Kotthaus and J. Kuhn [ 13 ] G.L Shaw, D Sdverman and R Slansky, Phys Lett. B 94 (1980) 57 [ 14 ] M Splra, Diploma Thesis, RWTH Aachen (1989) [ 15 ] Ch Berger et al, Aachen Report PITHA (1989) [16] F M Renard, Phys Lett B 116 (1982) 264
447
PHYSICS LETTERS B
21 September 1989
M E A S U R I N G S T A T I C Q U A R K P R O P E R T I E S AT L E P ~" A. D J O U A D I a n d M. S P I R A Instttut fur Theorettsche Phystk, R WTH Aachen, D-51 O0 Aachen, FRG
Received 1 June 1989
The measurement of the hadromc Z branching ratios and jet angular distributions can set stringent limits on static quark properties, radn and anomalous (Z) magnetic moments For heavy quarks, in particular the third generation, e+e - anmhdatlon is the only method to measure these quantRles. Bounds on radn of the order of 10-16 cm down to 10-17 cm can be expected from LEP experiments. The radn and magnehc moments can be interrelated in phenomenologlcal models for quark substructures
1. Praeliminaria Theoretical speculations on an intrinsic finite size o f quarks, a non-zero radius and an a n o m a l o u s magnetic m o m e n t , have been p u r s u e d for a long time [ 1 ]. The proliferation o f quarks and leptons revived the idea o f quark and lepton substructures - following the old philosophy to derive complex structures for simple building blocks. However, no d y n a m i c a l m o d e l has emerged so far in which the very small size o f preonlc b o u n d states could be reconciled with nearly zero-mass, a p r o b l e m related to Helsenberg's uncertainty principle. Stringent b o u n d s on the size o f electrons a n d m u o n s have been set by m e a s u r e m e n t s o f the a n o m a l o u s magnetic m o m e n t s which agree on a level o f u n p r e c e d e n t e d accuracy with the Q E D predictions. A n y genuine nonpointlike structure o f these particles is restricted to energy scales above a n d length scales below the following values [ 2 - 4 ]: Ae>~76GeV,
Re~<0.6×10-15cm,
A,>~720GeV,
R,~0.7×10-16cm.
These b o u n d s are d e r i v e d from J ( g - 2 ) e/~ = 6 X 1 0 - ~~/ 17 × 10- 9 with a quadratic mass d e p e n d e n c e ( m l / A ) 2 as suggested by chiral symmetry. Preon interchange m o d e l s induce contact terms in Bhabha scattering amplitudes, defining a particle size R = x / / ~ f r A ~t i where geff is the novel effective strong coupling constant on the preon level [ 5 ]. P E T R A a n d P E P experiments (see ref. [ 6 ] for a c o m p i l a t i o n ) set limits o f A~t>~½-2TeV
and
Re~<10-15-10-16cm
on these quantities. To estimate the radius, the strong coupling has been chosen = 1. The b o u n d s d e p e n d on the specific current × current form o f the contact interactions. The analysis o f q u a r k - q u a r k scattering in the C E R N pp colhder gave limits [ 7 ] o f A~t >_-½ TeV
and
Rq ~< 1 0 - l~ c m
for light-quark radn, d e r i v e d from b o u n d s on the contact interactions. Bounds on finite-size properties o f leptons a n d quarks o f the first two generations cannot readily be extrapolated to the t h i r d generation. In fact, a variety o f compositeness m o d e l s have been investigated in which the t h i r d generation plays a special role (see e.g. ref. [ 8 ] ). The large top quark mass m a y seem indicative for possibly distinctive properties o f this generation. This idea has spurred analyses o f the x magnetic m o m e n t in lower Supported m part by the West German Bundesmlmstermm fur Forschung und Technologle. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
443
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
energy e+e - collisions [ 9 ]. LEP, however, is a unique facility to study also the static properties of b quarks, besides x and all the other quarks and leptons. The large number of these quarks in Z decays - 1.5 × 106 b13 pairs in 107 Z decays offers an excellent opportunity to look into the structure of these particles. In this note we shall describe the phenomenological basis for the analysis of static quark (and lepton) properties in the e+e - energy range around the Z ~. We will give rough estimates of the limits on quark radii and anomalous (Z) magnetic moments to be expected from LEP. Finally, a model calculation will be presented in which radii and anomalous magnetic moments are linked to finite-size effects caused by diquark exchanges at the quark-Z vertex.
