- Email: [email protected]
Constraints on extended neutral gauge structures
3 June 1999
Physics Letters B 456 Ž1999. 68–76
Constraints on extended neutral gauge structures Jens Erler 1, Paul Langacker
2
Department of Physics and Astronomy, UniÕersity of PennsylÕania, Philadelphia, PA 19104-6396, USA Received 24 March 1999 Editor: M. Cveticˇ
Abstract Indirect precision data are used to constrain the masses of possible extra Z X bosons and their mixings with the ordinary Z. We study a variety of Z X bosons as they appear in E6 and left-right unification models, the sequential Z boson, and the example of an additional UŽ1. in a concrete model from heterotic string theory. In all cases the mixings are severely constrained Žsin u - 0.01.. The lower mass limits are generally of the order of several hundred GeV and competitive with collider bounds. The exception is the Zc boson, whose vector couplings vanish and whose limits are weaker. The results change little when the r parameter is allowed, which corresponds to a completely arbitrary Higgs sector. On the other hand, in specific models with minimal Higgs structures the limits are generally pushed into the TeV region. q 1999 Elsevier Science B.V. All rights reserved.
The possibility of additional neutral gauge bosons, Z X s, is among the best motivated types of physics beyond the Standard Model ŽSM.. They are predicted by most unifying theories, such as Grand Unified Theories ŽGUTs., left-right unification, superstring theories and their strong coupling generalizations. In many cases their masses remain unpredicted and may or may not be of the electroweak scale. In the context of superstring models, however, which are much more constrained than purely field theoretical models, they are often predicted to arise at the electroweak scale, as we will discuss below. In this paper we consider six different types of Z X bosons: 1. The Zx boson is defined by SO Ž10. ™ SUŽ5. = UŽ1.x . This boson is also the unique solution to
the conditions of Ži. family universality, Žii. no extra matter other than the right-handed neutrino, Žiii. absence of gauge and mixed gaugergravitational anomalies, and Živ. orthogonality to the hypercharge generator. In the context of a minimal SO Ž10. GUT, conditions Ži. and Žii. are satisfied by assumption, while Žiii. and Živ. are automatic. Relaxing condition Živ. allows other solutions Žincluding the ZL R below. which differ from the Zx by a shift proportional to the third component of the right-handed isospin generator. 2. The Zc boson is defined by E6 ™ SO Ž10. = UŽ1.c . It possesses only axial-vector couplings to the ordinary fermions. As a consequence it is the least constrained of our examples. 3. The Zh boson is the linear combination 3r8 Zx y 5r8 Zc . It occurs in Calabi-Yau compactifications w1x of the heterotic string w2x if E6 breaks directly to a rank 5 subgroup w3x via the Hosotani mechanism w4x.
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E-mail: [email protected] E-mail: [email protected]
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 5 7 - 8
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
4. The ZL R boson occurs in left-right models with gauge group SU Ž3. C = SU Ž2. L = SU Ž2. R = UŽ1.By L ; SO Ž10. and is defined through the current JL R s 3r5 w a J3 R y 1rŽ2 a . JByL x. J3 R couples to the third component of SUŽ2.R , B and L coincide with baryon and lepton number for the ordinary fermions, a s g R2 rg L2 cot 2u W y 1 , where g L, R are the SUŽ2.L, R gauge couplings, and u W is the weak mixing angle. 5. The sequential ZSM boson is defined to have the same couplings to fermions as the SM Z boson. Such a boson is not expected in the context of gauge theories unless it has different couplings to exotic fermions than the ordinary Z. However, it serves as a useful reference case when comparing constraints from various sources. It could also play the role of an excited state of the ordinary Z in models with extra dimensions at the weak scale. 6. Finally we consider a superstring motivated Zstring boson appearing in a specific model w5x ŽTable 1. based on the free fermionic string construction with real fermions. This model has been investigated in considerable detail w6x with the goal of understanding some of the characteristics of Žweakly coupled. string theories, and of contrasting them with the more conventional ideas such as GUTs. While this specific model itself is not realistic Žfor example it fails to produce an acceptable fermion mass spectrum. the predicted Zstring it contains is not ruled out. Its coupling strength is predicted and so are its fermion cou-
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plings. It is particularly interesting in that its couplings are family non-universal. While this may induce problems with too large flavor changing neutral currents through a violation of the GIM mechanism w7x, we will not address this issue here. Another important observation is that such a Zstring can be naturally at the electroweak scale w8x. The basic reasons are strong restrictions on the superpotential, and that in a given model the sectors of supersymmetry ŽSUSY. breaking, and its mediation are usually not arbitrary. In string models one also expects bilinear terms in the Higgs superpotential to vanish at tree level Žotherwise they should be of the order of the Planck scale. and to be generated in the process of radiative symmetry breaking. This is linked to the top quark Yukawa coupling driven symmetry breaking and typically involves extra Higgs singlets which are predicted in many string models. In all cases, there is a relation between the mixing angle u between the ordinary Z and the extra Z X from the diagonalization of the neutral vector boson mass matrix, M02 y MZ2 1 y r 0rr 1 tan2u s 2 s , Ž 1. 2 X r 0rr 2 y 1 MZ y M 0 where MZ and MZ X are the physical boson masses, and M0 is the mass of the ordinary Z in the absence of mixing. The second equality in relation Ž1. uses the neutral and charged boson mass interdependence which reads at tree level, MW Ma s . Ž 2. ra cos u W
(
Table 1 Multiplet t
Ž b. L tR bR Ž ud . L , Ž cs . L u R , cR d R , sR
Ž ntt . L tR
Ž nm . L m
mR
Ž nee . L eR
100 Q
X
y71 q133 y136 q68 y6 q3 q74 y130 y65 q9 y204 q9
As in the case of the SUŽ3. C = SUŽ2.L = UŽ1. Y model, Ý Ž t i2 y t 32i q t i . <² fi :< 2
r0 s
i
Ý 2 t 32i <² fi :< 2
,
Ž 3.
