First-Order Electroweak Phase Transition in the Standard Model with a Low Cutoff

0.28 MB·English

by  C. Grojean

Saclay T04/084, MCTP-04-37, ANL-HEP-PR-04-63,EFI-04-23
First-Order Electroweak Phase Transition in the Standard Model with a Low Cutoff
Christophe Grojeana,b, G´eraldine Servanta,c,d, James D. Wellsb
(a) Service de Physique Th´eorique, CEA Saclay, F91191 Gif-sur-Yvette, France
(b) MCTP, Department of Physics, University of Michigan, Ann Arbor, MI 48109
(c) High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60539
(d) Enrico Fermi Institute, University of Chicago, Chicago, IL 60637
Westudythepossibilityofafirst-orderelectroweakphasetransition(EWPT)duetoadimension-
sixoperatorintheeffectiveHiggspotential. IncontrastwithpreviousattemptstomaketheEWPT
strongly first-order as required by electroweak baryogenesis, we do not rely on large one-loop ther-
mally generated cubic Higgs interactions. Instead, we augment the Standard Model (SM) effective
theory with a dimension-six Higgs operator. This addition enables a strong first-order phase tran-
sition to develop even with a Higgs boson mass well above the current direct limit of 114 GeV.
The ϕ6 term can be generated for instance by strong dynamics at the TeV scale or by integrating
5
outheavyparticleslikeanadditionalsingletscalarfield. Wediscussconditionstocomplywithelec-
0
troweak precision constraints, and point out how future experimental measurements of the Higgs
0
2 self couplings could test theidea.

n
a Baryogenesis and the Standard Model: The ob- processes in the broken phase would actually require
J
served large baryon asymmetry requires natural law to mH < 35 GeV. However, the current limit on the Higgs
2
obey three principles: baryon number violation, C and boso∼nmassiswellabovethatatm >114GeV[11],and
1 H
CP violation, and out-of-equilibrium dynamics [1]. In the SM fails to be an adequate theory for baryogenesis.

2 the Standard Model (SM), baryon number violation can As the hopes for a SM solution to baryogenesis faded
v occurthroughtheelectroweaksphaleron[2,3],whichisa other ideas have been pursued [12]. One of the most
9 non-perturbative saddle-point solution to the field equa- promising ideas presented in the last decade is from su-
1
tions attainable at high temperatures. These solutions persymmetry. If the superpartner to the top quark is
0
allow transitions to topologically distinct SU(2) vacua lighter than about 150GeV, a first-order EWPT can be
7
0 with differing baryonnumber. inducedfromlarge-enoughcubicinteractionsintheHiggs
4 potential. This scenario is getting a thorough test as
C is already violated in the SM as well as CP, as ev-
0 searchesforthelighttopsuperpartnerarerapidlyclosing
idenced in the Kaon and B-meson systems. Neverthe-
/
h less, it has been thought [4] that CP violation from the the viable parameter space for this solution [13]. Recent
p ideasto extendthe particlespectrummay helpresurrect
Kobayashi–Maskawa phase is too suppressed to play a
-
electroweak baryogenesis in supersymmetry [14].
p dominantroleinbaryogenesis,althougharecentwork[5]
e suggestsawaytocircumventthiscommonview. Wenote
h Low-scale cutoff theory: In this work, we focus on
alsothathigherdimensionaloperatorscouldwellprovide
v: the desired CP violation [6]. a single Higgs doublet model and we study how the dy-
i namics of the EWPT can be affected by modifying the
X In this letter, we focus on the last main challenge for
SMHiggsself-interactions. Incontrastwithpreviousap-
r the viability of SM baryogenesis [7]: the requirement of proaches initiated by ref. [15], we do not rely on large
a
out-of-equilibrium dynamics. This would be present in
cubic Higgs interactions. Instead, we allow the possibil-
the SM if there was a strong first order EWPT. In this
ity of a negative quartic coupling while the stability of
case, bubbles of the non-zero Higgs field vev nucleate
thepotentialisrestoredbyhigherdimensionaloperators.