2. Facta Assuming CP invariance, the Z-quark vertex is determined by three independent form factors
~
l"GuMz"~1/21"
--Z-
=t~)
lf2
)
(1)
tflTs'+2-~mq au'k~'+fAT"y''
q
which in the pointlike limit reduce to the familiar coupling constants of the standard model. Introducing the quark radii R1,A and the anomalous quark magnetic moment ~2 t¢ for couplings to the Z, and assuming both to be small, the form factors can be expanded in the following way A =Vq( l +- sl R l s2)
, f2=Vqa: , f a = a q ( l +_ ~IR A2s ) ,
(2)
with
vf=+_l-4ersinZOw,
af=_+l,
f=e,q
and
_+lforup/down
in electroweak standard model. From this general form we derive the cross section for e +e---,Z~q~l
da dcosO-
G 2 (v~ +a2) 32rc
M~s [ ( s - M 2 ) 2 + (SFz/Mz) 2 ~ ~( l +cos20)fl[f~ + fl2ffA--2f~f2 + f2]
(1-fl 2 + 3 (1- c o s 2O)fl\----~f~
(1-2fl2)fA +½(s/4m2q_4fl2)f~)+3cosO2fl2(flfA_fafA)]
'
(3)
fl= (1 -4m2/s)~/2 being the quark velocity in the final state for a non-zero quark mass mq. The cross section (3) is supposed to be compared to experimental data after QED, genuine electroweak and QCD corrections, as defined in the standard model, are substracted. Data purified in this way may then be used to set limits on the radii and the anomalous magnetic moments. (It has been tacitly assumed that the electron can be treated pointlike on the level of accuracy that can be envisaged for measurements of quark substructures. This assumption would not be valid if the vertex ( 1 ) were affected more by possible internal structures of the Z boson and less by that of the fermions; in this case we could not allude to the g - 2 measurements as an argument for deriving ( 3 ), and electron-Z form factors would have to be introduced in addition. ) Three independent measurements determine the cross section (3). As such we may choose the total cross section (equivalently the partial Z ~ q(:l decay width, eventually normalized to Z ~ IX+ Ix- to achieve higher precision), the forward-backward asymmetry of the q jet, and the Otq parameter defined by dN/dcos0oc 1 + aqCOS20+ .... Expressed in terms of the form factors, the various observables can be written as ~ Form factors for electrons and muons have been discussed m ref [ 10] ~2 x need not c o m o d e with g - 2 for photon couplings
444
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
GuM3z F ( Z ~ q d l ) = 8x/~Tr [½fl(3--fl2)f~ + fl3f2A--fl(3--2fl2)Af2 + l f l ( M z2/ 4 r n 2q + 2 - 4fl 2)f~2]
Aq~
aq
3 2Veae 2fl2(fllfA--f2fA) Ve +ae2 lfl(3--/32)f2 +/33f2A--fl(3--2f12)f~f2+ ~l f l ( M z2/ 4 m q 2 + 2-- 4f12)f~
-- _ _ 4 2
= f12
f2 + f 2 - - 4 f l f2 + ( 4-- M27 / 4mZq )f~2 (2-- fl2)f 2 +fl2fzA --4(1 -- fl2)f~ f2 + (1 -4f12+M2z/4m2q)f~2 "
(4)
If the quark mass can be neglected and just the leading terms are kept, the observables simplify considerably. The derivations from the Born terms G,M3z 3 2Veae 2Vqaq F~(Z-,qCl)= 8x/~rc (vZq+a~) , A q = 4 VZe+ a ~ V2q+a2q can be cast into form F(Z--,qCi) =FB(Z--,qdl) (1 _+IM~z/~2-g),
Aq =A~(1 _+-~M~zAR2 - AK),
aq = 1 - - 4 g ,
(5)
where the average radii/~2, etc., are given by R- 2 = V"2q R l2+ a q" 2R A ,2
~2 AR 2 = ( V q - a~2 q ( R j2 - R ~ )
g=K.Sq:[1--(M2/8m2q)tC] ,
A x = x [ a q-2+ v q-2 ( M z / 82 m q ) X: ]
,
2 2 g = x . b 2 [ 1 + (Mz/8mq)lC]
(6)
with vq(aq)~2 ~2 =v2(a2)/(vq2 +aq2). Note that the parameter Ogqis not affected by the quark radn in this approximation but is related directly to the anomalous magnetic moments. The same expressions hold for lepton final states, in particular the x, after eliminating the color factor Nc = 3 and altering the electroweak charges appropriately. Anticipating experimental accuracies up to + 4% for heavy quarks and _+ 1% for all quarks, quark radii can be probed down to R~
10-16-10-17
cm
and anomalous (Z) magnetic moments down to K~ 10-2-10 -3 ' Thls promises not only improvements of one or two orders of magnitude over existing limits (see e.g. refs. [ 9,1 1-1 3 ] ) but an exciting premiere for the third generation b quarks. Similar estimates hold for the z lepton.