i
where t i Ž t 3 i . is Žthe third component of. the weak isospin of the Higgs field f i . r 0 s 1 if only SUŽ2. Higgs doublets and singlets are present, in which case M0 would be known independently. Nondegenerate SUŽ2. multiplets of extra fermions and scalars affect the W and Z self-energies at the loop level, and therefore contribute to the T parameter w9x. They can arise, for example, in E6 models, and in their
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Table 2 Z pole precision observables from LEP w14,15x and the SLC w16–18x. Shown are the experimental results, the SM predictions, and the pulls. The SM errors are from the uncertainties in MZ , ln MH , m t , a Ž MZ ., and a s . They have been treated as Gaussian and their correlations have been taken into account. The first set of measurements is from the Z line shape and leptonic forward-backward asymmetries, A F B Ž l . s 3r4 A e A l . The hadronic, invisible, and leptonic decay widths are not independent of the total width, the hadronic peak cross section, and the R l s G Žhad.rG Ž l q l y ., and are shown for illustration only. The second set represents the quark sector, where R q s G Ž qq .rG Žhad., and A q s 4r3 A FL RB Ž q . is a function of the effective weak mixing angle of quark q. The third set is a variety of polarization and forward-backward asymmetries sensitive to the leptonic weak mixing angle, where analogous definitions apply. For details see Ref. w19x Quantity
GroupŽs.
Value
Standard Model
Pull
MZ wGeVx GZ wGeVx G Žhad. wGeVx G Žinv. wMeVx G Ž l q l y . wMeVx s had wnbx Re Rm Rt A F B Ž e. AF B Ž m. A F B Žt .
LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP
91.1867 " 0.0021 2.4939 " 0.0024 1.7423 " 0.0023 500.1 " 1.9 83.90 " 0.10 41.491 " 0.058 20.783 " 0.052 20.789 " 0.034 20.764 " 0.045 0.0153 " 0.0025 0.0164 " 0.0013 0.0183 " 0.0017
91.1865 " 0.0021 2.4957 " 0.0017 1.7424 " 0.0016 501.6 " 0.2 83.98 " 0.03 41.473 " 0.015 20.748 " 0.019 20.749 " 0.019 20.794 " 0.019 0.0161 " 0.0003
0.1 y0.8 – – – 0.3 0.7 1.2 y0.7 y0.3 0.2 1.3
Rb Rc R s, drR Ž dquqs. A F B Ž b. A F B Ž c. A F B Ž s. Ab Ac As
LEP q SLD LEP q SLD OPAL LEP LEP DELPHI q OPAL SLD SLD SLD
0.21656 " 0.00074 0.1735 " 0.0044 0.371 " 0.023 0.0990 " 0.0021 0.0709 " 0.0044 0.101 " 0.015 0.867 " 0.035 0.647 " 0.040 0.82 " 0.12
0.2158 " 0.0002 0.1723 " 0.0001 0.3592 " 0.0001 0.1028 " 0.0010 0.0734 " 0.0008 0.1029 " 0.0010 0.9347 " 0.0001 0.6676 " 0.0006 0.9356 " 0.0001
1.0 0.3 0.5 y1.8 y0.6 y0.1 y1.9 y0.5 y1.0
A L R Žhadrons. A L R Žleptons. Am At A eŽ QL R . At Ž Pt . A e Ž Pt . s 2l Ž Q F B .
SLD SLD SLD SLD SLD LEP LEP LEP
0.1510 " 0.0025 0.1504 " 0.0072 0.120 " 0.019 0.142 " 0.019 0.162 " 0.043 0.1431 " 0.0045 0.1479 " 0.0051 0.2321 " 0.0010
0.1466 " 0.0015
1.8 0.5 y1.4 y0.2 0.4 y0.8 0.3 0.5
presence r 0 should be replaced by r 0rŽ1 y a Ž M Z . T .. If the Higgs UŽ1.X quantum numbers are known, as well, there will be an extra constraint, g 2 MZ2 usC , Ž 4. g 1 MZ2X where g 1,2 are the UŽ1. and UŽ1.X gauge couplings with 5 g2 s sin u W 'l g 1 . Ž 5. 3
(
0.2316 " 0.0002
l s 1 if the GUT group breaks directly to SUŽ3. = SUŽ2. = UŽ1. = UŽ1.X , while in general l is still of O Ž1.. We will quote our results assuming l s 1, but our limits also apply to 'l sin u and MZ Xr 'l for other values of l Žwe always assume MZ X 4 MZ .. Similar to r 0 ,
Ý t 3 i Qi <² fi :< 2 X
Csy
i
Ý t 32i <² fi :< 2 i
Ž 6.
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Table 3 Non-Z pole precision observables from Fermilab w29–33x, CERN w14,34x, and elsewhere. The second error after the experimental value, where given, is theoretical. The SM errors are from the inputs as in Table 2. The various quantities R are cross section ratios from n-hadron scattering, where the CHARM w35x results have been adjusted to CDHS w36x conditions, and can be directly compared. The g V , A are effective four-Fermi couplings from n-e scattering w37x, and QW denotes the weak charge appearing in APV w38–41x Quantity
GroupŽs.
m t wGeVx MW wGeVx MW wGeVx
Value
Tevatron Tevatron q UA2 LEP
173.8 " 5.0 80.404 " 0.087 80.37 " 0.09
171.4 " 4.8 80.362 " 0.023
Ry Rn Rn Rn Rn Rn Rn
NuTeV CCFR CDHS CHARM CDHS CHARM CDHS 1979
0.2277 " 0.0021 " 0.0007 0.5820 " 0.0027 " 0.0031 0.3096 " 0.0033 " 0.0028 0.3021 " 0.0031 " 0.0026 0.384 " 0.016 " 0.007 0.403 " 0.014 " 0.007 0.365 " 0.015 " 0.007
0.2297 " 0.0003 0.5827 " 0.0005 0.3089 " 0.0003
g Vn e g Vn e g nA e g nA e
CHARM II all CHARM II all
y0.035 " 0.017 y0.041 " 0.015 y0.503 " 0.017 y0.507 " 0.014
QW ŽCs. QW ŽTl.
