Physics Letters B 269 ( 1991 ) 361370 NorthHolland
PHYSICS LETTERS B
Constraints on the BESS model from precision electroweak data. Specialization to technicolor and extended technicolor R. Casalbuoni
a,b S. D e
Curtis
b D.
D o m i n i c i c,b, F. Feruglio d,~ and R. Gatto d
a Dipartimento di Fisica, Universitgt di Firenze, 150125 Florence, Italy b INFN, Sezione di Firenze, 150125 Florence. Italy c Dipartimento di Fisica, Universith di Camerino, 162032 Cameroni, Italy DOpartement de Physique Th?orique. Universit? de Genbve, CH1211 Geneva 4, Switzerland
Received 30 July 1991
We perform an update of the bounds on the parameters of the BESS model, which describes a possible strong electroweak breaking, by including latest experimental informations. The obtained bounds can be compared with predictions from special cases, corresponding to technicolor and extended technicolor. In spite of some ambiguities, it appears that conventional QCDscaled technicolor with NTCtechnicolors and Nd technidoublets is excluded at 90% CL for NTcNd larger than ~ 12.
1. Introduction It is i m p o r t a n t to test the idea of a strong interacting sector as responsible for electroweak symmetry breaking. The BESS model [ 1,2 ] provides for a rather general frame which allows one to translate new precision electroweak data into numerical b o u n d s on relevant parameters of the proposed strong sector. A characteristic feature of the strong sector would be the occurrence of resonances in the TeV range, most importantly spin1 resonances. One expects electroweak selfenergy corrections to be affected by the occurrence of such resonances and the BESS approach, based on the s t a n d p o i n t of custodial symmetry and electromagnetic gauge invariance, allows for simple inclusion of the essential aspects. One i m p o r t a n t specialization of BESS is to technicolor theories, of which BESS, for particular parameters, should reproduce the main features, roughly in the same way as p dominance reproduces some m a i n feature of QCD. In this note, besides performing the general BESS update from recent electroweak data, we shall consider specialization to technicolor in detail. A m a i n ~ Partially supported by the Swiss National Foundation. Address after 30 September 1991: Dipartimento di Fisica, UniversitY.di Padova, 135131 Padua, Italy.
conclusion from the update will be the exclusion at 90% CL of conventional technicolor (with an assumed QCDscaled dynamics) with Nvc technicolors and Nd technidoublets for NvcNd larger than ~ 12. We shall also c o m m e n t on the role of extended technicolor, which does not influence the conclusion. In the m i n i m i z a t i o n program we shall first introduce a .~2 function of the total Z width, of the hadronic width, of the leptonic width, of the m u o n forwardbackward asymmetry, and of m w / m z . At a second step zpolarization data and data for forw a r d  b a c k w a r d asymmetry for b are added. The cesium atomic parity violation data are also included. The analysis uses available full oneloop radiative correction programs. This brings in a dependence, through the radiative corrections, from the parameters as, mtop and a cutoffA which can be thought to correspond to mH of the standard model. We allow these parameters to vary within the relevant ranges. The limitations on the BESS parameters coming from the first set of data we have m e n t i o n e d (FT, Fh, F~, A ~B, m w / m z ) restrict rather strongly the parameters g / g " and b (see figs. 1 and 2). For small b values (such as expected for instance from an extended technicolor origin of such a coupling) the allowed values o f g / g " turn out to lie in a very narrow range including the standard model limit. Adding to the
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above set of data the available data from z polarization, A ~,ol, essentially gives no additional constraints, whereas adding the available A 78 results in a stronger bound (see fig. 3) on g/g" at fixed b (a result that could, however, be strongly affected by a shift in the central value of A bB). We discuss at length the case b = 0 but including an axialvector field chosen such as to satisfy the current algebra constraints (Weinberg sum rules). This serves for specialization to technicolor and to obtain bounds on the product NvcNd (see figs. 4 and 5, where horizontal lines corresponding to NrcNd=3 and 12 are shown). The variation with respect to the radiative correction parameters as, mtop and A can be quantitatively discussed and allows for the derivation, based on Z widths, m w / m z , and A~B, of an upper bound of 0.1130.118 for g/g" at 90% CL. Translated in conventional technicolor this corresponds to NxcNd <~1213.1. The atomic parity cesium data make this result stronger by giving g/g" <~0.092 and correspondingly NTcNd~<7.9. A further strengthening occurs when data on z polarization and b forwardbackward asymmetries are included. This results in g/g" <~0.07 (including Cs data)0.08 (without Cs data) corresponding respectively to NxcNd <~4.66 (see fig. 6 and table 1 ). In section 2 we recall some of the principle features of BESS and of the contribution of spin1 bosons to electroweak selfenergies. In section 3 we discuss how the experimental data, from LEP and C D F / U A 2 restrict the BESS parameter space. In section 4 we discuss how by a convenient restriction of the BESS parameter space one can get a description of the vector and axialvector resonances of technicolor, including extended technicolor, in a way particularly relevant for LEP physics.
