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Z. Phys. C - Particles and Fields 52, 283-288 (1991) Zeitschrift
F~, r t i c~s f~r Physik C
and F 4cls 9 Springer-Verlag 1991
Higgs s-channel production in e +e- collisions below the W + W-
threshold* G. Alexander ~, C. Milst6ne l, W. Hollik 2
1 Department of Physics and Astronomy, Tel-Aviv University,
69978 Tel-Aviv, Israel 2 Max-Planck-lnstitut ffir Physik und
Astrophysik, Werner-Heisenberg-Institut ffir Physik, P.O. Box
401212, W-8000 Mfinchen, Federal Republic of Germany
Received 6 May 1991
Abstract. The cross section for direct neutral Higgs pro-
duction in the reaction e+e - ~H~ is calculated in the EcM energy
range of 40 to 160 GeV and compared to the corresponding
Electro-Weak (EW) process e+e - --,(Z, 7 )~f f . Neglecting
radiation effects, a signal of the order of 10 4 to 10 3 over the
EW can be expected outside the Z ~ region for the Minimal Standard
Model Higgs in its decay to a bb state. For Mn > ]/s - Mz, the
s-channel Higgs formation can surpass the Bjorken Bremsstrahlung
process and thus may afford at LEP 2 a realistic search method for
high mass, say __> 100 GeV Higgs, given enough luminosity. For a
non-Standard Model Higgs, in some cases, significantly higher
signals are expected. The effects of initial state radiation and-
machine energy resolution are evaluated and the gain in using
longitudinal polarized electron beams is discussed.
1 Introduction
The Higgs mechanism [1], which is responsible for the non-zero
mass values of the gauge particles, quarks and leptons, plays a
central and crucial role in the Standard Model (SM) [2]. So far
however, intensive searches for a) the Higgs particle have failed
to detect it. In the past these studies were based mainly on the
decay of the light " , ,~ bosons [3], like n and K, and on
radiative decays [3, 4] of bound qq states like the Y (9.46). More
recently, ex- periments at SLC and LEP have and are searching for
Higgs emission from the gauge vector boson Z ~ produced in e+e -
collisions via the so called Bjorken mechanism [5] (diagram in Fig.
la). With his method, which is mainly b) effective in the search of
Higgs particles with mass values much lower than Mzo, no evidence
for the presence of a \e* neutral Higgs has so far been found in
the mass range of ) N0 MeV to ,-~ 50 GeV [6]. It is further
estimated that even with 107 Z ~ decay events the detection of
Higgs is limited in this method to masses less than about ~60 GeV
[7].
* Supported in part by the Israel Ministry for Science and Tech-
nology
In its second phase LEP 2 is expected to reach the energy value
of ]/s~-200 GeV. At this energy, well above M z, the search for
Higgs bosons via the Bjorken mechanism is still useful as long as
Mn< = 1 Is - M z. For higher Mn values the Bjorken mechanism
cross section is rapidly decreasing. For example, at l fs = 120 GeV
already for a Higgs with a mass as low as 60 GeV the cross section
is very small namely around 1 10 3 pb [8]. For this reason it is of
interest to explore other possibilities to observe neutral Higgs
such as in the s-channel formation of Higgs in e + e- collisions
(diagram in Fig. 1 b) which in the case of LEP could cover mass
values up to < 200 GeV. Higgs with mass values above twice the
mass of the W gauge boson, are best searched for through their
decay into W or Z pairs [3, 7]. For the search of Higgs with lower
mass values, the bb is the preferred decay channel. Here one should
note that the current lower mass limit for the t- quark, as given
by the CDF Collaboration, is > 89 GeV with 95% c.1. [9] so that
its production threshold in e +e- collisions is well above that of
the W + W- pair.
In this paper we study the relative contributions of the Higgs
s-channel diagram to that of the Z, 7 Electro-
Z ~
F
Z~ , 7 @ HIGGS
Fig. 1. a Diagram of a Z ~ formed in e+e annihilation emitting a
Higgs boson, (Bjorken Mechanism). b The s-channel Z, y and Higgs
contributions to the reaction e+e - ---,ff
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Weak Gauge Bosons process and discusss possible ex- perimental
consequences, in particular in view of the planned LEP program of
precision measurements. Our study is confined to the energy range
from 40 GeV, avail- able to the TRISTAN e+e - collider in Japan, up
to the energy of 160 GeV, below the W + W- threshold, which will be
accessible to the LEP collider at CERN in its 2 "d operation phase.
In Sect. 2 we present the main formulae used and the assumptions
made. Our results for the SM Higgs formation is discussed in Sect.
