Constraints on physics in and beyond the Minimal Standard Model from LEP experiments

16 Pages·2005·1.63 MB·English

Z. Phys. C - Particles and Fields ,65 355-370 (1992)
Partides
ritZefitschrift kisyhP C
dna F tds
(cid:14)9 Springer-u 1992
Constraints on physics
in and beyond the Minimal Standard Model
from LEP experiments
Peter Renton
Particle and Nuclear Physics Laboratory, University of Oxford, Oxford OX1 3RH, UK
Received 8 January 1992
Abstract. The results of fits to the LEP data, taken up 2, L 3 3 and OPAL ,4- have led to spectacular con-
to the end of 1990, on cross-sections for hadron and firmation of the predictions of the Minimal Standard
lepton final states and lepton forward-backward charge Model (MSM). However, the implications of these results
asymmetries are presented. The fits are first of all per- on the possible restrictions to models beyond the MSM
formed to extract the parameters of the improved Born have been less extensively investigated.
approximation. Within the context of the Minimal Stan- These fall into essentially two categories. The first,
dard Model (MSM), the constraints on ,s~ mt and mn and one of the most important questions in particle
from these LEP data, and with the inclusion of other physics, is to ascertain the precise nature of the custodial
LEP and electroweak data, are examined. The results S U )2( symmetry-breaking which gives rise to the particle
favour a top-quark mass in the region of about 041 GeV, masses. In the MSM, this is achieved by the introduction
and a relatively light Higgs mass. However, the upper of a complex doublet of scalar fields with weak hyper-
limit on m~ from such fits is sensitive to the value of charge Y= 1/2. After spontaneous symmetry-breaking
the strong coupling constant ~s(Mz)~s. Some differences this gives rise to a single neutral scalar particle H ~ the
between the data and the MSM are observed, although Higgs boson. Negative searches at LEP have established
these are not yet at a level of significance to indicate a lower limit in excess of 05 GeV on the mass of H ~
physics beyond the MSM. Lower limits on the mass In the Minimal Supersymmetric extension to the
of a possible second Z ~ are examined. Results are then Standard Model (MSSM), the searches at LEP 001 have
presented in terms of the variables ,S T and .U These excluded the lightest CP-even scalar state h ~ almost up
variables provide a general framework for the descrip- to the present kinematical limit of -~45 GeV. However,
tion of electroweak radiative effects, and are thus suitable such limits have been established using tree-level formu-
for the investigation of physics beyond the MSM. The lae, and in particular assuming that h ~ is lighter than
variables are defined such that S = T= U = 0 corresponds the CP-odd pseudoscalar A ~ Recently, a re-evaluation
to the MSM prediction for mt=140GeV and t~m of radiative corrections in the light of a potentially heavy
= 300 GeV. A fit to the LEP data on cross-sections for top-quark 5- has shown that the masses of the Higgs
hadron and lepton final states and lepton forward-back- particles can receive considerable upward shifts with re-
ward charge asymmetries gives S=-0.44+_0.91 and spect to those extracted using the tree-level formulae.
T=-0.15+0.52. With the addition to the fit of other The hierarchy of masses, and consequently the decay
LEP and electroweak data, the values obtained are schemes, can also be changed. When these effects are
S= -1.03_+0.66, T= -0.46_+0.41 and U= -0.13+0.94. taken into account, the limits established using the tree-
These results are only just compatible with the assumed level formulae are somewhat weakened, in a way which
central MSM values S=T=U=O. Some constraints depends on the values of both the top-quark and squark
from these results on models beyond the MSM are dis- masses. Consequently, it may well prove difficult to rule
cussed. out this minimal SUSY model at LEP 200.
In the MSM approach, the new (i.e. Higgs) sector
is weakly interacting. Fundamental scalar fields are in-
troduced and the effects are calculable perturbatively.
1 Introduction An alternative approach is that symmetry-breaking ar-
ises from strongly interacting effects in the new sector.
The very precise results which have been obtained from
Although technicolour and extended technicolour mod-
the cross-section and asymmetry measurements from the
els (see, e.g. 6) have considerable theoretical difficulties,
four experiments at LEP, namely ALEPH 1, DELPHI
it is nevertheless of considerable interest to test the pre-
e-mail RENTON @ UK.AC.OX.PH.V1 dictions of these models wherever possible.
653
The second category is the possible existence of new st?
fundamental particles (quarks, leptons, gauge-bosons, s2Fz 2 , (2)
1 3
SUSY particles etc.). The effects of heavy particles with +
weak isospin can possibly be seen through radiative cor-
rections, since the effects of such particles do not decou- where
plc from low energy interactions at a scale q2< .2zM The
observation of such effects requires, however, precise and 12 ~ yF~F
reliable data. M r? )3(
In this note an analysis is described of fits made to
the LEP data on the cross-section line shapes and lepton is the pole cross-section, defined in terms of the Z ~ mass
forward-backward charge-asymmetries. In Sect. 2, the Mz, total width Fz and the partial widths eF and rF for
results of fits to the parameters of the improved Born Z~ *-- e + e- and Z~ ~ f f (f + e) respectively. The term
approximation are presented. This is followed by a dis- 1( + 3 c~Q}/4 )er is a QED correction. For the cross-section
cussion of the constraints on %c ~m and m~, within the to all possible hadronic (i.e. quark) final states, the pole
context of the MSM, using the above and additional cross-section is
LEP data on z-polarisation and quark forward-back-
12 ~ ~F hF
ward~ asymmetries, as well as other electroweak data.*
ao= MZ Fz z , (4)
In Sect. 2.6 the limits on a possible second Z ~ are exam-
ined. The results of the improved Born approximation
where hF is the hadronic width of the Z ~ In addition
fits serve as a starting point for the use of a more general
to the Z ~ term, there are direct photon and ~-Z ~ inter-
framework in which electroweak radiative effects can be
ference terms. These are relatively small, and are taken
formulated. In Sect. 3, the variables S, T and U (or Sw)
into account in the analysis below; for the interference
used in this approach are discussed. In Sect. 4 the results
term the MSM is used, except where specifically indicat-
of fits to these variables are presented, and the implica-
ed to the contrary. It should be noted that the contribu-
tions of these fits are discussed in Sect. .5 A detailed
tion of weak boxes can violate the factorisation property
review of results prior to the LEP experiments can be
between the initial and final states implicit in the above
found, for example, in 7 and 8.
formalism; however, these can be neglected at the current
level of precision.