Boulder Oxford q Seattle
y72.41 " 0.25 " 0.80 y114.8 " 1.2 " 3.4
is another function of vacuum expectation values ŽVEVs., where QXi are the UŽ1.X charges. For minimal cases, the functions C are given explicitly in Table III of Ref. w10x. Similarly, for the fermion couplings of the various Z X s we refer to Table II of the same work. There is the possibility of an extra gauge invariant term, mixing the field strength tensors of the hypercharge and the new gauge bosons. We do not consider such a term here, since it is expected to be small in typical models 3. The phenomenology of gauge kinetic mixing has been reviewed in Ref. w21x; constraints on leptophobic Z X bosons can be found in Ref. w22x; Ref. w23x are detailed reviews on extra neutral gauge bosons at colliders; and for related topics in neutral current physics see Ref. w24x.
3 It was shown in Ref. w12x that a relatively large kinetic mixing term can be generated at the loop level when Higgs doublets from a 78 representation of E6 are employed. However, restriction to Higgs doublets from 27 and 27 representations yields much smaller effects w13x.
Standard Model
0.3859 " 0.0003 0.3813 " 0.0003 y0.0395 " 0.0004 y0.5063 " 0.0002 y73.10 " 0.04 y116.7 " 0.1
Pull 0.5 0.5 0.1 y0.9 y0.2 0.2 y1.7 y0.1 1.1 y1.0 – y0.1 – y0.1 0.8 0.5
After the Z X properties have been specified, the new contributions to the precision observables can be computed using the formalism presented in Refs. w10,25x. The new Z X boson is treated as a small perturbation to the SM relations. The most important effect is the modification of the MZ –MW –sin2u W interdependence via Z–Z X mixing. Z pole observables are affected by the modification of the Z couplings to fermions, which is another manifestation of the Z X admixture. This change in couplings is also relevant for the low energy observables from neutrino scattering and atomic parity violation ŽAPV.. For these there will be additional effects from Z X exchange, and Z–Z X Žg –Z X . interference. Such effects can be neglected at the Z pole, but the interference terms are relevant for the cross section measurements at LEP 2. They give interesting constraints on MZ X w27x practically independent of the mixing parameter u . These constraints are complementary to the direct search limits at Fermilab w28x, where additional assumptions about possible exotic decay channels have to be specified. In our analysis we use the data as of ICHEP 98 at Vancouver. It includes the very precise Z pole mea-
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Table 4 Mass limits win GeVx on extra Z X bosons and constraints on Z–Z X mixing for two classes of Higgs sectors. The upper part of the table allows r 0 as a free fit parameter and corresponds to a completely arbitrary Higgs sector. The lower part assumes r 0 s 1, but is arbitrary otherwise. The first Žsecond. numbers correspond to the 95 Ž90.% CL lower mass limits. Below this we show the central values and the 95% lower and upper limits on sin u . Also shown are the central values and 95% limits for r 0 as a fit parameter. Finally we indicate the minimal x 2 for each model. The last column is included for comparison with the standard case of only one Z boson. All results assume MZ F MH F 1 TeV Zx
Zc
Zh
ZL R
ZSM
Zstring
SM
sin u sin umin sin uma x r0 r 0min r 0max 2 xmin
551Ž591. y0.0006 y0.0022 q0.0020 0.9993 0.9931 1.0010 27.62
151Ž162. q0.0004 y0.0015 q0.0021 0.9974 0.9923 1.0017 27.52
379Ž433. y0.0010 y0.0058 q0.0019 0.9979 0.9931 1.0017 27.34
570Ž609. q0.0002 y0.0010 q0.0022 0.9995 0.9917 1.0013 27.71
822Ž924. y0.0015 y0.0040 q0.0008 0.9982 0.9933 1.0018 26.83
582Ž618. y0.0002 y0.0011 q0.0008 0.9996 0.9986 1.0011 27.34
0.9996 0.9985 1.0017 28.37
sin u sin umin sin uma x 2 xmin
545Ž582. y0.0003 y0.0020 q0.0015 28.43
146Ž155. q0.0005 y0.0013 q0.0024 28.09
365Ž408. y0.0026 y0.0062 q0.0011 28.17
564Ž602. q0.0003 y0.0009 q0.0017 28.22
809Ž894. y0.0019 y0.0041 q0.0003 27.43
578Ž612. y0.0002 y0.0011 q0.0007 27.82
28.79
r 0 free
r0 s 1
surements from LEP and the SLC, which are close to being finalized; the W boson and top quark mass measurements, MW and m t , from the Tevatron run I, and further MW determinations from LEP 2; results from deep inelastic n-hadron scattering at CERN and Fermilab; n-electron scattering; and atomic parity violation. The low energy measurements in neutrino scattering and APV are very important in the presence of new physics, and in particular, for the Z X bosons discussed here. They offer complementary information about Z X exchange and interference effects, which are suppressed at the Z pole. The Z pole observables are summarized in Table 2 and the non-Z pole observables in Table 3. For more details Table 5 95 Ž90.% CL lower mass limits on specific Z X bosons as they appear in models of unification. Assumed are minimal Higgs structures and r 0 s1. Note that s is defined differently for the Zc and Zh models, and the Zx and ZL R models, respectively, as explained in the text. In particular, the versions of the Zx and ZL R models most often considered correspond to s s 0
s s0 s s1 s s5 s s`
Zx
Zc
Zh
ZL R
1368Ž1528. 643Ž688. 1210Ž1314. 1464Ž1601.
1181Ž1275. 146Ž156. 1393Ž1581. 1810Ž2039.
470Ž498. 1075Ž1235. 1701Ž1948. 1985Ž2277.
1673Ž1799. 925Ž987. 980Ž1076. 1537Ž1711.
and further references we refer to our recent reviews w19x. The theoretical evaluation uses the FORTRAN package GAPP w42x dedicated to the Global Analysis of Particle Properties. GAPP attempts to gather all available theoretical and experimental information from precision measurements in particle physics. It treats all relevant SM inputs and new physics parameters as global fit parameters. For clarity and to minimize CPU costs it avoids numerical integrations throughout. GAPP is based on the MS renormalization scheme which demonstrably avoids large expansion coefficients. In Tables 4 and 5 and Fig. 1 we present our main results. We list lower limits on Z X boson masses for a variety of cases. Note, that the new physics, i.e., the Z X s and the extra Higgs bosons, decouple and that the SM Ž MZ X s `, u s 0. is well within the allowed regions of Fig. 1. As a consequence a rigorous Bayesian integration over the MZ X probability distribution diverges 4 . Therefore, we approximate the 95 Ž90.% CL limits by requiring Dx 2 s
4 The Bayesian confidence integral in Eq. Ž4.13. of Ref. w43x is not well-defined unless a non-trivial Jacobian is implicitly included.