2. The BESS model and the vector boson selfenergies The vector resonances of the BESS model are bound states of a strongly interacting sector. In this sense they are similar to ordinary p vector mesons, or to the techni 9 particle of technicolor theories [ 3 ]. Due to their composite nature, the V particles are then expected to mix to the photon and to the W and Z vector bosons. From this, a nontrivial behavior under 362
31 October 1991
the electromagnetic gauge group U ( 1 )era is expected (see ref. [4] ). Using this fact and the requirement that the electroweak p parameter be equal to 1 at tree level, one can easily construct the most general mixing term of the V particles with the ordinary vector bosons. By defining 3 ~/~= Z " i=1
3 ,, l i t
=
• l i   i lg'~z Wu,
i=1
~ =ig'" ½r3/~u
(2.1)
one has ~M =  ¼v2[tr( Y ~  ~ )2 + a tr( YC/'+.~ 2 Y/) 2 ] , (2.2) where v and a are free parameters. The first term is nothing but the usual mass term for the W and Z fields, whereas the second one is the only mixing term compatible with the properties we have just discussed. The lagrangian (2.2) must then be supplemented by kinetic YangMills terms for all the three vector boson fields, with a gauge coupling g" for the field V. As far as the interactions with fermions are concerned, one must specify the current J to which the new triplet of states Vcouples. If we assume J = JL (see ref. [1] for a more general discussion), the U ( 1 )era gauge invariance requires the interaction lagrangian to be
g
l~.jL+g,~jv+l
b
g"~"JL.
(2.3)
The parameter b specifies a possible direct coupling of the fermions to the new gauge vector bosons. However, it must be stressed that, even for b = 0, a coupling of the physical V particles to fermions is present due to their mixing with the physical WeinbergSalam vector bosons. At the zeroth order in the weak couplings the V mesons are degenerate in mass and
M ~ = ~vZag "2 .
(2.4)
The complete list of couplings to fermions can be found in refs. [1,2]. The parameter space of the model is given by (g, g', v, My, g", b). We trade of (g, g', v) for (aem, Gv, mz) and therefore we remain with (My, g", b). In turn, the parameter v can be reexpressed in terms of mw, and the expressions of (g, g', mw) in terms of (aem, GF, mz) can be found in ref. [ 2 ].
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The case b = 0 is particularly simple from a theoretical point of view. In fact this is the case of heavy particles with no direct coupling to the fermions (the coupling coming only through the mixing to W and Z). We can therefore use for the discussion a general approach discussed recently in refs. [5,6]. This approach just describes effects of new physics due to heavy particles with no direct coupling to fermions. Therefore, all their effects regarding LEP physics are concentrated in modifications o f the selfenergies of the ordinary vector bosons. The main idea is to parametrize the selfenergies through an expansion in q2/A2, where A is the scale o f new physics. The relevant parameters turn out to be three, and they can be related to physical observables such as the ratio row~ mz (measured at C D F / U A 2 [ 7 ] ), the leptonic width, and the forwardbackward asymmetry in leptons. The expansion of the selfenergies is according to
the correlators of the vector and of the axialvector currents [ 5 ]. The parameter z is free. We will discuss later how z is fixed in the special case of technicolor.