2 Fits using the improved Born approximation The fermionic partial width and forward-backward
charge asymmetry at the Z ~ pole can be parameterised
2.1 Theoretical formalism in the improved Born approximation in terms of effective
vector and axial vector couplings yV( and Af) of the
Z ~ to fermions by 11:
It has been shown that, to a very good approximation,
the results of the full electroweak calculations of the
Gu Mz 3 ~N ~p( + I()}A + I()D~QO + ,)DCQ~
cross-sections and leptonic forward-backward asymme- (5)
tries can be reproduced by the parameterisation implicit
in the improved Born approximation (e.g. 9-11). ** Ini-
3 2V~Ae 2VFAr A)es '
tial state purely QED corrections can be described by (6)
"YFA )2zM( -- 4 e'~ 2 +.4~ ~V + A} -~
a radiator function ,~H such that the observed cross-
section for e § e- --,ff can be written,
where G u is the Fermi coupling constant measured in
muon decay. cN is the number of colour degrees of free-
o-~ = I S S(wG f t )'Hi(s, t S)"~Fint ds', (1) dom, and has the value 1 for leptons and 3 for quarks.
o The QCD correction DCQ6 is zero for leptons, and for
quarks is given by 12
where s' is the square of the invariant mass of the final
state fermions and Fire describes the interference of initial 6oc D = ~/~7c + 1.409 )~/s~C( 2 - 12.805 )~J~c( ,3 )7(
and final state radiation, as well as QED box-diagrams.
The term wCa is the modified Born cross-section for where cq(Mz)~s(=~s) is the strong coupling constant
e + e- ~ff, which can be expressed in terms of the line- in the MS renormalisation scheme. The term 1( + )DEQ6
shape parameters in an (almost) model independent way, is a QED correction factor, with DEQ6 = 3 e Q}/4 .er Note
as that this factor appears in (2) to avoid double counting
this correction in (1). In the MSM, propagator diagrams
lead to an increase of the partial width yF with increasing
* Only final data are used in the fits; preliminary conference data mt (approximately quadratically) and to a decrease with
are not included
m~ (approximately logarithmically). However, for f= b,
** In fact, in the way in which ZFITTER 11 si used in this
there are also large vertex correction terms, from loop
paper there si no approximation at all made in this formalism,
diagrams involving b, t and W particles, which lead to
with respect to the full electroweak calculation. However, the Z ~ --
interference term and the small corrections to the forward-back- an overall decrease of bF with increasing .rm The term
ward asymmetry in )6( are computed assuming the MSM, so that r~A se is the residual contribution to the asymmetry from
these small terms are not strictly model independent photon exchange, the imaginary part of the propagator
753
Table .1 Comparison of the results from the global fit to the data the second box, quantities which can be derived directly from this
directly to the results of a weighted average of the parameters fit are given. In these, R=Fh/Fe and F~,~=Fz-Fh-3F .e ~N si the
quoted by the individual experiments. eF refers to the electron or number of light neutrino species and si extracted from the above
muon partial widths; that for the z lepton is about 0.2 MeV smaller. quantities, together with the ratio F~/Fe=l.993_+0.001 from the
In the first box the results of the five-parameter fit are given. In MSM
Parameter Global fit Weighted mean MSM
Mz 91.175 _+0.021 GeV 91.175 _+0.021GeV
zF 2.486 _+0.010GeV 2.487 _+0.010GeV 2.484 _+0.010 GeV
% 41.24 _+0.22 nb 41.34 _+0.23 nb 84.14 +_ 0.04 nb
eV 2 0.00103 +_ 0.00039 0.00118 +_ 0.00040 0.0011 +_ 0.0002
e~if 2 0.2491 3100.0+_ 0.2493 3100.0+_ 0.2506 +_ 0.0009
reeP 0.9964 2500.0___ 0.9972 +0.0052 1.002 +0.003
sin fle02 0.2339 +0.0030 0.2328 +0.0029 0.2333 +0.0013
hF 1.738 _+0.010GeV 1.739 __0.010GeV 1.733 +0.008 GeV
F e 83.10 +0.40 MeV 83.24 _+0.41 MeV 83.6 -I-0.3 MeV
R 20.91 _+0.12 20.91 _+0.12 37.02 _ 0.05
,,n~F 499 +_ 8 MeV 497 +_ 8 MeV 1.005 _ 8.1 MeV
F/,,nIF e 6.012 +0.093 5.970 _+0.094 5.979 +0.004
~N 10.3 +_ 50.0 00.3 +_ 50.0 0.3
and weak boxes. Allowing for variation of m t and mn experiments* to extract these parameters. The fits were
within the ranges <__98 <__tnr 250 GeV and 50_-< mn made using the ZFITTER program 1- lJ *% and the re-
____ 1000 GeV, this term has a value of 0.0018 for leptonic sults of the fits are given in Table 1. The systematic errors
final states, 0.0010 for u and c-quark final states, and quoted by the experiments were taken into account in
0.0004 for d and s-quark final states, where the range the construction of the Z 2 function. These can be appli-
of values is +_0.0001. For b-quarks this term decreases cable to either all measurements of a particular data
from 0.0004 at small ~m to 0.0001 at mt~-250 GeV. set (or sets) of a given experiment, or, in the case of
It can be seen from the above equations that the the theoretical uncertainty on the luminosity, to all cross-
cross-sections and lepton asymmetries can be described, section data. The common error of +0.3% on the lumi-
assuming lepton universality, by the five parameters Mz, nosity is included directly in all the fits in this paper.