3. Resolving the vertices. A phenomenological model There is no theory at the present time that reconciles the small lepton/quark masses with the tiny radn corresponding to the very large composlteness scales in energy. Nevertheless, one might formulate phenomenologlcal models in which static properties of the known quarks and leptons are traced back to the exchange of novel particles which are suggested in composite models. A nearly infinite variety of such models can be thought of, but we concentrate on just one typical example m which an isoscalar, charge - ~, color triplet, spin-zero dlquark D couples to two quarks with strengths 2 L,R ( f o r left/right handed components) in an SU (3) × SU (2) × U ( 1 ) lnvarlant form LD = ("].L/~bL +'~Rt~bR)D+h.c.
(7)
(the fields are antisymmetrlzed in the color indices). Similarly to the third generation, the interaction density may be defined for the first two generations. 445
Volume 228, number 3
PHYSICS LETTERS B
T
21 September 1989
_
--z--
+ -z-15
b Fig 1.
The two vertex diagrams shown in fig. 1 introduce finite-size effects and anomalous (Z) magnetic moments. Anticipating a mass range of several hundred GeV for mD >> mt, the radii and the magnetic moment are given by ~3 122+2212 R2=2n 4~r 1 2R+2L 2 2
R2-2n x=
4n
2eD Ut --3~-b x w + - v b ( L - 4 ) +
vt 1 2e_pD~xw+__~(L m 2 --3ab ab
1 2R2L mbmt ( 2 v ~ ut Xw+-(L-i) 4n m2 Vb
)
+
U~ b
((L-7
4)+ -a~
)
'
(L_7)
ab
122+22m~(eDxw+lVt--~at ~ ,
n
4n
m2
\V b
6
vb
(8)
/
where the abreviations m2 - L = l o g mt2
2 2 2 R --2 L (= 22+22 , Xw=Sln20w
have been used. (The signs in eq. (2) are to be identified with the sign of the sum between the bars in R21 and R 2. ) eD = -- ½is the diquark charge. Thus an experimental bound o f R ~<1 × 10 - 16cm would translate into a compositeness scale ofA = x/~R - 1>~1 TeV, corresponding to a diquark mass of rnD~>200x/~-/4n GeV. The predicted value for the (Z) anomalous magnetic moment depends less dramatically on the assumption whether the theory is chiral or not (2 R2L= 0 or 0) than for ( g - 2 ) e [2,1 5,1 6] because the mass ratio mJmt is not expected to deviate significantly from O ( 1 0 - ' ), in contrast to the ratio mc/ms or even mc/mt from dilepton and leptoquark exchanges. We have investigated variants of this model in which the effects of additional form factors at the diquark vertices are estimated and the internal boson-fermion loop masses are chosen close to each other. The overall picture, however, does not change significantly [ 14 ]. 4. Summa
It has been shown in this brief note that LEP is a unique facility to probe the static properties of quarks, in particular heavy quarks of the third generation. There is no other way to set limits on their radii and anomalous magnetic moments. Furthermore, for hght quarks and x the existing hmits can greatly be improved. (Note that the radius R used here is the physical particle radius; it is not plagued by the ambiguities introduced in the definition of a radms by the unknown coupling constants in contact terms. ) We had assumed from the outset that electrons could be considered pointlike for the experimental limits we are envisioning for x and quarks. This assumption has been inferred from the ( g - 2 ) e measurements. However, if experimental analyses would reveal deviations from the values F ( Z ~ e + e - ) etc. expected m the standard model, it would be a natural consequence m the frame of composite models to question the polntlike behawor of the Z boson ~tself. ~3 The expressions for arbitrary ratios m~/m2 have been reported m ref. [ 14]. 446
Volume 228, number 3
PHYSICS LETTERS B
21 September 1989
Acknowledgement W e a r e m d e b t e d t o P . M . Z e r w a s f o r h i s c o n t i n u o u s h e l p d u r i n g all t h e stages o f t h i s w o r k .
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