H~jU(q 2 ) =  i g U V ( A i j + q Z F ij) +q~'q" t e r m s ,
FT = 2.485 _+0.009 G e V ,
(2.5)
where the indices i, j run over W, Z and photon fields. The combinations of the parameters A q and F ° related to the observables are el =
Azz
m~ 

Aww m2w ,
F h = 1.740+0.009 G e V ,
F~= 83.31 _+0.40 M e V ,
~3=(g/g") 2 .
(2.6)
and [11] (3.1b)
m w / m z = 0.8807 _+0.0031 .
(3.1c)
In these notations the forwardbackward asymmetry in leptons at the Z peak is given by
g2vgZA
A~vB= 3 (g2 +g2A)2 "
(3.2)
(2.7)
It is straightforward to enlarge the vector BESS model to include a triplet of axialvector resonances [9]. Assuming the same gauge coupling g" for the axialvector and vector resonances, one has to introduce two more parameters: the mass MA of the axialvector resonances, and a quantity z associated to the mixings between the vector V and the axialvector A with W and Z [ 9 ]. For the e's one obtains
e3=(1z2)(g/g") 2 .
(3.1a)
and [7]
and e3 is related to the parameter S of ref. [ 5 ] by e3=O~emS/4sin20. For the BESS model we find [8] (at the first order in q 2 / M 2 )
el=l~2=0,
In the present analysis of the BESS model, which in this section will be restricted to the pure vector case, we will use the data as obtained by the LEP and C D F / UA2 experiments. The mass of the Z was fixed at the value m z = 91.174 GeV. First we will present the result obtained by minimizing a Z 2 function which is the sum of four contributions: the total width Fz, the hadronic width l"h, the leptonic width F~, and the ratio (gv/gA) obtained from the muons forwardbackward asymmetry. The averages of the experimental values of these quantities are [ 10]
gv/gA = 0.072 _+0.011 , e2   F w w  F 3 3 ,
cos 0 E3 = sin 0 F3o
el=E2=0,
3. Allowed regions for the BESS parameters
(2.8)
The axialvector resonances contribute with an opposite sign with respect to the vector particles. This is easily understood by noticing that e3 can be expressed in terms o f the combination (HvvHaA) o f
Later on we will extend this analysis by including recent data on the zpolarization asymmetry, obtained by A L E P H [ 12 ], and on the bottom forwardbackward asymmetry averaged over the four LEP Collaborations [ 13 ]. In order to have a precise evaluation o f widths and asymmetries in BESS one has to be very careful. We recall that BESS has no elementary scalars and it is a nonrenormalizable theory. Radiative corrections for BESS can be defined only if one consider this model as a cutoff theory. A similar problem practically affects all the calculations made in similar contexts, like in technicolor theories. It is true that these are renormalizable theories, but because of our ignorance o f the dynamics the best we can do is to define their low363
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energy limits as effective theories. One then encounters p r o b l e m s similar to ours if one ignores the scalars (Higgslike m o d e s ) . Effectively one has a lowenergy theory with only pseudoGoldstones, which is equivalent to have some nonlinear a model (cf. Q C D ) . However, these considerations suggest the definition o f the k i n d o f c u t o f f t h e o r y we want to deal with. In fact, a characteristic feature o f BESS is that it is based on a nonlinear a  m o d e l which can be thought o f as the Higgs model in the m n ~ limit, at some formal level. Appelquist and Bernard [ 14 ] had shown that in the nonlinear a model, considered as a cutoff theory, the leading term in the cutoff, in the oneloop a p p r o x i m a t i o n , corresponds exactly to the leading term in log mH o b t a i n e d at oneloop renormalization o f the Higgs lagrangian. Therefore, a possible strategy to be followed at oneloop level is to consider for BESS the same full oneloop radiative corrections as in the standard m o d e l ( S M ) . However, in this context mH must be interpreted as a cuto f f A rather than as the mass o f an elementary particle. Because the standard lore, in theories where the electroweak s y m m e t r y is broken strongly, is that the mass scale o f new physics should be a r o u n d 1 TeV, we shall put A = 1 TeV in the following. However, we will point out how our results d e p e n d on the cutoff by varying it between 1 TeV and 2 TeV. F r o m the point o f view o f radiative corrections we should also worry about loops o f heavy vector bosons. However, one can show that they are o f order C~wwith respect to the modifications that BESS induces on the SM at tree level. Therefore, these contributions can be safely ignored. The results o b t a i n e d by fitting the d a t a given in eq. (3.1) are illustrated in figs. 1, 2, where we have considered variations o f the allowed region in BESS par a m e t e r space with c~s, mtop and A. In fig. 1 we plot the allowed region, at 90% CL, in the plane (b, g/g" ), by fixing M y = 1000 GeV, mtop= 140 GeV, letting C~sfree to vary between 0.11 a n d 0.13, and A between 1 TeV and 2 TeV. The variations with A are between the bands illustrated in the figure, where the curves at the left of the b a n d s c o r r e s p o n d to A = 1000 GeV. We see for decreasing c~s, a n d / o r increasing A, the allowed region shifts to the right o f the plane. This implies that, fixing the other variables, the upper limit on g/g" decreases for decreasing c~s, a n d / o r increasing A. In fig. 2 we show how the al364
31 October 1991
g/g" 0.050 0.025 0.000 0.025 0.20 . . . . I . . . . I . . . . q . . . .