Fz, Oo, ~2 and A~. Alternative parameters to ~ and The errors given in Table 1 are the combined statistical
Ae are the variables Peff and the effective weak mixing and systematic errors (added in quadrature). The error
angle sin 2 0elf (see e.g. 11, 13), defined as on Mz includes the estimated +_ 0.02 GeV error on the
overall LEP energy scale, which is significantly larger
x,= )8(
than the +_ 0.005 GeV statistical error from the fit. The
error on Fz includes a +_ 0.005 GeV contribution, arising
from the point-to-point energy uncertainty. This correc-
= 1 - 4 sin 2 0err. (9)
Ae tion is treated in the same way by all of the LEP experi-
ments. All of these corrections are described in detail
Fits to the data in terms of these five (or equivalent)
parameters are, essentially, model independent fits to the * It was checked that the appropriate fit results, for each of the
experiments separately, could be reproduced to reasonable accura-
data; i.e. they are independent of the validity of the
cy. For the data of 4, a subtraction of all the t-channel effects
MSM. As discussed in Sect. 2.2, other quantities which
for the e + e- ~ e + e- interaction has been made using the ALIBA-
are useful in the interpretation of the data can be derived
BA program 14. A common luminosity error, arising from theo-
from these five parameters. These are R--Fh/Fe, the ratio retical uncertainty of the e + e- ~e § e- process, of 0.3% has been
of the hadronic and leptonic partial widths, and the 'in- assigned to all experiments. Where the quoted error on this quan-
visible' partial width F~nv=Fz--Fh-3Ft. The quantity R tity was larger than 3.0 %, the additional error (subtracted in quad-
rature) was added (in quadrature) to the experimental error on
is independent of the luminosity measurements. In
the luminosity. In this way this additional error was treated as
Sect. 2.5, fits made within the context of the MSM are
uncorrelated between the experiments. There is an additional com-
described. The free parameters used for the MSM fits
mon systematic error on the t-channel subtraction of the e + e-
are Mz, mr, mn and as. The five parameters of the model e § e- cross-section. This si performed using the ALIBABA pro-
independent fit are all specified, within the MSM, by gram 14, and the quoted uncertainty is +0.5%. The size of the
these four variables. statistical errors si based on the expected number of entries in
a given bin, as determined by the fit. In this way the bias due
to downward fluctuations in bins with small statistics is avoided
** The electroweak corrections in this package are all calculated
2.2 Fits to the LEP hadron and lepton data
using the on-shell renormalisation scheme, except for the QCD
corrections where the MS scheme si used. To the order of the
Fits have been made to the cross-sections and lepton present calculations, the QCD and electroweak corrections decou-
forward-backward charge asymmetries of the four LEP ple
358
Table 2. Correlation matrix corresponding to the five-parameter Table 3. Results of a nine-parameter fit to the LEP hadronic and
fit given in Table 1 leptonic cross-section and leptonic forward-backward charge asym-
metry data. Lepton universality was not imposed in the fit and
Parameter Mz Fz " orC R 2 Aft the results for the effective vector and axial-vector couplings are
given separately for each lepton species. In the fit the signs of
Mz 1.000 0.082 0.009 0.075 0.012 V,, A e, Au and ~A were constrained to be negative. The treatment
F z 0.082 1.000 --0.257 0.015 0.587 of common errors is the same as that for Table 1
~o 0.009 - 0.257 1.000 0.000 0.274
p2 0.075 0.015 0.000 1.000 -0.313 Parameter Fitted value
A~ 0.012 0.587 0.274 -0.313 1.000
Mz 91.176 __0.021 GeV
Fz 2.486 + 0.010 GeV
ao 41.21 _+0.22 nb
in 14. The correlation matrix for the five-parameter
+ O.009
fit, including both the statistical and systematic error -0.019
-0.013
contributions (except the common systematic errors on
Mz and zF arising from the LEP energy scale), is given Ae - 0.499 + 0.002
in Table .2 Also shown in Table 1 are the results of the - 0.002
weighted averages of the quantities given by each of the
uP - 0.060 + 0.034
experiments (or calculated from quantities given by the -0.070
experiment). In this averaging procedure, any common
A~ - 0.497 + 0.014
systematic errors are removed before computing the
- 0.004
weighted average, and then added back in quadrature
after averaging. The common systematics are 6Mz= -0.107 +0"051
+0.02GeV, 6Fz=_+0.005GeV and 6ao= +0.3%, as - 0.091
discussed above. The resulting common errors on ,hF J~ -0.488 +0.030
tF and At 2 are 0.25%. - 0.009
Comparison of the parameter values obtained from
the global fit to those obtained from the weighted aver-
ages shows, in general, good agreement.* The values
ing the interference term does not, in general, mimic
used for the weighted averages are the ones quoted from
models beyond the MSM, it gives an indication of both
the experiments themselves, and do not necessarily come
the stability of the results and the current sensitivity to
from an equivalent fit to that described above; for exam-
physics beyond the MSM being observed through the
ple, some numbers are from fits to the cross-sections
interference term.
only, whereas others include lepton asymmetry measure-
ments. The small differences observed in Table 1 are at-
tributable to the differences in the information used in
3.2 Universality of sgnilpuoc
the respective fits. It should be noted that the overall
2~)
probability for the five-parameter fit given in Table 1
In the global fits described above, lepton universality
is 98%, indicative perhaps of some overestimation of
was imposed. A fit to the cross-section data in terms
some of the systematic errors. If the statistical errors
of Mz, ,zF hF and the individual leptonic widths ,~F ~F
only are used in the fit, then the central values of the
and ~F leads to only very small changes in Mz, zF and
parameters are essentially unchanged (the largest change
hF and gives =~F 82.9_+0.5 MeV, F,= 8.0+_2.38 MeV and
is that % is increased by 0.10 nb), and the corresponding
=~F 82.8 +_ 0.1 MeV; in agreement with the hypothesis of
Z 2 probability is 80%.
lepton universality. The results of a nine-parameter fit
In the second box in Table ,1 the results for other
to all the cross-section and lepton forward-backward
quantities which can be extracted directly from the five-
asymmetry data are given in Table .3 Again it can be
parameter fit are shown. Again reasonable agreement
seen that the data support the postulate of lepton univer-
between the two procedures is obtained.
sality.