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Fig. 1. 90% CL contours for various Z X models. The solid contour lines use the constraint r 0 s 1 Žthe cross denotes the best fit location for the r 0 s 1 case., while the long-dashed lines are for arbitrary Higgs sectors. Also shown are the additional constraints in minimal Higgs scenarios for several VEV ratios as discussed in the text. The lower direct production limits from CDF w28x are also shown. They assume that all exotic decay channels are closed, and have to be relaxed by about 100 to 150 GeV when all exotic decays Žincluding channels involving superparticles. are kinematically allowed w28x.
3.84 Ž2.71., which is motivated by a reference univariate normal distribution. Similarly, we define the 90% allowed region in the Ž MZ X ,sin u .-plane by Dx 2 - 4.61, here referring to a bivariate Gaussian. The first set of fits in Table 4 is for the most general case of completely arbitrary Higgs sector, while the second set is for r 0 s 1. Below the mass limits we show the best fit values and 95% lower and upper limits on the mixing parameter sin u . Results on r 0 and the x 2 minimum are also shown. For comparison we have included the SUŽ2. = UŽ1. case in the last column. We note that the ZL R boson is equivalent to the Zx boson with a non-vanishing kinetic mixing term, sin x y 2
X B mn Zmn .
It can be absorbed by a shift of the gauge couplings proportional to the third component of the righthanded isospin generator and a rescaling of the
coupling ratio l. In models with given Higgs structure the parameter C ™ C y 53l g sin x is shifted, as well, while r 0 is unaffected. The limits on masses and mixings of the Zx and ZL R bosons, shown in Tables 4 and 5 Žwith l s 1., are indeed quite similar. Our mass limits on extra Z X bosons are somewhat stronger than those from a recent analysis w43x of Z X s in supersymmetric E6 models w44x. The differences are due to a slightly different and more recent data set in our analysis, different implementations of radiative corrections, different statistical methods 5,
(
5 The difference goes beyond the more common choices of Bayesian versus frequentist kind of approaches. The authors of Ref. w43x choose to allow three fit parameters, of which only two are independent. This is a problematic procedure when parameter estimation is desired and renders confidence intervals ambiguous.
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J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
Fig. 1 Žcontinued..
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
and our alternative evaluation of the photonic vacuum polarization effects w45x with advantages for global fits. Moreover, Ref. w43x assumes the SUSY inspired range for the Higgs mass, MH - 150 GeV w46x. Table 5 lists results for specific Higgs charge assignments as they occur in ‘‘minimal’’ models: For the Zc and Zh bosons, we assume an SUŽ2. Higgs singlet with a large VEV s to ensure MZ X 4 MZ , and in addition a pair of Higgs doublets with quantum numbers as in the 5 q 5 of SUŽ5. appearing in the 27 of E6 . They receive VEVs Õ and Õ, where the combination Õ 2 q Õ 2 is fixed by the measured value of the Fermi constant. Thus, these models are described by an extra parameter s s < ÕrÕ < 2 which is analogous to tan2b in supersymmetric extensions of the standard model. The Zx model does not depend on the ratio < ÕrÕ < so that C s 2r '10 is predicted. If we add another Higgs doublet with quantum numbers like the SM leptons and VEV x, we have the extra parameter s s < x < 2rŽ Õ 2 q Õ 2 ., and then C g 1r '10 wy3,2x. However, in those models of SUSY in which this Higgs doublet is identified with the superpartner of a SM lepton doublet, one has to require x s s s 0 to avoid severe problems with charged-current universality. The Higgs content of the Zx model can be lifted to an appropriate Higgs structure for a ZL R model ŽLR 1., transforming under SUŽ2.L = SUŽ2.R = UŽ1.By L as Ž2,2,0. q Ž2,1,1r2. q Ž1,2,1r2.. The same definitions and remarks apply, except that here C g 3r5 wy1ra , a x. Another possibility ŽLR 2. is a Higgs sector transforming as Ž2,2,0. q Ž3,1,1. q Ž1,3,1. which results in C s 3r5 a . There are no SUŽ2. triplets in a 27of E6 , but one might find them in string models without an intermediate GUT group, if they are realized at higher Kac-Moody levels Ž k ) 1.. In the context of SUSY the Higgs fields carrying B y L charge have to be supplemented by extra fields with opposite charge to cancel anomalies and Žin the case of LR 1. to generate fermion masses. There is no analog of a minimal Higgs sector for the sequential ZSM . In the Zstring model r 0 s 1 is predicted, and even the Z X mass and the Z–Z X mixing can be calculated in principle. Universal high scale boundary conditions yield too large a value for
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u and are excluded w13x. Non-universal boundary conditions on the soft SUSY breaking terms can yield acceptable mixing. In any case, the concrete realization of the soft terms depends strongly on the SUSY breaking and mediation mechanisms. We parametrize our lack of understanding by allowing an arbitrary Higgs sector Žexcept for r 0 s 1.. In conclusion, indirect constraints from high precision observables on and off the Z pole continue to play an important role for searches for new physics, and in particular for extra gauge bosons. The obtained mass limits are competitive with current direct searches at colliders with the highest attainable energies. Moreover, no assumptions about the absence of exotic decay channels are necessary 6 . The indirect constraints are even much stronger in specific models with known Higgs structure, where lower limits are typically in the 1 to 2 TeV range. Finally, Z–Z X mixing effects are severely constrained to the sub per cent level in all cases.
Acknowledgements This work was supported in part by US Department of Energy Grant EY–76–02–3071.
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6 However, we ignored contributions from possible exotic states to the S parameter. The potentially much larger contributions to T are accounted for in our fits with r 0 allowed.