0.15
,~
0.050 0.20
"
0.15
0.10
0.10
,~,
,~
S
0.05
,
,~,,
0.05
J
" ~ 0.00
....
0.050
i
,
0.025
~ I t ,
0.000
" ,
' '~ ,
,
,
0.025
,
,
,
0.00
0.050 b
Fig. 1. The allowed region at 90% CL in the (b, g/g" ) plane for Mv = 1000 GeV, mtop= 140 GeV. The continuous and dashed lines correspond respectively to ~s = 0.11 and o~s= 0.13. The shadowed regions show how the allowed region varies when A is varied between 1000 GeV and 2000 GeV. The curves to the left correspond to A= 1000 GeV. The origin (corresponding to the standard model limit) is inside the allowed region. The analysis is based on the data on Fx, Fh, F~, gv/gA, and mw/mz [see eq. (3.1)]. lowed region in the plane (b, g/g" ) varies when the top mass is varied between 110 GeV and 170 GeV, for Mv = A = 1 TeV and c~s= 0.12. As easily expected, the effect of mtop is similar to the effect o f C~s, that is, the upper limit on g/g" decreases by decreasing mtop. The allowed region in the plane (b, g/g" ) is practically i n d e p e n d e n t o f the value o f M y as soon as Mv >> mw.z. This is a characteristic feature o f the BESS model, for which the V particle does not decouple in the M v  , ~ limit (at fixed g " ) , not even at b = 0 . This can be seen from eq. (2.2), which shows that the mixing term between V and W, Z is proportional to c~, that is to M 2 at fixed g" [see eq. (2.4) ]. As a result, the couplings o f W and Z are m o d i f i e d by terms of order g/g" in a way which practically does not d e p e n d on Mv ( i f Mv >> mw.z). We notice also that for b < 0 the upper limit on gig" is smaller than for b>~0. As m e n t i o n e d at the beginning of this section, we have also considered possible modifications o f the previous picture coming from the inclusion o f recent
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g/g"
.......... i II'':°2°
0.050 020
PHYSICS LETTERS B
0.025
0.000
0.025
0.050
31 October 1991
g]g" 0.i0
0.05
0.00
0.05
0.10
020
020
0.15
0.15
0.15
0.15
o o
OlO
OlO
o o
0.05
0.05
0.05
0.05
o.oo
ooo t
ooo 0.050
0.025
0.000
0.025
0.050 b
Fig. 2. The allowed region at 90% CL in the (b, g/g" ) plane for M v = 1000 GeV, %=0.12, and A = 1000 GeV. The continuous and dashed lines correspond respectively to mtop = 110 GeV and mtoo= 170 GeV. The origin (corresponding to the standard model limit) is inside the allowed region. The data used are the same as for fig. 1.
data on the x polarization asymmetry A pol and on the bottom forwardbackward asymmetry AbB. The experimental values are [ 12 ] A~o I =0.152_+0.045 ,
(3.3a)
and [131 A~B =0.125 + 0 . 0 2 4 .