The interference term in the above fits was computed
The compatibility of the Z ~ parameters determined
using the MSM. The five-parameter fit given in Table 1
separately from quarks and leptons has also been investi-
was repeated with an additional parameter, which al-
gated. The results of separate fits to the hadronic and
lowed the interference term to vary by a multiplicative
leptonic cross-section data are given in Table .4 The er-
factor K, such that K = 1 corresponds to the MSM as-
rors quoted do not include the common errors on Mz
sumption (the results of which are given in Table .)1 This
and ,zF but do include the other systematic errors. It
six-parameter fit gave a value of K = 1.022 +_ 0.068. The
can be seen from Table 4 that the parameters extracted
values and errors on the other parameters are unchanged
from the leptonic final state cross-sections are compati-
(to the number of significant digits given in Table ,)1
ble with the hadronic final state data, and that the errors
except for -o o which increased to 41.25 nb. Although seal-
from the hadronie data are considerably smaller. For
* This conclusion is consistent with that of a working group set example, the error on the leptonic width ~F is reduced
up by the LEP experiments to investigate the combination of cross- from 8.0 MeV to 0.4 MeV by the inclusion of the hadron-
section data ic data. If the forward-backward charge asymmetry data,
953
0.25
Table 4. Comparison of the results of fits to the hadronic and lep-
tonic cross-section data. The errors do not include the common
errors on Mz and .zF In the fit to the leptonic data the MSM
value Fh= 1.733 GeV was imposed. If eF rather than rc ,0 si used
in the leptonic fit, then the fit gives Fe=83.6_+0.8 MeV
0.26
Parameter Hadronic fit Leptonic fit
Mz 91.175 500.0+_ GeV 91.172_+0.015 GeV
F z 2.4854- 0.009 GeV 2.5024- 0.027 GeV
th
rc o 41.33 -+0.22 40.83 -+0.54
32.0
= 002 6eV
Table 5. Results of a fit to the LEP leptonic cross-sections and
forward-backward charge asymmetries. The errors do not include
the common errors on Mz and zF
0.22 I L , ~ , I , , J ,
Parameter Fitted value 098 00.1 20.1
M z 91.173 510.0+_ GeV Pelf
Fig. .1 Results of the fit to the LEP hadron and lepton cross-section
F z 684.2 + 0.027 GeV
~2 4000.0+_0100.0 and lepton forward-backward asymmetry data to the variables
~z 5200.0+_0942.0 sin ffo02 and poff, together with the 68% and 95% confidence level
contours. The solid region is that expected in the MSM, as ex-
plained in the text
as well as the cross-section data, are included in the
fit to the lepton data, then the results obtained are as R
given in Table 5. Comparison with the fit in Table 1
shows that the error on .~2 is reduced by a factor two 212
with the inclusion of the hadron data with, as expected,
no change in the error on ~2.
2.4 Comparison of the data with the MSM
20.8
Also shown in Table 1 are the values expected in the
MSM. Within the MSM there are two unknown masses,
, , . 07)
mt and ran. In addition, the calculation of hF requires
the use of c~s(Mz)~. The values given are computed us-
ing the electroweak libraries available within the ZFIT- 20./* , L = I I h I , I I t t I I
TER package, and use the parameter values Mz 605 /.1.0 /.1.5 /*2,0
= 91.175 +_ 0.021 GeV, mt= 140_+ 40 GeV and c~s(Mz s~m)m o~( {nb)
=0.114_+0.007.* The errors on mt and es(Mz)~s were
Fig. 2. Results of the fit to the LEP hadron and lepton cross-section
taken to be Gaussian. The Higgs mass mn was taken
and lepton forward-backward asymmetry data to the variables R
to be uniformly distributed between 50 and 1000 GeV. eF/hF= and rc ,o together with the 68% and 95% confidence level
It can be seen that all the quantities are in agreement contours. The solid region is that expected in the MSM, computed
with the MSM values. The number of light neutrinos for the range of values of ,tor( mn and )s~ given in the text. The
values of (m, mn and )s~ at the extremes of the region are shown
is compatible with ~N = 3.
In Fig. 1, the fitted values of Peff and sin 2 0elf are
shown, together with the 68% and 95% confidence level
played in Fig. 1, as a solid region, are the expectations
contours, defined such that both parameters lie inside
of the MSM. The solid region corresponds to 89<mr
the contour at the quoted probability level. Also dis-
< 200 GeV, 50 < mu < 1000 GeV and 0.107 < ~s < 0.121. It
can be seen that the main variation is that due to ~;
* The value of m t corresponds to that determined below in this
section; it should be noted, however, that the sensitivity of the the variation with mn giving rise to the width of the
LEP data to tm comes from quantities which are essentially quadra- band, with the lower and upper parts corresponding to
tic in mr The value of c~s(Mz)~ si taken from studies of the proper- mR= 50 and 1000 GeV respectively. The quantities poff
ties of the final state hadrons in Z ~ decays by the LEP experiments and sin 2 0eel are strongly correlated in the MSM, so that
15-18. The central value quoted si the mean value of e~(Mz s~)m
the error on sin 2 0eff can be considerably reduced if the
for the four experiments, and the error assigned is an estimate
MSM constraint is imposed (see Sect. 2.5). However, care
of the total error, including the statistical and systematic errors
as well as the theoretical uncertainty arising from the scale depen- must be exercised in the use of such a quantity in the
dence search for physics beyond the MSM.