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Physics Letters B 456 Ž1999. 68–76
Constraints on extended neutral gauge structures Jens Erler 1, Paul Langacker
2
Department of Physics and Astronomy, UniÕersity of PennsylÕania, Philadelphia, PA 19104-6396, USA Received 24 March 1999 Editor: M. Cveticˇ
Abstract Indirect precision data are used to constrain the masses of possible extra Z X bosons and their mixings with the ordinary Z. We study a variety of Z X bosons as they appear in E6 and left-right unification models, the sequential Z boson, and the example of an additional UŽ1. in a concrete model from heterotic string theory. In all cases the mixings are severely constrained Žsin u - 0.01.. The lower mass limits are generally of the order of several hundred GeV and competitive with collider bounds. The exception is the Zc boson, whose vector couplings vanish and whose limits are weaker. The results change little when the r parameter is allowed, which corresponds to a completely arbitrary Higgs sector. On the other hand, in specific models with minimal Higgs structures the limits are generally pushed into the TeV region. q 1999 Elsevier Science B.V. All rights reserved.
The possibility of additional neutral gauge bosons, Z X s, is among the best motivated types of physics beyond the Standard Model ŽSM.. They are predicted by most unifying theories, such as Grand Unified Theories ŽGUTs., left-right unification, superstring theories and their strong coupling generalizations. In many cases their masses remain unpredicted and may or may not be of the electroweak scale. In the context of superstring models, however, which are much more constrained than purely field theoretical models, they are often predicted to arise at the electroweak scale, as we will discuss below. In this paper we consider six different types of Z X bosons: 1. The Zx boson is defined by SO Ž10. ™ SUŽ5. = UŽ1.x . This boson is also the unique solution to
the conditions of Ži. family universality, Žii. no extra matter other than the right-handed neutrino, Žiii. absence of gauge and mixed gaugergravitational anomalies, and Živ. orthogonality to the hypercharge generator. In the context of a minimal SO Ž10. GUT, conditions Ži. and Žii. are satisfied by assumption, while Žiii. and Živ. are automatic. Relaxing condition Živ. allows other solutions Žincluding the ZL R below. which differ from the Zx by a shift proportional to the third component of the right-handed isospin generator. 2. The Zc boson is defined by E6 ™ SO Ž10. = UŽ1.c . It possesses only axial-vector couplings to the ordinary fermions. As a consequence it is the least constrained of our examples. 3. The Zh boson is the linear combination 3r8 Zx y 5r8 Zc . It occurs in Calabi-Yau compactifications w1x of the heterotic string w2x if E6 breaks directly to a rank 5 subgroup w3x via the Hosotani mechanism w4x.
'
'
1 2
E-mail: [email protected] E-mail: [email protected]
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 5 7 - 8
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
4. The ZL R boson occurs in left-right models with gauge group SU Ž3. C = SU Ž2. L = SU Ž2. R = UŽ1.By L ; SO Ž10. and is defined through the current JL R s 3r5 w a J3 R y 1rŽ2 a . JByL x. J3 R couples to the third component of SUŽ2.R , B and L coincide with baryon and lepton number for the ordinary fermions, a s g R2 rg L2 cot 2u W y 1 , where g L, R are the SUŽ2.L, R gauge couplings, and u W is the weak mixing angle. 5. The sequential ZSM boson is defined to have the same couplings to fermions as the SM Z boson. Such a boson is not expected in the context of gauge theories unless it has different couplings to exotic fermions than the ordinary Z. However, it serves as a useful reference case when comparing constraints from various sources. It could also play the role of an excited state of the ordinary Z in models with extra dimensions at the weak scale. 6. Finally we consider a superstring motivated Zstring boson appearing in a specific model w5x ŽTable 1. based on the free fermionic string construction with real fermions. This model has been investigated in considerable detail w6x with the goal of understanding some of the characteristics of Žweakly coupled. string theories, and of contrasting them with the more conventional ideas such as GUTs. While this specific model itself is not realistic Žfor example it fails to produce an acceptable fermion mass spectrum. the predicted Zstring it contains is not ruled out. Its coupling strength is predicted and so are its fermion cou-
'
(
69
plings. It is particularly interesting in that its couplings are family non-universal. While this may induce problems with too large flavor changing neutral currents through a violation of the GIM mechanism w7x, we will not address this issue here. Another important observation is that such a Zstring can be naturally at the electroweak scale w8x. The basic reasons are strong restrictions on the superpotential, and that in a given model the sectors of supersymmetry ŽSUSY. breaking, and its mediation are usually not arbitrary. In string models one also expects bilinear terms in the Higgs superpotential to vanish at tree level Žotherwise they should be of the order of the Planck scale. and to be generated in the process of radiative symmetry breaking. This is linked to the top quark Yukawa coupling driven symmetry breaking and typically involves extra Higgs singlets which are predicted in many string models. In all cases, there is a relation between the mixing angle u between the ordinary Z and the extra Z X from the diagonalization of the neutral vector boson mass matrix, M02 y MZ2 1 y r 0rr 1 tan2u s 2 s , Ž 1. 2 X r 0rr 2 y 1 MZ y M 0 where MZ and MZ X are the physical boson masses, and M0 is the mass of the ordinary Z in the absence of mixing. The second equality in relation Ž1. uses the neutral and charged boson mass interdependence which reads at tree level, MW Ma s . Ž 2. ra cos u W
(
Table 1 Multiplet t
Ž b. L tR bR Ž ud . L , Ž cs . L u R , cR d R , sR
Ž ntt . L tR
Ž nm . L m
mR
Ž nee . L eR
100 Q
X
y71 q133 y136 q68 y6 q3 q74 y130 y65 q9 y204 q9
As in the case of the SUŽ3. C = SUŽ2.L = UŽ1. Y model, Ý Ž t i2 y t 32i q t i . <² fi :< 2
r0 s
i
Ý 2 t 32i <² fi :< 2
,
Ž 3.