3 and compared to the Electro-Weak (EW) contributions. In the same
sec- tion the gain which one may obtain from the use of po- larized
electron beams will be discussed. As for the search of non-Standard
Model neutral Higgs, we present the minimum requirements for its
detection in terms of its coupling strength to fermions. The
initial state radiation and machine energy resolution effects on
the Higgs sig- nals are discussed in the summary and conclusions
given in Sect. 4.
2 The method and formalism
The cross section for e+e ~f f , described by the sum of the Z,
y and the Higgs processes shown in Fig. 1 b, is given by:
[] Idol idol da do. + ~- + d-~-- ~ y,Z Higgs d~ Int '
where the first term is the known Electro-Weak cross- section.
The cross section for the s-channel SM Higgs diagram, o.n, is given
by:
do" Arc 3,~3 2 2 H-- He 64 n2s #j ge gf (s-- M~) 2 + M~F~2
2,
where ~s = Ecm , M H and F H are the mass and width of the Higgs
boson, N c is the color factor which is equal to 1 for leptons and
to 3 for quarks. The velocity of the outgoing particle flk is equal
to
flk = ~/1 - 4 m2k/s.
The parameter gk is the Higgs coupling to a pair of k/~ fermions
which in the Standard Model is given by:
gk = 21/4 ~GFFmk,
where G F is the Fermi coupling constant. The correspond- ing
partial decay width is then equal to:
F k = Nc ~ Mz_tfl 3 .
The interference contribution between the Z, y and the Higgs
diagrams is given by:
= Nc memsve VsgegsRe [Dz' ~ D*] Pe Pf COS 0 2n2S
where Pk is the momentum of the particle k, D is the
Breit-Wigner amplitude, 1 / ( s - Mz+ iMF ) and vsis the
Electro-Weak vector coupling constant
213 - 4 Qf sin 2 0 w vf - 4 sin Ow cos Ow
First to note is that the interference term is proportional to
cos 0 and thus cancels out when one considers inte- grated cross
sections over a symmetric polar angle range. Since the Higgs
coupling is proportional to the particle mass, the Higgs production
in the e +e- collision (Fig. I b) is strongly suppressed by the
initial state small electron mass. Thus if there is at all a hope
to observe the Higgs particle effect in its formation in e+e
annihilation, one has to try and enhance as much as possible its
signal relative to that of the Z, y diagram. Keeping this goal in
mind we notice the following:
1) the decay rate of the Higgs particle to the final state f f
is proportional to the mass squared of the particle f 2) the effect
of the Higgs diagram will be maximal when
]~=M H 3) the differential cross-section do./dcosO is basically
proportional to 1 + cos20 for the Z, y diagrams, apart from a small
asymmetry term, whereas it is flat in cos 0 for the Higgs
production. 4) A suppression of the Z, ? amplitude contribution can
be obtained by a proper combination of longitudinal po- larized e +
and e- beams.
As a consequence we have considered the effect of the Higgs
diagram in e + e-collisions leading to the heaviest known lepton
pair r + r - , and quark pair bb. Current lower limits on the
t-quark are put at about 89 GeV and thus are beyond the energy
range considered in this work. In addition, we limit the polar
angular domain to the range of - 0.5 < cos 0 < 0.5. This cut
enhances by ~25% the relative Higgs contribution. Note that the
loss in sta- tistics due to this restriction is not too severe
since in any case an acceptance of [cos 0 1 < 0.8 is typical for
exper- iments at colliders. Finally we adopt the method of
searching for Higgs by studying the e+e - collisions as a function
of lf~ so that one can consider the Higgs effect at the peak of its
Breit-Wigner shape where its cross sec- tion is equal to
do.]peaks 3 ]~3 _2g2 1 /"e~f H~H =N c He f~:~e f _
64 n2Ft 2 M 2 Fto2t '
at M~/= ]/s.
The possibility to equip the SLC and LEP electron col- liders
with longitudinal polarized beams has been recently studied [10, 11
] and it is expected that they will be avail- able in the not too
far future. In the case that the lon- gitudinal polarization of the
electron beam Pe L- , and that of the positron beam PeL+ are equal,
the relation between
beams a , and polarized the cross-section for unpolarized v
beams o.v, for a vector ( Je= 1-) s-channel process is,
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285
up to corrections of order re~E, given by [10]:
o2= (1 . -P~ P~+)
Thus the total measured cross section is then equal to:
=oS(1-et where crs is the contribution from a scalar ( J~ '=0 +)
s- channel process like that of a Higgs boson. From this follows
that in the case that P# = P#+ = 1 the (Z, y) con- tribution may be
completely eliminated. In a circular col- lider however the
theoretical maximum value attainable for P# is ~-0.92 [12] so that
an ---80% suppression of the s-channel (Z, y) contribution can be
achieved in com- parison to that of the Higgs. In practice however,
a value around PLy-0.7 is a more realistic one having an effective
suppression of -~50%. In the following we will present our results
for the annihilation of unpolarized electron beams. The gain
provided by using polarized beams can readily be derived from these
results.