360
The quantities % and R are rather insensitive to the This value is somewhat larger than the value X---M)zM(se
unknown parameters of the MSM, and are therefore a = 0.114 (cid:127) 0.007 obtained from a study of the properties
good test of the validity of the model itself. In Fig. 2, of the final state hadrons (15-18), and also with the
the 68% and 95% confidence level contours for oO and value es(Mz)~ = 0.109 + 0.008 obtained from deep in-
R are shown (extracted from a four-parameter fit to the elastic lepton-nucleon scattering and prompt photon
cross-section data, the other two parameters being Mz data 22. The rather large vMue of cq(Mz)~ from the
and Fz), together with the MSM expectations defined LEP cross-section data stems essentially from the fact
as for Fig. .1 The MSM values most compatible with that the fitted value of R is larger than the MSM value
the data correspond to the smallest values of tm and (see Table .)1 R can be written 12
mu (in the quoted ranges) and the largest value of .s~
R = Ro 1( + ,)DCQ6 (15)
Combinations of the above variables can be con-
structed which are sensitive to different physics possibili- where Ro is the value of R without QCD corrections,
ties beyond the MSM. For example, the variable 19 and is calculated in the MSM to be Ro=19.95_+0.01,
and DCQ6 is given by (7). Solving the above equation
270
for R, using the global fit value from Table ,1 gives ~8
)01(
T v = R 59 Mz )zM(~o = 0.147 -t- 0.018; in agreement with the four-parameter
fit value for s8 in (14). An approximate estimate of the
is defined in such a way that, to a good approximation, s~-4 and qc 5- terms in DCQ6 has been made recently 23.
the tnr dependence is contained in the Zbbvertex correc- The effect of these terms would be to reduce the predicted
tion term, which is always negative for tm larger than value of R by about 0.027 for a value of 8s=0.147. In
the limit of 89 GeV set by p/5 collider data 20. Using Fig. ,3 the 68% confidence level contours for ~c s against
the MSM parameters described above, the predicted m t are shown for two values of the Higgs mass, Hm = 50
value is Tv=0.513+0.003, with a 99% confidence level and 1000GeV respectively, together with the fitted
upper limit Tv < 0.521. A five-parameter fit to the cross- values. The lower mr/value leads to lower values of ~c
section and lepton asymmetry data in terms of Mz, ,zF and mt.
R, ~2 and T v gives If the constraint ~ = 0.114 + 0.007 is added to the fit,
as a penalty in the 2~) function, then the value of ,m
Tv = 0.526 + 0.008, (11)
obtained is
with the other parameters as given in Table .1 The value
+35+21
of Tv is thus (just) compatible with the MSM expectation. m t = 125_ 43- 20 GeV, (16)
If future results give a value of Tv significantly above
the MSM upper bound, then the authors of reference where the central value of tnr is for me = 300 GeV, and
19 argue that this would be indicative of the existence the second error is the variation due to changing Rm
of a second Z ~ The results of explicit fits to a possible to 50 GeV (lower value) and to 1000 GeV (upper value).
second Z ~ are discussed in Sect. 2.6. The value ~= 0.120 +0.006 is also obtained.*
A variable complementary to Tv is
450 ~F 0.20
)21(
Tr R+ 59 Mzc~(Mz) "
0.18
This variable is essentially unaffected by any new Z ~
mixing of either E 6 or left-right symmetry origin 21. m H = 1000 fieV
A five-parameter fit to the cross-section and lepton 0.16 ~
asymmetry data in terms of Mz, ,zF R, ~2 and ~T gives
~T = 1.959 + 0.006, (13) */1.0
which is in good agreement with the MSM value 7~
0.12 ~ . . . . . . . . . . .
= 1.954 + 0.004.
m H : 05 VeG
0.10 i I I I I
2.5 Fits within the MSM
OS 001 150 200
fieV
A four-parameter fit was made to the combined cross- i t
Fig. 3. Results of the fit to the LEP hadron and lepton cross-section
section and asymmetry data to determine Mz, rot, m~
and lepton forward-backward asymmetry data to the variables 8 s
and 8s.* The fit favours low values of ~m and Hm (i.e.
and ,~m for mH=50 GeV (shown by a star) and m~= 1000 GeV
values close to the imposed lower bounds of 50 GeV (shown by a solid circle). Also shown is the 68% confidence level
in both cases), and gives a value of contour
r)
a~(Mz)~x =0.143 +0.017. (14) * The probability of this fit, and of all the other fits discussed
in this paper, are in excess of 95%. This stems from the low Z 2
* nI lla of eht fits ni this section eht value of Mz obtained si -nesse values of the LEP lineshape and lepton asymmetry data, as dis-
yllait degnahcnu from that nevig ni Table 1 cussed in Sect. 2.2
361
More precise values of mt and mn can be achieved as a 2~) penalty function in a fit within the MSM, since
by including further LEP measurements as constraints the value is predicted from the parameters of the fit.
in the fit. The following measurements were used. *** The dependences of ~R and Qw on mt and mn are given,
(a) v-polarisation. The weighted mean of the measured for example, in 40. Including these data in the fit, and
values by the ALEPH 24 and OPAL 25 collabora- also the constraint on ,s~ gives
tion, namely P~=-0.124+_0.040, was used as a con-
137+22+ 81 GeV, (18)
straint in the fit. mt 22
= - 25 -
(b) Forward-backward charge asymmetry ~vbA for b-
quarks. This has been measured using semileptonic b- where again the errors have the same meaning as in
decays by ALEPH 26 (~ = e, #), DELPHI 27 (f = #), (16). The value of s~ resulting from both of these fits
L3 28 ((=e, #) and OPAL 29 (~=#). The average is ~c = 0.120 +_ 0.006. It should be noted that the correla-
value of the mixing parameter for the LEP experiments tion between s~ and mt is such that increasing s~ leads
30, 31 is 2=0.144_+0.023. This value is then used to to smaller .tm The results of the fits can be directly trans-
correct the average observed asymmetry by a factor 1/ formed into values for the on-shell weak-mixing angle
(1-2Z) to obtain the corrected asymmetry AbB=0.123 sin 20w = 1 -Mw/Mz, 2 2 M w and A ros, giving
+__ 0.024.