(3.3b)
The value for A ~oJ was given by the A L E P H Collaboration in ref. [12], whereas the value o f AbB is an average over the four LEP Collaborations [ 13 ]. Fig. 3 shows the region allowed by the single observables A ~B, A ~ol and A ~B, as given in eqs. ( 3 . 1 ) (3.3). From this figure we see that the preliminary result on the x polarization asymmetry tends to give no additional constraints on the region of the plane (b, g / g " ) allowed by the data considered in eq. (3.1). On the other hand the present value o f A bB tends to shift the physical region towards the positive b values. In this way, at any fixed b value, we have a much stronger upper b o u n d on g / g " than before. We notice
0.10
~
" 0.05
. 0.00
" ooo 0.05
0.10 b
Fig. 3. Comparison of the three (90% CL) allowed regions, between the corresponding lines, in the (b, gig" ) plane, for A = 1000 GeV, mtop= 140 GeV, from the single observables A~,ol (continuous lines), A~vB (dashdotted) and AbB (dashdouble dotted). The shaded region is already excluded by the data ofeq. (3.1).
that this result is strongly dependent on the actual central value ofA bB, which, at present, is affected by a relatively large error. For b = 0 it is very easy to extend the previous analysis to the case in which a triplet of axialvector resonances is present in addition to the V triplet. We have seen at the end of section 2 that all the effects of such particles are indeed a simple modification of the e3 parameter by a factor ( 1  z 2). Because all the new physics (apart from radiative corrections which do not depend on g" ) is contained in the e3 term, one translates the previous results by simply multiplying the bounds obtained for ( g / g " ) 2 by a factor 1/(1  z 2). Therefore, the presence o f an axialvector resonance will generally make the bounds on g / g " more loose.
4. The relation to technicolor and extended technicolor theories
As illustrated in section 2, BESS is a model describing in a rather general way vector resonances; we can then apply it to describe the vector resonances of a theory like technicolor. About this point we recall 365
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that the idea of hidden gauge symmetries [15 ], in terms of which BESS can be formulated [ 1 ], was successfully used to describe the ordinary strong interacting vector resonances, like p etc. [ 16 ]. We will describe how the technicolor vector (and axialvector) resonances can be described by a convenient choice of the BESS parameters. Technicolor was one of the earliest suggestions [ 3 ] to provide for an alternative to the spontaneous symmetry breaking mechanism of the SM, based on elementary scalars, which is considered as theoretically unsatisfactory. The onedoublet model has a single techniquark doublet, of charge +½ and ½ for anomaly cancellation. The onefamily model [ 17,18 ] has four doublets (3 colors+ 1 lepton), anomaly cancellation going as in quarklepton families, with flavor symmetry S U ( 8 ) ® S U ( 8 ) and SU(NTc) of technicolor. Technifermion condensates break flavor into diagonal SU(8), giving 63 Goldstone bosons, three of which provide the W, Z longitudinal degrees of freedom. Generation of masses for ordinary fermions has led to extended technicolor and is generally associated with difficulties because of the limitations on flavorchanging neutral currents. Subsefjquent proposals have been related to walking technicolor [ 18,19 ]. PseudoGoldstones are in general a very sensitive and potentially dangerous feature, especially in the original technicolor models. They are absent in the onedoublet model. However, in the onefamily model some of the 63 pseudoGoldstones would be low in mass and are expected to the produced as charged scalar pairs from e+e . Also, virtual pseudoGoldstones would contribute to radiative corrections [5,20,21 ]. The overall technicolor contribution to electroweak radiative corrections will, however, result from a number of effects, among which one of the most ambiguous comes from technipions. Therefore, if one realistically believes in the full S U ( 8 ) N S U ( 8 ) onefamily model, or in a more complicated model, one will have to qualify all the quantitative statements for technicolor because of the ambiguites related at least to technipion masses and calculation of their virtual contributions. A detailed analysis under various assumptions can be found in ref. [21 ]. The dynamics in technicolor theories is usually believed to be roughly readable from QCD by simply scaling the fundamental scale AQCD to ATC In QCD, 366