from the symmetric vacuum and as they expand, parti-
We adda ϕ6 non-renormalizableoperatorto the SM po-
cles in the plasma interact with the phase interface in
tential, and show that it can induce a strong first-order
a CP-violating way. The CP asymmetry is converted
phase transitionsufficient to drive baryogenesis[16]. We
intoabaryonasymmetrybysphaleronsinthesymmetric
havenumericallycheckedthataddinghigherorderterms
phaseinfrontofthebubblewall[8]. Oneofthestrongest
in the potential suppressed by the same cutoff scale will
constraints on EW baryogenesiscomes from the require-
give corrections of a few percent at most to the ratio
ment that baryons produced at the bubble wall are not
ϕ(T ) /T that we computed analyticallywhile restrict-
c c
washed out by sphaleron processes after they enter the h i
ing ourselves to operators of dimension six or less.
brokenphase. Imposingthatsphaleronprocessesaresuf-
The most general potential of degree six can be writ-
ficiently suppressed in the broken phase at the critical
ten, up to an irrelevant constant term, as
temperatureleadstotheconstraint ϕ(T ) /T >1. This
c c
bound is very stable with respect toh modiificati∼ons of ei- v2 2 1 v2 3
ther the particle physics or of the cosmologicalevolution V(Φ)=λ Φ†Φ + Φ†Φ (1)
− 2 Λ2 − 2
as was reviewed in [9]. In the SM, the EWPT is first (cid:18) (cid:19) (cid:18) (cid:19)
order if m < 72 GeV [10] and to suppress sphaleron where Φ is the SM electroweak Higgs doublet. At zero
H
2
temperaturetheCP-evenscalarstatecanbeexpandedin ϕ=0 is degenerate with that at ϕ=0 is
6
terms of its zero-temperature vacuum expectation value
Λ4m4 +2Λ2m2 v4 3v8
ϕ = v 246GeV and the physical Higgs boson H: T2 = H H 0 − 0. (3)
h i 0 ≃ c 16cΛ2v4
Φ=ϕ/√2=(H +v )/√2. 0
0
At zerotemperature we canminimize eq. (1) to find λ The vacuum expectation value of the Higgs field at the
and v in terms of physical parameters mH and v0. We critical temperature in terms of mH, Λ and v0 is
find two possibilities 3 m2 Λ2
ϕ2(T ) =v2 = v2 H . (4)
h c i c 2 0 − 2v2
0
case m2 v2 λ
H We can see from eqs. (3) and (4) that for any given m
1 any mH v02 m2v2H2 >0 thereisanupperboundonΛtomakesurethatthephasHe
0 transitionis firstorder (i.e. v2 >0), and there is a lower
2 m2 < 3v04 v2 1 2Λ2m2H m2H <0 c
H 2Λ2 0 − 3v04 −2v02 bound on Λ to make sure that the T = 0 minimum at
(cid:16) (cid:17) ϕ = 0 is a global minimum (i.e. T2 > 0). These two
6 c
combine to give the important equation
Note that, up to an irrelevant constant, the potential is
unchanged by the parameter transformation: λ λ v2 √3v2 v2
and v2 v2(1 4Λ2λ/(3v2)). So, Case 2 is ac→tua−lly max 0 , 0 <Λ<√3 0 (5)
physical→ly equiva−lent to Case 1. And in the rest of the mH m2H +2m2c! mH
gplaopbearl,mwienimreustmricotftohuerspeolvteesnttiaolλas>lon0g.asϕm=2H v>0vi04s/Λth2e, twhhaetrtehmeccoe=ffivci0ent(4cpyint2+th3egt2he+rmg′a2l)/m8as≈si2s0p0osGiteivVe.ifNaontde
p
otherwise ϕ = 0 is a deeper minimum. The dynamics only if Λ>√3v2/ m2 +2m2. Thus when Λ saturates
0 H c
of the EWPT depends only on the values of mH and Λ. the lower bound in eq. (5), the critical temperature is
p
It should be noted that, for the region of the parameter either vanishing orinfinite when m is smaller orbigger
H
space where a first order EWPT occurs, the value of the than m respectively. At m =m and Λ=v2/m , the
c H c 0 c
quarticcouplingattheoriginwillbenegative,incontrast criticaltemperatureisnotuniquelydefinedbutthisisan
to the SM scenario. artifactofourapproximations. Aroundthatpointhigher
We approximate finite temperature effects by adding ordertermsinthethermalpotential,liketheT2ϕ4/(4Λ2)
a thermal mass to the potential V(ϕ,T) = cT2ϕ2/2+ terms or Tϕ3 terms mentioned earlier, will resolve the
V(ϕ,0), where c is generated by the quadratic terms singularity. These higher order terms will in particular
(T2m2 where i denote all particles that acquire a ϕ- give corrections to the bounds (5) delineating the first
i
dependentmass)inthehigh-T expansionoftheone-loop order phase transition region.
thermal potential Figs. 1 and 2 plot contours of constant T and v /T ,
c c c
respectively, in the Λ vs. m plane. These results are
H
1 m2 v2
c= 4y2+3g2+g′2+4 H 12 0 , (2) encouragingandmotivateafullone-loopcomputationof
16 t v2 − Λ2 the thermal potential. Such an analysis is underway.