i
where t i Ž t 3 i . is Žthe third component of. the weak isospin of the Higgs field f i . r 0 s 1 if only SUŽ2. Higgs doublets and singlets are present, in which case M0 would be known independently. Nondegenerate SUŽ2. multiplets of extra fermions and scalars affect the W and Z self-energies at the loop level, and therefore contribute to the T parameter w9x. They can arise, for example, in E6 models, and in their
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Table 2 Z pole precision observables from LEP w14,15x and the SLC w16–18x. Shown are the experimental results, the SM predictions, and the pulls. The SM errors are from the uncertainties in MZ , ln MH , m t , a Ž MZ ., and a s . They have been treated as Gaussian and their correlations have been taken into account. The first set of measurements is from the Z line shape and leptonic forward-backward asymmetries, A F B Ž l . s 3r4 A e A l . The hadronic, invisible, and leptonic decay widths are not independent of the total width, the hadronic peak cross section, and the R l s G Žhad.rG Ž l q l y ., and are shown for illustration only. The second set represents the quark sector, where R q s G Ž qq .rG Žhad., and A q s 4r3 A FL RB Ž q . is a function of the effective weak mixing angle of quark q. The third set is a variety of polarization and forward-backward asymmetries sensitive to the leptonic weak mixing angle, where analogous definitions apply. For details see Ref. w19x Quantity
GroupŽs.
Value
Standard Model
Pull
MZ wGeVx GZ wGeVx G Žhad. wGeVx G Žinv. wMeVx G Ž l q l y . wMeVx s had wnbx Re Rm Rt A F B Ž e. AF B Ž m. A F B Žt .
LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP LEP
91.1867 " 0.0021 2.4939 " 0.0024 1.7423 " 0.0023 500.1 " 1.9 83.90 " 0.10 41.491 " 0.058 20.783 " 0.052 20.789 " 0.034 20.764 " 0.045 0.0153 " 0.0025 0.0164 " 0.0013 0.0183 " 0.0017
91.1865 " 0.0021 2.4957 " 0.0017 1.7424 " 0.0016 501.6 " 0.2 83.98 " 0.03 41.473 " 0.015 20.748 " 0.019 20.749 " 0.019 20.794 " 0.019 0.0161 " 0.0003
0.1 y0.8 – – – 0.3 0.7 1.2 y0.7 y0.3 0.2 1.3
Rb Rc R s, drR Ž dquqs. A F B Ž b. A F B Ž c. A F B Ž s. Ab Ac As
LEP q SLD LEP q SLD OPAL LEP LEP DELPHI q OPAL SLD SLD SLD
0.21656 " 0.00074 0.1735 " 0.0044 0.371 " 0.023 0.0990 " 0.0021 0.0709 " 0.0044 0.101 " 0.015 0.867 " 0.035 0.647 " 0.040 0.82 " 0.12
0.2158 " 0.0002 0.1723 " 0.0001 0.3592 " 0.0001 0.1028 " 0.0010 0.0734 " 0.0008 0.1029 " 0.0010 0.9347 " 0.0001 0.6676 " 0.0006 0.9356 " 0.0001
1.0 0.3 0.5 y1.8 y0.6 y0.1 y1.9 y0.5 y1.0
A L R Žhadrons. A L R Žleptons. Am At A eŽ QL R . At Ž Pt . A e Ž Pt . s 2l Ž Q F B .
SLD SLD SLD SLD SLD LEP LEP LEP
0.1510 " 0.0025 0.1504 " 0.0072 0.120 " 0.019 0.142 " 0.019 0.162 " 0.043 0.1431 " 0.0045 0.1479 " 0.0051 0.2321 " 0.0010
0.1466 " 0.0015
1.8 0.5 y1.4 y0.2 0.4 y0.8 0.3 0.5
presence r 0 should be replaced by r 0rŽ1 y a Ž M Z . T .. If the Higgs UŽ1.X quantum numbers are known, as well, there will be an extra constraint, g 2 MZ2 usC , Ž 4. g 1 MZ2X where g 1,2 are the UŽ1. and UŽ1.X gauge couplings with 5 g2 s sin u W 'l g 1 . Ž 5. 3
(
0.2316 " 0.0002
l s 1 if the GUT group breaks directly to SUŽ3. = SUŽ2. = UŽ1. = UŽ1.X , while in general l is still of O Ž1.. We will quote our results assuming l s 1, but our limits also apply to 'l sin u and MZ Xr 'l for other values of l Žwe always assume MZ X 4 MZ .. Similar to r 0 ,
Ý t 3 i Qi <² fi :< 2 X
Csy
i
Ý t 32i <² fi :< 2 i
Ž 6.
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Table 3 Non-Z pole precision observables from Fermilab w29–33x, CERN w14,34x, and elsewhere. The second error after the experimental value, where given, is theoretical. The SM errors are from the inputs as in Table 2. The various quantities R are cross section ratios from n-hadron scattering, where the CHARM w35x results have been adjusted to CDHS w36x conditions, and can be directly compared. The g V , A are effective four-Fermi couplings from n-e scattering w37x, and QW denotes the weak charge appearing in APV w38–41x Quantity
GroupŽs.
m t wGeVx MW wGeVx MW wGeVx
Value
Tevatron Tevatron q UA2 LEP
173.8 " 5.0 80.404 " 0.087 80.37 " 0.09
171.4 " 4.8 80.362 " 0.023
Ry Rn Rn Rn Rn Rn Rn
NuTeV CCFR CDHS CHARM CDHS CHARM CDHS 1979
0.2277 " 0.0021 " 0.0007 0.5820 " 0.0027 " 0.0031 0.3096 " 0.0033 " 0.0028 0.3021 " 0.0031 " 0.0026 0.384 " 0.016 " 0.007 0.403 " 0.014 " 0.007 0.365 " 0.015 " 0.007
0.2297 " 0.0003 0.5827 " 0.0005 0.3089 " 0.0003
g Vn e g Vn e g nA e g nA e
CHARM II all CHARM II all
y0.035 " 0.017 y0.041 " 0.015 y0.503 " 0.017 y0.507 " 0.014
QW ŽCs. QW ŽTl.