31 October 1991
vector dominance has revealed itself to be a useful concept leading to results comparing very well with the experimental data. Therefore it is natural to assume that vector dominance works as well in technicolor theories as in QCD. In this spirit one can specialize the BESS model to technicolor, as taken in a vector dominance approximation. Strictly speaking, BESS would correspond to a technicolor model involving a single technidoublet. However, we will take here the simplest assumption of neglecting dynamical contributions from technipions. By doing so one can specialize BESS also to technicolor models involving more than one technidoublet. To translate BESS parameters into such technicolor specialization, we recall that for SU (Nvc) of technicolor one scales directly from QCD the techni 9 mass
( 3 ~'/2(4"~ 1/2 MoT ~, = Mo \ ~TC,]
\~dJ
'
( 4.1 )
where Na is the number of technidoublets and Mo is a scale parameter roughly of order 1 TeV. As repeatedly said, technicolor dynamics is obtained by scaling QCD, so the Kawarabayashi, Suzuki, Fayazzuddin, Ryazzuddin (KSFR) relations are supposed to be valid also in technicolor. The first KSRF relation for technicolor would read 2
g~oTc = 2 N o f Tcg~T,~TC~VC
(4.2)
relating the 7OTC coupling to the technipion decay coupling constant and to the PTCXVCXTCthreelinear coupling. This relation is automatically satisfied in BESS, where gvv =  ½°¢g" v2
(4.3)
for the 7V coupling, g v ~ =  14o~g"
(4.4)
and v is the decay coupling constant of the Goldstone re. So this relation does not impose any condition on the BESS parameters c~ and g". On the other hand the second KSRF relation, M 2O T C  2 N d ~ x C : g m2c ~ T c ~ c
(4.5)
requires c~= 2 in the BESS model by using eqs. (2.4) and (4.4). No restrictions are imposed on g" and by comparing the technip mass expression [eq. (4.1) ] to the BESS expression for Mv [eq. (2.4) ], one can
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establish the following relation between g" and the product NTcNd: g
g , , = Mo N / v
24
(4.6)
NTcNd "
TO c o m p a r e the e x p e r i m e n t a l b o u n d s on BESS with those for technicolor the following relation is useful: g _mw /NvcNd g" MoN/ 6
(4.7) '
In technicolor theories one also expects axialvector resonances. To translate this part in BESS one can recall from ref. [16] that in the hidden gauge symmetry a p p r o a c h to Q C D including axialvector resonances, one should take the p a r a m e t e r z, i n t r o d u c e d in eq. (2.8), equal to ½. This follows from vector d o m i n a n c e and the Weinberg sum rules. Furthermore one has the Weinberg mass relation M 2 =
2g~v. To complete the technicolor translator for this specialized BESS we consider the role o f the b p a r a m eter o f BESS, characterizing a direct coupling o f V to the fermions as described in eq. (2.3). The fourfermion interaction o f extended technicolor, o f the form
1 A~iTC (~TLcV" • 2'['~/TC 1 L ) ( ~L 1 L) , v r 7 , ' ~r~ur
(4.8)
would induce a similar direct coupling o f PTC tO fermions, through technifermion loops, given by an effective interaction
31 October ! 991
ent authors [ 5,6 ]. We shall aim here at a quantitative statement at a high confidence level. By fixing, in BESS, as we have said, c~= 2, z = ½ and considering the case b = 0, we remain with the only p a r a m e t e r g" (the V mass is now fixed by g" ), or equivalently with NTcNd from eq. (4.6). In figs. 4, 5 we then show the 90% CL region in the plane (mtop, g / g " ) by varying c~s. The upper curves correspond to oq equal to 0.11 and 0.13, as o b t a i n e d from LEP data. Also shown are the regions o b t a i n e d by combining the LEP data with the data from a t o m i c parity violation in cesium [22,23]. In fig. 5 we show the region o b t a i n e d by increasing the cutoffA up to 2 TeV, for c~s=0.11, 0.13. In the figures we show also the lines corresponding to NTcNd equal to 3 and 12 (for NTc= 3, N d = 1, 4). The figures show explicitly the behavior discussed at the end o f section 3 regarding the variation of the upper limit on g / g " with c~s, mmp and A. The limit decreases, at fixed mtop, by decreasing C~s a n d / o r increasing A, and increases with mtop, at fixed c~ and A, except that the region completely closes after a critical value o f mtop. It is interesting to notice that
g/g" 100 0.20 . . . .
150 i . . . .