(cid:18) 0 (cid:19)
g and g′ are the SU(2) and U(1) gauge couplings, Sphaleron solution: We compute the sphaleron so-
L Y
andy isthetopYukawacoupling. TheT2m2 termsalso lution of this effective field theory, eq. (1), by starting
t i
generateaT-dependentcontributiontotheHiggsquartic with the ansatz [3]
coupling of the form T2ϕ4/(4Λ2). In the following, we
2i v 0
have discarded this contribution to keep our analytical Waσadxi = f(ξ)dUU−1, φ= 0 h(ξ)U ,
study simple. We havecheckedthatitdoes notalterour i −g √2 1!
resultsbymorethanafewpercentinthephysicallyinter-
esting regionwhere a strongly first-order EWPT occurs. 1 z x+iy
where ξ = gv r and U = and,
0
There is also a cubic Higgs interaction induced by finite r x+iy z !
−
temperature effects (crucial in supersymmetric baryoge- as usual, v 246GeV. We compute only the SU(2)
0
≃
nesis) but it has a smaller role in our discussion, and sphaleron as corrections from U(1) are expected to be
Y
should tend to make the EWPT slightly stronger first- small [3, 18]. The functions f and h are solutions to two
order. While perturbation theory is expected to break coupled nonlinear differential equations:
downathightemperature,itsvalidityhasbeenconfirmed
d2f ξ2
by lattice calculations in the regime where the EWPT is ξ2 = 2f(1 f)(1 2f) h2(1 f)
dξ2 − − − 4 −
stronglyfirst-order,intheSM[10]aswellasinitssuper-
d dh λ
symmetric extension [17]. We therefore expect that the ξ2 = 2h(1 f)2+ ξ2(h2 1)h
value of ϕ(T ) /T givenby ournaive tree levelanalysis dξ dξ − g2 −
h c i c (cid:20) (cid:21)
be not too different from its actual value. 3 v2
+ 0 ξ2h(h2 1)2
The critical temperature Tc at which the minimum at 4g2Λ2 −
3
FIG. 1: Contours of constant Tc from 0 to 240 GeV. The
shaded blue region satisfies the bounds of eq. (5). Above it,
the EWPT is second order and the critical temperature is
FIG. 3: Sphaleron energy at zero temperature in units of
no more given by eq. (3) but instead by Tc2 = (2Λ2m2H − 4πv /g=4.75 TeV.
3v4)/4cΛ2. 0
0
partpermillionbeforedeclaringthatasolutionhasbeen
obtained. This procedure is computer intensive.
Afterobtainingthesphaleronsolutionwecomputethe
sphaleron energy at T = 0 (shown in Fig. 3) according
to the equation
4πv ∞ 8 1
E = 0 dξ 4f′2+ f2(1 f)2+ ξ2h′2+
sph g ξ2 − 2
Z0 (cid:18)
λ v2
h2(1 f)2+ ξ2(h2 1)2+ 0 ξ2(h2 1)3 (.6)
− 4g2 − 8g2Λ2 −
(cid:19)
ItdiffersfromtheSMvaluebythelastterm,whichtends
to make the sphaleron energy slightly smaller (by only a
few percent). A similar conclusion was also reached in
FIG.2: Contoursofconstantvc/Tcfrom1to∞. Theshaded
the MSSM [19].
blueregion satisfies theboundsof eq.(5).