Boulder Oxford q Seattle
y72.41 " 0.25 " 0.80 y114.8 " 1.2 " 3.4
is another function of vacuum expectation values ŽVEVs., where QXi are the UŽ1.X charges. For minimal cases, the functions C are given explicitly in Table III of Ref. w10x. Similarly, for the fermion couplings of the various Z X s we refer to Table II of the same work. There is the possibility of an extra gauge invariant term, mixing the field strength tensors of the hypercharge and the new gauge bosons. We do not consider such a term here, since it is expected to be small in typical models 3. The phenomenology of gauge kinetic mixing has been reviewed in Ref. w21x; constraints on leptophobic Z X bosons can be found in Ref. w22x; Ref. w23x are detailed reviews on extra neutral gauge bosons at colliders; and for related topics in neutral current physics see Ref. w24x.
3 It was shown in Ref. w12x that a relatively large kinetic mixing term can be generated at the loop level when Higgs doublets from a 78 representation of E6 are employed. However, restriction to Higgs doublets from 27 and 27 representations yields much smaller effects w13x.
Standard Model
0.3859 " 0.0003 0.3813 " 0.0003 y0.0395 " 0.0004 y0.5063 " 0.0002 y73.10 " 0.04 y116.7 " 0.1
Pull 0.5 0.5 0.1 y0.9 y0.2 0.2 y1.7 y0.1 1.1 y1.0 – y0.1 – y0.1 0.8 0.5
After the Z X properties have been specified, the new contributions to the precision observables can be computed using the formalism presented in Refs. w10,25x. The new Z X boson is treated as a small perturbation to the SM relations. The most important effect is the modification of the MZ –MW –sin2u W interdependence via Z–Z X mixing. Z pole observables are affected by the modification of the Z couplings to fermions, which is another manifestation of the Z X admixture. This change in couplings is also relevant for the low energy observables from neutrino scattering and atomic parity violation ŽAPV.. For these there will be additional effects from Z X exchange, and Z–Z X Žg –Z X . interference. Such effects can be neglected at the Z pole, but the interference terms are relevant for the cross section measurements at LEP 2. They give interesting constraints on MZ X w27x practically independent of the mixing parameter u . These constraints are complementary to the direct search limits at Fermilab w28x, where additional assumptions about possible exotic decay channels have to be specified. In our analysis we use the data as of ICHEP 98 at Vancouver. It includes the very precise Z pole mea-
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Table 4 Mass limits win GeVx on extra Z X bosons and constraints on Z–Z X mixing for two classes of Higgs sectors. The upper part of the table allows r 0 as a free fit parameter and corresponds to a completely arbitrary Higgs sector. The lower part assumes r 0 s 1, but is arbitrary otherwise. The first Žsecond. numbers correspond to the 95 Ž90.% CL lower mass limits. Below this we show the central values and the 95% lower and upper limits on sin u . Also shown are the central values and 95% limits for r 0 as a fit parameter. Finally we indicate the minimal x 2 for each model. The last column is included for comparison with the standard case of only one Z boson. All results assume MZ F MH F 1 TeV Zx
Zc
Zh
ZL R
ZSM
Zstring
SM
sin u sin umin sin uma x r0 r 0min r 0max 2 xmin
551Ž591. y0.0006 y0.0022 q0.0020 0.9993 0.9931 1.0010 27.62
151Ž162. q0.0004 y0.0015 q0.0021 0.9974 0.9923 1.0017 27.52
379Ž433. y0.0010 y0.0058 q0.0019 0.9979 0.9931 1.0017 27.34
570Ž609. q0.0002 y0.0010 q0.0022 0.9995 0.9917 1.0013 27.71
822Ž924. y0.0015 y0.0040 q0.0008 0.9982 0.9933 1.0018 26.83
582Ž618. y0.0002 y0.0011 q0.0008 0.9996 0.9986 1.0011 27.34
0.9996 0.9985 1.0017 28.37
sin u sin umin sin uma x 2 xmin
545Ž582. y0.0003 y0.0020 q0.0015 28.43
146Ž155. q0.0005 y0.0013 q0.0024 28.09
365Ž408. y0.0026 y0.0062 q0.0011 28.17
564Ž602. q0.0003 y0.0009 q0.0017 28.22
809Ž894. y0.0019 y0.0041 q0.0003 27.43
578Ž612. y0.0002 y0.0011 q0.0007 27.82
28.79
r 0 free
r0 s 1
surements from LEP and the SLC, which are close to being finalized; the W boson and top quark mass measurements, MW and m t , from the Tevatron run I, and further MW determinations from LEP 2; results from deep inelastic n-hadron scattering at CERN and Fermilab; n-electron scattering; and atomic parity violation. The low energy measurements in neutrino scattering and APV are very important in the presence of new physics, and in particular, for the Z X bosons discussed here. They offer complementary information about Z X exchange and interference effects, which are suppressed at the Z pole. The Z pole observables are summarized in Table 2 and the non-Z pole observables in Table 3. For more details Table 5 95 Ž90.% CL lower mass limits on specific Z X bosons as they appear in models of unification. Assumed are minimal Higgs structures and r 0 s1. Note that s is defined differently for the Zc and Zh models, and the Zx and ZL R models, respectively, as explained in the text. In particular, the versions of the Zx and ZL R models most often considered correspond to s s 0
s s0 s s1 s s5 s s`
Zx
Zc
Zh
ZL R
1368Ž1528. 643Ž688. 1210Ž1314. 1464Ž1601.
1181Ž1275. 146Ž156. 1393Ž1581. 1810Ž2039.
470Ž498. 1075Ž1235. 1701Ž1948. 1985Ž2277.
1673Ž1799. 925Ž987. 980Ž1076. 1537Ž1711.
and further references we refer to our recent reviews w19x. The theoretical evaluation uses the FORTRAN package GAPP w42x dedicated to the Global Analysis of Particle Properties. GAPP attempts to gather all available theoretical and experimental information from precision measurements in particle physics. It treats all relevant SM inputs and new physics parameters as global fit parameters. For clarity and to minimize CPU costs it avoids numerical integrations throughout. GAPP is based on the MS renormalization scheme which demonstrably avoids large expansion coefficients. In Tables 4 and 5 and Fig. 1 we present our main results. We list lower limits on Z X boson masses for a variety of cases. Note, that the new physics, i.e., the Z X s and the extra Higgs bosons, decouple and that the SM Ž MZ X s `, u s 0. is well within the allowed regions of Fig. 1. As a consequence a rigorous Bayesian integration over the MZ X probability distribution diverges 4 . Therefore, we approximate the 95 Ž90.% CL limits by requiring Dx 2 s
4 The Bayesian confidence integral in Eq. Ž4.13. of Ref. w43x is not well-defined unless a non-trivial Jacobian is implicitly included.