)C( Charge asymmetry in hadronic Z ~ decays (QvB). The sin 2 0 w = 0.2285 + 0.0027 + 0.0024
(19)
value measured by the ALEPH collaboration 32 is - 0.0026 - 0.0021'
(QFB) = --0.0084 + 0.0023, and that by the DELPHI col-
+0.13+0.11
laboration 33 is (QFB) = -- 0.0076 ___ 0.0019.**
Mw = 80.08 _ 0.14- 0.12 GeV, (20)
Each of these measurements was converted into a mea-
surement of sin 2 0elf (as defined for leptons), using the
~0~7
predictions of ZFITTER, and then used as a constraint dros = 1
/2GF M 2 sin 20w
in the fit. The effects of the QCD corrections to Aqa
are thus taken into account by this fitting procedure.
This method is a convenient 'book-keeping' procedure; = 0.052 + 0.008 + 0.007 (21)
-- 0.008 -- 0.006'
however, the same results are obtained if the fits are
formulated in terms of the directly measured quantities. where again the errors have the same meaning as those
Including these LEP data, and the constraint on ~, on tm in (16). Alternatively, the results can be expressed
gives in terms of the effective weak mixing angle defined for
leptons (with Peff given by the MSM), giving
+ 30+20 GeV, (17)
tm = 142_ 37 - 20
+ 0.0007 + 0.0008
where the errors have the same meaning as in (16). sin 2 0elf = 0.2339 _ 0.0007 - 0.0010" (22)
Additional constraints can be added by including
other electroweak data in the fits. The most precise data It should be stressed, however, that the results given
are: in (19) to (22) are just transformations of variables, with
no additional information content.
(a) Measurements of the Wboson mass Mw (or the ratio
If mn is allowed to be a free parameter in the fit
Mw/Mz), which has been measured by the CDF -34
to all the above data, then the fit gives
and UA2 35 collaborations. Combining these mea-
surements, together with the LEP measurement of Mz,
115 +32
gives Mw = 80.14+0.27 GeV. mt = -- 27 GeV, (23)
(b) The ratio ~R of the neutral to charged current cross-
sections in deep-inelastic v-nucleon scattering by the 50 +353
mn = GeV, (24)
CDHS 36 and CHARM 37 collaborations. The
theory error, arising from both the uncertainty on cm
where the value of mn obtained is the imposed minimum
and from other sources, is included in the analysis be-
value allowed in the fit. Figure 4a shows the results of
low.
this fit, together with the 68% confidence level contour
)c( Measurement of parity violation in atomic Caesium,
for m t and mn. In Fig. 4b and c the variation of 2(~ as
which gives for the weak charge Qw = -71.04 _+ 1.81 38,
a function of both mn and mt is shown. This X 2 is, for
39.
a given value of mt (or mn), the minimum for any values
The constraint from Mw can be incorporated directly of Mz, s~ and Hm (or mr). At the 95% confidence level,
mr< 391 GeV. It can also be seen from Fig. 4 that the
* LEP data on other quantities (e.g. bF and )~F are not yet at a
level of precision where they provide significant additional con- data prefer a low value of mn, particularly if the top-
straints quark is relatively light. However, care must be exercised
** The error given for the ALEPH result includes the quoted 20% in the interpretation of the errors on ran, since the depen-
theoretical systematic uncertainty, which stems mainly from the
dence on mn in the fits is mainly logarithmic. In particu-
dependence of the charge separation on the fragmentation parame-
lar, a large contribution to the Z ,2 and to the sensitivity
ters of the Monte Carlo(s). The part of the error due to fragmenta-
tion uncertainties is treated as a common systematic error when to mn, comes from the measurements of the forward-
combining the ALEPH and DELPHI results on this quantity backward asymmetry for b-quarks, where the measured
263
and ~s=0.130+0.010. Thus the conclusion that the fit
favours a light Higgs is unchanged, but the magnitude
of the upper error is sensitive to the assumptions made
2000
on ~s"
The existing data have more constraining power on
mn than would be expected from the present accuracy
1000
of the data. To illustrate this, a fit was made with the
values of all electroweak data shifted to correspond to
005 the MSM 'central' values mr= 140 GeV, mH=300 GeV
and ~s(Mz)~s = 0.114, but leaving the errors on all quan-
003 tities at their current values. The fit gives
200 +47
mt= 140_ 53 GeV, (25)
001
and errors on m~ which encompass the entire range 50
to 1000 GeV for m~. Thus, in the evaluation of the im-
a 50 I provements which can be obtained with more precise
50 100 150 200 electroweak data, the comparison of the improved errors
~m leVI with these latter errors should be made.