3 The s-channel Higgs production
Throughout our work here we use in our calculations the value of
sin20 w= 0.237 and discuss the sensitivity of our results to this
particular choice. Furthermore we present our results without the
initial QED radiation effects. For the Z, 7 diagram they are only
of importance around
= Mzo where anyhow the relative Higgs signal is ex- tremely
small and undetectable. The effect on the Higgs signal coming from
initial QED radiation and machine energy resolution will be
elaborated in the conclusion section.
3.1 The Standard Model Higgs
In the case of the Standard Model Higgs its partial decay width
to a given final state fermion is well determined and so is also
its total width which is shown in Fig. 2 as a function of its mass
M~/. In the Higgs mass range of 40 to 160 GeV, the width spans
between ~ 2 to ~ 8 MeV linearly proportional to M H as it is far
away from bb threshold.
The Higgs cross-section cr11(e+ e - - f f ) and 9 C rz ,~(e+e-~f
f ) , integrated over the range Icos01
< 0.5, are shown respectively in Figs. 3 and 4 as a func-
tion of EcM for the final states r-lepton and b-quark pairs. The
contribution of the interference term at cos 0 = 1 be- tween the Z,
7 and the Higgs diagrams, is shown in Fig. 5 as a function of ECM.
First to note is the fact that the interference terms value is
smaller than the Higgs con- tribution by 6 order of magnitudes
outside the Z ~ mass energy region and by about 3 order of
magnitudes on the Z peak and thus can safely be ignored. The
relative con- tribution
R- anigg~(e+ e- ~ f f ) (e + e- - * i f )
of the Higgs diagram to that of the Z, ? is shown in Fig. 6 as a
function of EcM. As anticipated, this ratio is ex- tremely small on
the Z peak (~ 10 -7) however far below and far above the Z peak the
ratio R is approaching the value of 10 -3 for the production of the
b/~ quark pair. This level is nearly comparable to that considered
in pre- cision measurements with polarized beams at LEP [10].
g 8
7
I ~ , ~ I , , , , I _~_ , , , I , , , , I , ~ , t [ , , ~ , 4@
6@ 80 100 I~0 140 16@
MH~gg. (GeV) Fig. 2. The Standard Model Higgs total width, Ft~
~gg~ , as a function of its mass value MHigg s
-5
# I ,cos ,,o.5
b l """'"'"'"" 10-6 .............. , ...............
10 .7
,~ I 10 .9 4O
........................ bg
"7-+7 "-
60 80 100 120 140 160
Eo~ (GeV)
Fig. 3. The cross section ~s~ (e+e - ~ff) integrated over the
polar angle range -0.5 < cos0 < 0.5 as a function of ECM. The
solid line is for r +r- production and the dotted line for the
production of the b6 quark pair
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286
-s [ -
b 1
Icos ~1
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287
dependence of AR/R on sin 2 0 w varies tepidly with EcM in
particular around EcM = M z, where it reaches the maximum of ~ 1.6%
for the b# production. To note how- ever is that R is almost
independent of sin 2 0 w at Ec~ values below 60 GeV. Above 60 GeV
the sensitivity of the R value to the Electro-Weak mixing angle
dictates the need to reduce the measured error on sin 2 0 w to
levels below 10 3 before attempting to search for direct Higgs
production in e+e - annihilation. In fact, precise sin 2 0 w
determination with an error of A sin20w = +0.0003 should be
possible via a measurement of the Left-Right asymmetry (ALR) of the
reaction e+e - ~z~f f using longitudinal polarized beams [10].
3.2 The non-standard neutral Higgs
Due to the shortcoming of the Standard Model more general
theories have been proposed [7], among them the super-symmetry
(SUSY) derived models [13]. In those models one expects at least
two doublets of Higgs fields. The minimal super-symmetric model
(MSSM) has the lowest configuration namely two doublets. These cor-
respond to five physical Higgs states two of which are charged and
three are neutral. Out of these three neutral states, one is a CP =
-1 Higgs the other two are scalar Higgs, H 1 and H 2 with mass
values of MH 1 < Mz < MI~2 (at tree level).
The MSSM neutral Higgs formation in e+e - anni- hilation
followed by a decay to an f f pair (diagram given in Fig. 1 b) is,
as in the SM, proportional to the ratio
r#r# Pr= [rto~]2,
where/'toHt is the total decay width of the Higgs particle and F
f and Fin are respectively partial decay widths to the electrons
and the final state f f . The partial and total decay widths of the
non-Standard Model Higgs may in some cases be significantly higher
than those of the Stan- dard Model Higgs so that they are much more
accessible to an experimental observation. To illustrate the detec-
tion capabilities of a direct produced non-Standard Higgs in its
decay to r + r - or b/; pair, we show in Figs. 8a and 8b curves for
R=~THiggs/~Z,~, =0.1%, 1%, and 5%, as a function of Pr and EcM. As
expected, the better condi- tions for a Higgs detection are far
below or far above the Z mass value, where relative low values of
Pr suffice for adequate detection.