The sphaleron energy is a crucial quantity for EW
baryogenesis as the rate of baryon number viola-
subject to the boundary conditions f(0)=h(0)=0 and tion in the broken phase at Tc is proportional to
f( ) = h( ) = 1. To solve these differential equations e−Esph(Tc)/Tc [20]. Esph(Tc) is approximately given by
∞ ∞
it is necessary to first expand the solutions about their eq.(6)wherev0 isreplacedbyvc,andthisishowrequir-
asymptotic values as ξ 0 and ξ : ingthatsphaleronprocessesbefrozenleadstothebound
→ →∞ v /T > 1. Knowing whether the right-hand side of this
c c
f(ξ →0) = ξ2/a20 , h(ξ →0)=ξ/b0 inequa∼lity is 1 or 1.5 is crucial in deriving the resulting
f(ξ ) = 1 a exp( ξ/2) boundontheHiggsmass,andthisdepends,amongother
∞
→∞ − −
things, on the precise sphaleron energy. The fact that
h(ξ ) = 1 (b /ξ)exp( σξ)
∞
→∞ − − E is largerthanthe cutoff scalefora first-orderphase
sph
where σ 2λ/g2 and a ,a ,b and b are constants transition is not inconsistent with the calculation of the
0 ∞ 0 ∞
≡
to be determined. rate of baryon number violation at T . Indeed, E is
p c sph
Solving for the constants (a ,b ,a ,b ) is equivalent large because the sphaleron is an extended object, but
0 0 ∞ ∞
to solving the differential equations and the boundary its local energy density is always smaller than the cutoff
conditions. Wesolvethembychoosingrandomnumerical scale. While a largeamountofenergyhas to be pumped
values for a ,a ,b and b , shooting the solution from intothethermalbathtobuildasphaleronconfiguration,
0 ∞ 0 ∞
ξ = down to some intermediate ξ = ξ and also this does not involve any local physics beyond the cutoff
fit
∞
from ξ = 0 up to ξ = ξ . If both equations match scale.
fit
at ξ a solution has been found. In practice, we set
fit
up a χ2 fit to measure goodness of match, and require Precision electroweak constraints: Thetheorywe
that h, f and their derivatives match to better than one havepresentedaboveistheSMwithalow-scalecutoff. It
4
isminimalinthatnonewparticleshavebeenintroduced the operators can have substantially different conformal
toachievethedesiredout-of-equilibriumfirst-orderphase weights if the theory at the cutoff is a strongly coupled
transition needed for baryogenesis. However, this does theorywhereeachfieldgetslargeanomalousdimensions.
notmeanthatthe phenomenologyofthis model is indis- Perhaps this distinguishing property of the operators is
tinguishable from that of the SM. a key to the needed hierarchy.
The non-renormalizable operators of this theory can As a concrete example of a possible origin of the non-
significantly affect observables. If the only additional renormalizableHiggsself-interaction,wenotethata Φ6
| |
terms are those given by eq. (1), there would be no termcanbegeneratedbydecouplingamassivedegreeof
phenomenological constraints on this scenario to worry freedom. Forinstance,inamannersimilartoref.[14]we
about. However, a low-scale cutoff for other dimension- can consider a scalar singlet φ coupled to the Higgs via
s
six operators can be problematic for precision elec-
1 1
troweak observables [21]. As an example, let us consider ∆V = m2φ2+mφ Φ†Φ+ aφ2Φ†Φ. (8)
2 s s s 2 s
the followingfour dimension-sixoperatorssuppressedby
the cutoff scale Λ: Assuming that the mass of the singlet is higher than the
ǫ ǫ ǫ weak scale, integrating out this scalar degree of freedom
∆ = Φ(Φ†D Φ)2+ W(D Wa )2+ B(∂ B )2
L Λ2 µ Λ2 ρ µν Λ2 ρ µν givesrisetotheadditionalHiggsinteractions(wediscard
+ǫFν¯ γ P µe¯γαP ν . (7) for simplicity other terms, like Mφ3s or βφ4s, that could
Λ2 µ α L L e also be added, since they generically will not change the
conclusions):
The most sensitive precision electroweak observ-
aanbdlesΓZar.e sTinh2eθWeffp,ercmenWt ,shΓiflts =to tΓh(eZse o→bserlv+abl−le)s, Vnew =−2mm22|Φ|4+ a2mm42|Φ|6+O a2mm46|Φ|8 . (9)
∆ = sin2θeff,m (GeV),Γ (MeV),Γ (GeV) in- s s (cid:18) s (cid:19)
Oi { W W l Z }
duced by ∆ are We assume that m and m are of the same order to be
s
L
able to neglect the higher-order terms in the expansion.