200 i . . . .
250 0.20
015
0.15
2
1 Mp~ C PTc'JL A2Tc g~c~c~<"
(4.9)
0.10
Using eqs. (2.4) and (4.4) for the BESS model (o~=2), we find (for small b)
0.05
b~2
A~TTC "
.
;;;;.52
[illi[[ ...... , 
1[
0.10
0.05
(4.10)
W h e n translating the specialized BESS to technicolor, one sees that b can be interpreted as a p a r a m eter o f extended technicolor, its magnitude being expected to the inversely related to the square o f the extended technicolor scale. We can now use our previous results to get experimental b o u n d s on technicolor theories. The fact that electroweak data could already be limitative to technicolor theories has already been r e m a r k e d by differ
0.00
0.00 100
150
200
250 Illtop
Fig. 4. The allowed region, below the corresponding lines, at 90% CL in the (mtop, g/g" ) plane forA= 1000 GeV. The continuous and dashed lines correspond respectively to as=0.11 and ces= 0.13. For each case, the upper curves include only the LEP and CDF/UA2 data of eq. (3.1), whereas the lower curves include also data from atomic parity violation in cesium. The two horizontal lines indicate ~he values of (g/g") corresponding to technicolor models with NTcNa equal to 3 and 12. 367
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g/gpt
g/g" 100 0.20
. . . .
150 ~ . . . .
0.15
200 ~ . . . .
....4
12
250
1 O0 0.20
0.20
0.15
0.15
,e'"I""
0.10
0.05
0.05
0.00
0.00
/ 150
200
250 0.20
0.15
0.10
250
mtop
Fig. 5. The allowed region, below the corresponding lines, at 90% CL in the (mtop, g/g" ) plane for A= 2000 GeV. The continuous and dashed lines correspond respectively to c~=0.11 and cq=0.13. The curves include only LEP and CDF/UA2 data of eq. (3.1). The two horizontal lines indicate the values of (g/g") corresponding to technicolor models with N+cN,j equal to 3 and 12. the variations of the parameters as and A tend to give simply a translation of the allowed region in the plane (mtop, g / g " ) . Therefore, we get an absolute upper b o u n d at 90% CL for g / g " of about 0.1130.1 18, corresponding to N T c N d = 1213.1 [using only the data o f e q . (3.1) ]. By including the data from the atomic parity violation in cesium, we obtain a 90% CL upper b o u n d on g / g " ~ 0.092 corresponding to N x c N d ~ 7.9. Finally, fig. 6 shows the result of a fit obtained by including the full set of electroweak data, listed in eqs. (3.1) and (3.3), both with and without the inclusion of data on atomic parity violation. The effect of having included the data on the • polarization a n d b forw a r d  b a c k w a r d asymmetries is evident: the absolute upper b o u n d on g / g " gets considerably smaller. With (without) the inclusion of the cesium data, one gets, at 90% CL, g / g " ~ < O . 0 7 (0.08), corresponding to NvcNd ~<4.6 (6). This last point should be taken cautiously due to the fact that we are neglecting dynamical technipions. However, following the Cahn a n d Suzuki estimate [21 ] of the technipion contribution to E3, which 368
200 1 . . . .