Relation between partial decay widths of the minimal SM Higgs to
that of the MSSM ones show that even though there are situations in
which the individual SUSY particle decay widths are by far larger
than the corre- sponding SM ones, nevertheless the ratio Pr remains
practically the same when the chosen Higgs decay channel is a b#
pair. This is due to the fact that in the case where f is a
b-quark, to a good approximation Pr~Fe/F /and the increasing factor
in the numerator and denominator are equal. This will not be the
case when one considers the integrated cross-section for the
reaction e+e-~b# which is given by the expression
ao
H H H, .~ H H H H f O'(s) ds=l 'e 1' 6 II"tot--F e 1-" b I1"-' b
=re . - -oo
This of course means that the strategy should not be a search
for Higgs peaks but rather a seach for ECM regions where the
integrated number of b# pairs exceeds the ex- pected value from the
(Z, 7) s-channel diagram.
r," r."/(r
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288
4 Discussion and conclusions
The signal of an s-channel Standard Model Higgs for- mation in
e+e - is strongly dependent on its mass value and on its decay
mode. As expected the signal of a Higgs with a mass value in the
vicinity of M z is much too weak to be detected. However, much
stronger signals are ex- pected for masses far away from the Z ~
pole. Clearly for MHigg s below the W + W- threshold a search for
direct formation of Higgs in its decay to b6 state is the favored
one. On the other hand, from the experimental point of view b-quark
tagging is not a simple proposit ion so that the final state r + z
- may turn out to be a better identified channel. At EcM >= 165
GeV obviously the detection of a direct produced Higgs will be best
achieved through its decay to a pair of W + W- or Z~ ~ heavy gauge
bosons [14] which not far away from threshold will have a sig-
nificant higher branching ratio than the b 6 decay channel. Even if
one obtains the condition where the Higgs signal can be detected
above the background coming from the Electro-Weak process, it is
absolutely necessary to use very high luminosity colliders. For
example, if the Higgs mass is 80GeV then sM + - O'Higgs(e e - - *b
/~)=l when integrated over the polar angle range -0 .5 < cos 0
< 0.5. However, this cross section is still higher by one or two
orders of magnitude than the Bremsstrah- lung Higgs production in
the reaction e+e ~b6H[15].
In assessing the signal level of the SM Higgs in e +e-
collisions two additional decisive factors have to be con- sidered.
Namely, the effect of the initial state QED ra- diation and the
energy resolution of the e+e - collider both of which reduce the
height of a resonance like shape and in particular that of the
Higgs due to the fact that /~Higgs is only a few MeV wide. We
estimate that a sup- pression by a factor of 9 to 10 should be
expected from the initial state radiation in the case of the
Standard Model Higgs. In additon, for a typical value of the LEP
energy resolution of 5. 10-4)< ECM a further signal reduction of
~50% is to be expected. For these reasons it is rather important to
recover in part these signal suppression by the use of longitudinal
polarized electron beams which can enhance the relative Higgs
effect up to a factor ~ 5. In this connection it is also worthwhile
to note that some gain in the signal may already be obtained in the
anni- hilation of transverse polarized beams by making use of the
cos (2 qS) dependence of the differential cross section daz, ~/dr'2
which is absent in the Higgs exchange dia- gram.
It is rather obvious that as long as M H < ~fs - -M z, the
Bjorken process is the preferred one for Higgs de- tection.
However, for higher mass values, (M/~ > ]~ - -Mz) , the signal
from a direct produced Higgs is com- parable or even stronger than
that coming from the Bjor- ken mechanism. Thus the search for Higgs
with mass above 100 GeV in their direct production may be a prac-
tical proposition for the physics program at LEP 2, op-
erated with a multi-bunch very high luminosity scheme, where ~
is expected to reach the value of ~ 180 GeV. We furher note that
with the TR ISTAN e+e collider search for Higgs in this method may
be extended to Mu values up to ~ 60 GeV.
As for SUSY, or other types of non-Standard Model Higgs bosons,
the direct production signal may be sig- nificantly higher than
that of the SM Higgs. This signal is best observed when searching
for an excess of b-pairs in an energy interval integrated over the
Higgs width.
Acknowledgements. Many helpful stimulating discussions with the
late Prof. Y. Dothan, concerning the Standard Model and its ap-
plication to the Higgs sector, are much appreciated.
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