It is interesting to note what improvement on the
m~ limit would be obtained if a direct measurement on
tm became available. This, in fact, depends rather strong-
ly on the actual value of m,. For example, if the measured
value were mr= 100+ 10 GeV, then the fit with all the
data above, taken at its face value, together with the
HE 2
e3 constraint on ~s, gives m~=50 +85 GeV (i.e. at the im-
posed lower bound), whereas for mt= 140 + 10 GeV, the
1
+ 640
fit gives mn = 150 _ 100 GeV.
b 0 I i I I
05 001 002 003 005 0001 0002
Rm fieV
6.2 Constraints on a second Z ~ boson
t,
The existence of additional gauge-bosons is a common
feature to many extensions of the MSM. A new massive
3
neutral gauge-boson Zm in addition to the standard
gauge-boson Zs, will lead to modifications of the cou-
.c:
HE 2 plings of the Z ~ to matter and also to shifts in the masses
of the ordinary gauge bosons. In this section, the con-
straints that can be derived on the parameters of such
1 an additional massive neutral gauge-boson, using the
data discussed above, are examined. If there is an addi-
tional U(1) symmetry then, at tree-level and before sym-
c 0 I
05 001 051 002 metry breaking, the states Zs and ZN have terms in the
tier Lagrangian proportional to ZS(J3L--sin2OJEM) and
m t
Zs ,NJ coupling to the standard and additional neutral
Fig. 4a-c. a Results of the fit to the variables m t and nm to the
LEP hadron and lepton cross-section and lepton forward-back- currents respectively. After symmetry-breaking the states
ward asymmetry data, z-polarisation, b-quark asymmetry and had- Zs and ZN are mixed, leading to mass eigenstates
ron charge asymmetry, as well as measurements of ,wM ~R and
parity violation in atomic Cesium. The fitted value is shown by
a star and the 68% confidence level contour si also shown; b and \-- sin ( cos ( \ZN
c the variations of X 2- ni2X as a function of mn and tm respectively.
The data used are as for a. The procedure to obtain the curves with masses Mz and Mz,, related by
is explained in the text (
Mz~=( cosr sinr (27)
Mz, \- sin ~ cos ~ \MN"
value is some 1.5 a away from the MSM predictions on
the side favouring a light Higgs. Further, if a much looser The mass matrix can be written as
constraint on s~ is applied, namely s~ = 0.120 __+ 0.020,
= ( (28)
then the fit gives mt = 105 + 28 GeV, mn = 50 + 926 GeV ~ i 2 Mi '
363
which, after diagonalisation using (27), leads to the rela- In addition to the effects on p and the effective vector
tionships and axial-vector couplings, there are also Z ~ exchange
2 2 terms. However, from the Z ~ mass limits already estab-
tan2 r _ Ms - Mz (29) lished from the low energy neutral current data (7,
2 2
Mz, - Ms
and references therein), it is expected that the contribu-
and tions of such terms are generally small. However, the
6M 2 direct term is important for the low 2q measurements
tan ~ = 2 2 (cid:12)9 (30) of parity violation in atomic Caesium 43, and it is
Mz, -- Ms
taken into account in the analysis described below.
The MSM mass relationship between Mz and Mw be- An additional Z ~ boson will thus manifest itself, in
comes, at tree-level, cos20 =MZ/po M 2, where 0 is the LEP physics, by leading to shifts in the MSM fermionic
electroweak mixing angle and oP = 1 +A .MP The term partial widths and asymmetries. These shifts are ob-
A MP arises due to mixing, with A MP > 0 for Mz, > Mz. tained by inserting the expressions for Peff, V; ff and A) ,er
Additional contributions to Po could, in general, occur given by (32) to (34), into )5( and )6( for the fermionic
from Higgs multiplets larger than doublets. partial width and forward-backward asymmetry respec-
In the models considered here, the W mass and cou- tively, and retaining only terms first order in ,4 which
plings are unaffected (at tree-level) by the existence of is known to be small. The values of Mw and R,, pre-
a Z ~ For given values of Mw, mt and ran, the expected dicted from the measured parameters of the lightest Z ~
value of the Z ~ mass, before mixing, can be computed are also modified by the presence of a Z ~ as is the
in the MSM. Using the experimentally measured value value of Qw for atomic Caesium 43, 44.
for Mw, and allowing mt and mn to vary within the Fits have been made to the LEP cross-section data
ranges 100<mr< 180 GeV and 50<ran< 1000 GeV, the for hadrons and leptons, the forward-backward asymme-
Z ~ mass is predicted to be (using 11) tries for leptons and b-quarks, the hadron charge asym-
metry (QFB), the -c-polarisation ~P and the measured
Ms= 91.21 +0 2- +0.25 GeV, (31) values of Mw, ~R and Qw. The parameters used to form
_ . 2_0.29 the Z 2 for the fit (i.e. the five parameters discussed in
Table ,1 together with A~B, (QFB), ,~P Mw, R, and Qw)
where the first error is from the experimental uncertainty
are all expressed in the form X = MSMX + A X. The MSM
on Mw (see Sect. 2.5), and the second from the quoted
values are calculated for the assumed central values
variation in the MSM parameters. Comparing this value
mt=140GeV, mn=300GeV, ~s=0.114 and Mz
with the measured value given in Table ,1 it can be seen
= 91.175 GeV. The effect of changing the assumed MSM
that the mass shift at the Z ~ peak is ~<0.5 GeV.
values is discussed below. The shift A X is expressed in
The effective weak neutral current coupling to the
terms of the parameters of the model in question, and
lighter Z ~ (i.e. that studied at LEP) is of the form
the Z 2 is minimised with respect to the parameters of
Joc(J3L--sinZOJEM)COS~+Jusinr In terms of the im-
the model.* The Z ~ mass is also a parameter; however;
proved Born approximation, the effect of an additional
in all the fits discussed below the fitted value for Mz
Z ~ is thus to replace the MSM values for Peff, sV and
is essentially unchanged from that given in Table .1
As (for the coupling of a fermion f to the lighter Z ~
These fits are then used to obtain constraints on the
by
parameters of the Z ~ A complete discussion of the many
MSM
peff=,Oeff 1( +A ,)MP (32) existing models is not attempted here, but rather a few
models are examined in order to illustrate the improve-
V; eft-- ~eff (V/cos r V/sin r (33) ments in the constraints on the Z ~ parameters that the
present data permit.