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0.10
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0.10
t
100
3
i
150
200
i
r
l
0.05
0.00 25O
romp Fig. 6. The allowedregion, belowthe correspondinglines, at 90% CL in the ( mtop,g/g" ) plane for A= 1000 GeV. The continuous and dashed lines correspond respectively to as=0.11 and as = 0.13. The upper curves include the LEP and CDF/UA2 data of eq. (3.1), and the x polarization and b forwardbackward asymmetries of eq. (3.3), whereas the lower curves include also data from atomic parity violation in cesium. The two horizontal lines indicate the values of (g/g") corresponding to technicolor models with NTcNdequal to 3 and 12. gives about  0.002, the upper limit in NTcNd would rise about 2.5. In conclusion our analysis shows that the full set of data, given in eqs. (3.1) and ( 3.3 ) together with data from atomic parity violation in cesium provide for an upper b o u n d of NTcNd ~ 7.1, including uncertainties about the technipion contribution. This excludes the onefamily technicolor with NTc = 3 and Nd = 4. In table 1 we summarize our results by showing their dependence on the particular set of data employed in the analysis. Technipions uncertainties are those evaluated by Cahn and Suzuki [21 ], a n d they are taken as bad as possible within the limits of that analysis. Extended technicolor is not going to change the conclusions. Relying on the negative sign in eq. (4.10), one sees from figs. 1 and 2 that negative b values correspond to more stringent b o u n d s on g i g " , and so the difficulty with technicolor would be reinforced. More generally and more effectively, any reasonable order of magnitude for AETC shows that the
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Table 1 Upper limits on g/g" and NTcNo at 90% CL. Data A include FT, Fh, F~, A~vaand mw/mz. Data B include A and cesium atomic parity violation data. Data C include A and A~B,A;o~.Data D include C and cesium atomic parity violation data. Data
A B C D
Vector+ axialvector+ technipions
Vector + axialvector
(g/g" ) ul
( NvcNa) ul
(g/g" ) ul
(NTcNd)uJ
0.1130.118 0.092 0.08 0.07
1213.1 7.9 6 4.6
0.1240.129 0.106 0.095 0.087
14.415.6 10.5 8.5 7.1
expected values of b, when comparing with the bounds of figs. 1, 2, produce negligible changes in this particular analysis.
5. Conclusions We have included the latest LEP data to derive b o u n d s on the BESS model parameters. Electroweak radiative corrections are dealt with at full oneloop approximation, and variations in terms of 0% mtop, and a cutoffmass A equivalent to mH are considered. Specialization to c o n v e n t i o n a l QCDscaled technicolor is studied in detail. An absolute upper b o u n d that we find on the BESS parameter g / g " , independent of as, mtop, and A, translates into a b o u n d for the product NTcNd, of the n u m b e r of technicolors and technidoublets, NTcNa~<4.6. Considering ambiguities related to technipions, this b o u n d may become as large as 7.1, a n d one can still safely conclude that onefamily technicolor with NTC = 3 a n d Nd = 4 is excluded at 90% CL. Considerations on extended technicolor do not appear to affect this conclusion. Improvements in data will allow for still stronger conclusions, which depend strongly on the average experimental values, besides experimental errors. The general BESS model approach to possible strong electroweak breaking is thus limited by precision electroweak data within a definite range of values for its parameters, and it appears that naive Q C D extrapolations are not probably applicable to an electroweak strong sector.
Acknowledgement We would like to thank R. Barbieri and D. Nanopoulos for useful discussions and correspondence. We would also like to thank W. Hollik for providing us with a programme to compute the full oneloop standard model predictions for the widths.
Note added in proof At the 1991 L e p t o n  P h o t o n Conference new data from LEP 1 and C D F / U A 2 were reported. In particular, for the observables considered in eqs. (3.1) and (3.3), the new results are F z = 2 . 4 8 7 + 0.010 G e V , F h = 1.740+0.009 G e V , F~ = 83.20 + 0.40 M e V , A~B=0.0163+ 0.0036, A~B=0.126+0.022, Apo~= 0.134+0.035 (from ref. [24] ),
m w / r n z = 0.8789 + 0.0029 (from ref. [ 25 ] ). Using these new values the 90% CL upper limits on g/g" and NTcNd get more restrictive, and the first two columns of table 1 are changed as follows: for the set of data (A)
(g/g"),~,=O.093,
(NTcNd)uI= 8.1 ,
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for t h e set ( B )
(g/g")u~=O.083,
(NvcNo)ul=6.3,
for t h e set ( C )
(g/g")ul=O.070,
(NvcNd)u,=4.6,
a n d for t h e set ( D )
(g/g" )u~=O.064,
( NrcNd)u~= 3.8 •
A g a i n t h e t e c h n i p o n c o n t r i b u t i o n , as d i s c u s s e d w i t h i n t h e text, w o u l d rise u p t h e u p p e r l i m i t o n (NTcNo) o f a b o u t 2.5 units. W e see t h a t n e w d a t a r e i n f o r c e t h e conclusion that the onefamily conventional QCDs c a l e d t e c h n i c o l o r ( w h i c h c o r r e s p o n d s to
NxcNd=
12) is e x c l u d e d at 90% CL.
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