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
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Fig. 1. 90% CL contours for various Z X models. The solid contour lines use the constraint r 0 s 1 Žthe cross denotes the best fit location for the r 0 s 1 case., while the long-dashed lines are for arbitrary Higgs sectors. Also shown are the additional constraints in minimal Higgs scenarios for several VEV ratios as discussed in the text. The lower direct production limits from CDF w28x are also shown. They assume that all exotic decay channels are closed, and have to be relaxed by about 100 to 150 GeV when all exotic decays Žincluding channels involving superparticles. are kinematically allowed w28x.
3.84 Ž2.71., which is motivated by a reference univariate normal distribution. Similarly, we define the 90% allowed region in the Ž MZ X ,sin u .-plane by Dx 2 - 4.61, here referring to a bivariate Gaussian. The first set of fits in Table 4 is for the most general case of completely arbitrary Higgs sector, while the second set is for r 0 s 1. Below the mass limits we show the best fit values and 95% lower and upper limits on the mixing parameter sin u . Results on r 0 and the x 2 minimum are also shown. For comparison we have included the SUŽ2. = UŽ1. case in the last column. We note that the ZL R boson is equivalent to the Zx boson with a non-vanishing kinetic mixing term, sin x y 2
X B mn Zmn .
It can be absorbed by a shift of the gauge couplings proportional to the third component of the righthanded isospin generator and a rescaling of the
coupling ratio l. In models with given Higgs structure the parameter C ™ C y 53l g sin x is shifted, as well, while r 0 is unaffected. The limits on masses and mixings of the Zx and ZL R bosons, shown in Tables 4 and 5 Žwith l s 1., are indeed quite similar. Our mass limits on extra Z X bosons are somewhat stronger than those from a recent analysis w43x of Z X s in supersymmetric E6 models w44x. The differences are due to a slightly different and more recent data set in our analysis, different implementations of radiative corrections, different statistical methods 5,
(
5 The difference goes beyond the more common choices of Bayesian versus frequentist kind of approaches. The authors of Ref. w43x choose to allow three fit parameters, of which only two are independent. This is a problematic procedure when parameter estimation is desired and renders confidence intervals ambiguous.
74
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
Fig. 1 Žcontinued..
J. Erler, P. Langackerr Physics Letters B 456 (1999) 68–76
and our alternative evaluation of the photonic vacuum polarization effects w45x with advantages for global fits. Moreover, Ref. w43x assumes the SUSY inspired range for the Higgs mass, MH - 150 GeV w46x. Table 5 lists results for specific Higgs charge assignments as they occur in ‘‘minimal’’ models: For the Zc and Zh bosons, we assume an SUŽ2. Higgs singlet with a large VEV s to ensure MZ X 4 MZ , and in addition a pair of Higgs doublets with quantum numbers as in the 5 q 5 of SUŽ5. appearing in the 27 of E6 . They receive VEVs Õ and Õ, where the combination Õ 2 q Õ 2 is fixed by the measured value of the Fermi constant. Thus, these models are described by an extra parameter s s < ÕrÕ < 2 which is analogous to tan2b in supersymmetric extensions of the standard model. The Zx model does not depend on the ratio < ÕrÕ < so that C s 2r '10 is predicted. If we add another Higgs doublet with quantum numbers like the SM leptons and VEV x, we have the extra parameter s s < x < 2rŽ Õ 2 q Õ 2 ., and then C g 1r '10 wy3,2x. However, in those models of SUSY in which this Higgs doublet is identified with the superpartner of a SM lepton doublet, one has to require x s s s 0 to avoid severe problems with charged-current universality. The Higgs content of the Zx model can be lifted to an appropriate Higgs structure for a ZL R model ŽLR 1., transforming under SUŽ2.L = SUŽ2.R = UŽ1.By L as Ž2,2,0. q Ž2,1,1r2. q Ž1,2,1r2.. The same definitions and remarks apply, except that here C g 3r5 wy1ra , a x. Another possibility ŽLR 2. is a Higgs sector transforming as Ž2,2,0. q Ž3,1,1. q Ž1,3,1. which results in C s 3r5 a . There are no SUŽ2. triplets in a 27of E6 , but one might find them in string models without an intermediate GUT group, if they are realized at higher Kac-Moody levels Ž k ) 1.. In the context of SUSY the Higgs fields carrying B y L charge have to be supplemented by extra fields with opposite charge to cancel anomalies and Žin the case of LR 1. to generate fermion masses. There is no analog of a minimal Higgs sector for the sequential ZSM . In the Zstring model r 0 s 1 is predicted, and even the Z X mass and the Z–Z X mixing can be calculated in principle. Universal high scale boundary conditions yield too large a value for
'
'
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u and are excluded w13x. Non-universal boundary conditions on the soft SUSY breaking terms can yield acceptable mixing. In any case, the concrete realization of the soft terms depends strongly on the SUSY breaking and mediation mechanisms. We parametrize our lack of understanding by allowing an arbitrary Higgs sector Žexcept for r 0 s 1.. In conclusion, indirect constraints from high precision observables on and off the Z pole continue to play an important role for searches for new physics, and in particular for extra gauge bosons. The obtained mass limits are competitive with current direct searches at colliders with the highest attainable energies. Moreover, no assumptions about the absence of exotic decay channels are necessary 6 . The indirect constraints are even much stronger in specific models with known Higgs structure, where lower limits are typically in the 1 to 2 TeV range. Finally, Z–Z X mixing effects are severely constrained to the sub per cent level in all cases.
Acknowledgements This work was supported in part by US Department of Energy Grant EY–76–02–3071.
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