A}ff= p~ff(A} cos ~ + A~ sin ,)4 (34)
Two classes of models are considered:
where V 7 and Af are the vector and axial-vector cou-
(a) Extra U(1) in E 6 models. These comprise a set of
plings to the new vector-boson ZN (see, e.g., 9, 41, 42
models in which an extra U(1) group is contained in
and references therein*). The Zs couplings are defined
by a simple E 6 group (see, for example, 45 and references
therein). The extra Z ~ is a superposition of two colour
VT=ff3L_2Qfsin 2 0eff, M (35) singlet neutral generators Y' and Y", giving an additional
s I (36) neutral current NJ = cos 0 2 Jy, - sin 20 Jr,,. In the analysis
Af= rI3L .
described below the assumption Ng = g', for the couplings
The effective weak mixing angle sin 2 0~f is defined, for of the respective currents, is made. Thus, for a given
given values of ,rm Hm and ,~7c by value of 02, the model is described by the two parameters
A MP and .4 The vector and axial-vector couplings for
2 2
_sin 2 Mffe0 : 2ffeS 2ffeS 2ffec A PM, )73( these (superstring inspired) E 6 models are given in Ta-
ble 6 9, 41.
feC f -- Ser f
where feS 2 f =Sln (cid:12)9 2 0ef f is the MSM value.
* esuaceB the various models have couplings for the Z~ which
(cid:12)9 Note that different sign conventions for the axial-vector current are fermion type dependent, the analysis of a second Z ~ si not
exist in the literature llew suited to the methods described below in .tceS 3
463
Table .6 Vector and axial-vector couplings in E 6 models with an A lower limit on Mz,, from a search for direct Z ~
extra U(1) symmetry. The couplings are those of the fermion f production, has been obtained using the CDF collider
(= ,v ,g u and )d to the possible new gauge boson Zu, and are data 46. This limit depends on the Z ~ couplings to
written in the form )zOniszV+zOsoclV(Onis=~V and Af
= sin A(O 1 cos z0 + zA nis )20 quarks, and hence on 02, and on the set of structure
functions used. For the EHLQI structure functions 47,
f 111 zV 1A 2A the limit Mz, > 285 GeV, for any value of 02, is obtained.
Fits to the LEP and the other electroweak data dis-
cussed above give constraints on the parameters A MP
v
98(cid:1)- 98(cid:1)- and 4, as a function of ,zO as shown in Fig. .5 The bound-
aries of the shaded regions correspond to the 95% confi-
u o o 0
dence levels for A p~ and 4, and are valid for the MSM
assignments rot= 140 GeV, mH= 300 GeV and s8 =0.114.
The dashed lines are the 95% confidence levels when
the MSM parameters are allowed to vary within the
0.02
ranges 100<mt<180GeV, 50<mn<1000GeV and
la)
MOA
0.107<c~<0.121. It can be seen that these variations
considerably increase the allowed range for A PM, but
0.01 have less effect on .4
The above analysis is rather general, and does not
have the constraint A p~t>0 imposed, as would be the
0.00 case if there were only Higgs doublets. Negative values
of A MP could arise, for example, from a non-standard
Higgs structure. If it is further required that the Higgs
structure is such that mixing vanishes for large Mz,
10.0-
>~( Mz), then the relationships
(38)
_0.021 i i l i , I
-90 0 90
0 2 deg
50.0 A PM-~c2 --~M (39)
L )b(
are obtained (see e.g. 413). The variable c can be ex-
pressed as
i ii ii i ii i o oo!i c = -sin O ~ soc 20 + ~/~ 7~ sin 02 , (40)
with t/= 2V(/2V -~- ~2), where v and ~ are the Higgs expecta-
tion values and 0<q<0.5. In specific models, corre-
sponding to different patterns of symmetry breaking, the
values of zO and c are specified. For example, in the
t/(or A) model 02 = 0, and the Higgs structure restricts
c, to be in the range -0.64 < c, < -0.24, whereas in the
-0.05 i i I i i
-90 0 90 Z (or B) model, cos 0 2 =~83- and cx= --0.39; these values
0 2 ged of c being for the case of two Higgs doublets 7, 48.
The lower limits on Mz, are established as follows:
Fig. 5a, .b Limits on the variables a A MP and b ,~ as a function
For given values of zO and c, a fit is performed in which
of ,20 in 6E models. The boundaries of the shaded regions corre-
spond to the 95% confidence levels for A MP and ,~ and are valid Mz, and mu are fixed, and the minimum Z 2 with respect
for the MSM assignments ,VeG041=ttrr mH=300GeV and ~ to the variables Mz, tm and ~c is found. Such fits are
=0.114. The dashed lines are the 95% confidence levels when the performed for a series of values of Mz,, and the distribu-
MSM < 081 GeV, parameters <nm<05 are 0001allowed GeV to and vary 0.107< within 121.0<~g the ranges 100<mr tion of Z ,2 with respect to the minimum value of Z 2 for
any Mz,, is found. From the variation of Z 2 a lower
limit on Mz, is obtained. This procedure is repeated for
mu = 50 GeV and m~= 1000 GeV, and the most conser-
vative limit is taken as the lower limit on Mz,. At the
The coupling of a fermion f to the heavier Z ~ has
95% confidence level the limits found are
the form Vy f' = p~ff(- V/sin ~ + V~ cos 4) etc. It should
be noted that the couplings of the various fermions to t/(or A) model: Mz, >470 GeV c,= -0.24, (41)
the Z and Z ~ can be very different, leading to potentially
Mz, > 840 GeV c, = - 0.64, (42)
sizeable differences in the partial widths and asymme-
tries. Z (or B) model: Mz, > 700 GeV c z = -- 0.39. (43)