8.57 6.19 1.47 4.29 ˜ǫΦ Therefore,ifthemassscaleinthesingletsectorisaround
−
∆ 4.31 0.55 0.55 0.65 ˜ǫ a TeVa φ6 termas wellasa negativeφ4 termaregener-
% O = − − − −  W 
7.20 1.69 0.93 3.61 ǫ˜ atedin the Higgs potential. A smallfine-tuning between
(cid:18) O (cid:19)i  − −  B  the singlet-induced negative quartic coupling of order 1
 7.90 1.00 1.08 3.93 ǫ˜ 
 − −  F  and the initial positive quartic coupling of the Higgs po-
  
where ǫ˜ = ǫ (1TeV)2/Λ2. The experimental measure- tential would be needed to produce a total quartic cou-
i i
ments of these observables are [11] pling of order 0.1, as is required to be sitting in the
∼ −
desiredregionofthe(m ,Λ)plane. Meanwhile,thecus-
H
sin2θeff = 0.23150 0.00016 todialinvariantinteractionsofeq.(8)willnotleadtoany
W ±
m (GeV) = 80.426 0.034 of the dangerous operators eq. (7).
W
±
Γ (MeV) = 83.984 0.086
l ± Higgs self-couplings as test: Future colliders have
ΓZ(GeV) = 2.4952 0.0023 the opportunity to test this idea directly by experi-
±
mentally probing the Higgs potential. When a low-
We can compare the experimental values of the observ-
scale cutoff theory alters the Higgs potential with non-
ables with the dimension-six operator shifts induced by
renormalizableoperators,those same operatorswill con-
the cutoff scale Λ. The (Φ†D Φ)2 operator appears to
µ tribute to a shift in the Higgs self-couplings. Expand-
have the most substantial effect on the precision elec-
ing around the potential minimum at zero tempera-
troweak observables. This operator is a pure isospin
ture we can find the physical Higgs boson self couplings
breaking operator and is equivalent to a positive shift
( =m2 H2/2+µH3/3!+ηH4/4!+ )
in the T parameter in the Peskin–Takeuchi framework L H ···
(T 7.8˜ǫΦ). m2 v3 m2 v2
B≃ar−ring some nontrivial cancellations of multiple ǫi µ=3 vH +6Λ02, η =3 v2H +36Λ02. (10)
contributionsto the precisionelectroweakobservables,it 0 0
appears that ǫ <10−2 is necessary if Λ<1TeV. The The SM couplings arerecoveredas Λ . In Fig.4 we
Φ
other ǫ values ar∼e somewhat less constrain∼ed, but likely plot contours of µ/µ 1 in the Λ v→s. m∞ plane.
i SM H
−
need to be nearly as small also. Therefore, if this frame- No experiment to date has meaningful bounds on the
work is to be viable there must be a small hierarchy be- H3 coupling. It is estimated that for the Higgs masses
tween the 1/Λ2 coefficient of eq. (1) and the ǫ /Λ2 coef- in the range needed for the first-order phase transition
i
ficients of eq. (7). In the absence of a UV completion of presentedabove,ameasurementoftheH3couplingcould
the theory, this little hierarchy of high-dimensional op- be done to within a factor of one at the LHC at √s =
erators remains unexplained. We note in passing that 14TeV with 300fb−1 integrated luminosity [22]. This
5
[1] A.D.Sakharov,PismaZh.Eksp.Teor.Fiz.5,32(1967).
[2] N. S. Manton, Phys.Rev.D 28, 2019 (1983).
[3] F. R. Klinkhamer and N. S. Manton, Phys. Rev. D 30,
2212 (1984).
[4] M. B. Gavela, P. Hernandez, J. Orloff, O. Pene and
C. Quimbay,Nucl. Phys. B 430, 382 (1994).
[5] M. Berkooz, Y.Nir and T. Volansky,hep-ph/0401012.
[6] M.Dine,P.Huet,R.J.SingletonandL.Susskind,Phys.
Lett. B 257 (1991) 351.
[7] V.A.Kuzmin,V.A.Rubakovand M.E. Shaposhnikov,
Phys. Lett. B 155, 36 (1985).
[8] Firstdiscussionsofthiscommonlyacceptedscenariocan
befoundinA.G.Cohen,D.B.KaplanandA.E.Nelson,
FIG. 4: Contours of constant µ/µSM −1 in the Λ vs. mH Phys. Lett. B 245, 561 (1990).
plane. Thedashed lines delimit theallowed region definedin [9] G. Servant,JHEP 0201, 044 (2002).
eq. (5). [10] K.Kajantie,M.Laine,K.RummukainenandM.E.Sha-
poshnikov, Phys. Rev. Lett. 77, 2887 (1996); K. Rum-
mukainen, M. Tsypin, K. Kajantie, M. Laine and
constraint or measurement would be an interesting one M. E. Shaposhnikov, Nucl. Phys. B 532, 283 (1998);
for our scenario since a deviation by more than a factor F. Csikor, Z. Fodor and J. Heitger, Phys. Rev.Lett. 82,
of unity is possible. 21 (1999).
In the more distant future, a linear collider at √s = [11] LEP Collaborations, LEP Electroweak Working Group,
500GeV and 1ab−1 of integrated luminosity should be SLD Electroweak and Heavy Flavour Groups, “A com-
bination of preliminary electroweak measurements and
able to measure the coupling to within about 20% [23],
constraints on thestandard model,” hep-ex/0312023.
and a higher energy linear collider, such as CLIC with
[12] A. Riotto and M. Trodden, Ann. Rev. Nucl. Part. Sci.
√s=3TeV and 5ab−1 integrated luminosity, should be 49, 35 (1999).
able to measure the self-coupling to within a few per- [13] M. Carena, M. Quir´os and C. E. M. Wagner, Phys.
cent [24]. A few-percent measurement may also be pos- Lett. B 380, 81 (1996); J. M. Cline and G. D. Moore,
sible at the VLHC at √s = 200TeV with 300fb−1 inte- Phys.Rev.Lett.81(1998) 3315; M.Carena, M.Quir´os,
M. Seco and C. E. M. Wagner, Nucl. Phys. B 650, 24
grated luminosity [22].
(2003).
[14] J. Kang, P. Langacker, T. j. Li and T. Liu,
Conclusion: We have shownthat a strongfirst-order
hep-ph/0402086. A. Menon, D. E. Morrissey and
electroweak phase transition is possible within the SM
C. E. M. Wagner, hep-ph/0404184.
whenwetakeinto considerationthe effects ofaϕ6 Higgs [15] G. W. Anderson and L. J. Hall, Phys. Rev. D 45, 2685
operator with a low cutoff. Higgs masses well above the (1992).
114 GeV direct limit are possible within this framework. [16] Another analysis of this approach can be found in
The main experimental test of this idea is the altered X. m. Zhang, Phys. Rev. D 47, 3065 (1993). X. Zhang,
B.L.YoungandS.K.Lee,Phys.Rev.D51,5327(1995).
Higgs cubic self-coupling. The LHC should be able to
[17] K.Kajantie,M.Laine,K.RummukainenandM.E.Sha-
probe (1) corrections,but a high-energy linear collider
O poshnikov, Nucl. Phys. B 466, 189 (1996); F. Csikor,
will likely be required to measure the deviation at the
Z. Fodor, P. Hegedus, A. Jakovac, S. D. Katz and
tens of percent level accurately. A. Piroth, Phys. Rev. Lett. 85, 932 (2000). M. Laine
and K.Rummukainen,Nucl.Phys. B 597, 23 (2001).
[18] B.Kleihaus,J.KunzandY.Brihaye,Phys.Lett.B273,
100 (1991); Phys. Rev.D 46, 3587 (1992).
WethankJ.Cline,J.R.Espinosa,A.Hebecker,A.Nel-
[19] J.M.Moreno,D.H.OakninandM.Quir´os,Nucl.Phys.
son,M.Quiro´sandC.Wagnerforusefulcomments. This
B 483, 267 (1997).
work was supported by the Department of Energy and [20] P. Arnold and L. D. McLerran, Phys. Rev. D 36,
theMichiganCenterforTheoreticalPhysics. C.G.issup- 581 (1987). L. Carson, X. Li, L. D. McLerran and
ported in part by the RTN European Program HPRN- R. T. Wang, Phys. Rev.D 42, 2127 (1990).
CT-2000-00148 and the ACI Jeunes Chercheurs 2068. [21] V.Barger,T.Han,P.Langacker,B.McElrathandP.Zer-
G.S.wassupportedinpartbytheUSDepartmentofEn- was, Phys. Rev.D 67, 115001 (2003)
[22] U. Baur, T. Plehn and D. L. Rainwater, Phys. Rev. D
ergy, High Energy Physics Division, under contract W-
68, 033001 (2003)
31-109-Eng-38andalsobytheDavidandLucillePackard
[23] C. Castanier, P. Gay, P. Lutz and J. Orloff,
Foundation.
hep-ex/0101028.
[24] M.Battaglia,E.BoosandW.M.Yao,[hep-ph/0111276].