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Gauge boson masses dominantly generated by Higgs-triplet contributions?
Physics Letters B 306 (1993) 335-342 North-Holland
P H YSIC S I_ETT ER S B
Gauge boson masses dominantly generated by Higgs-triplet contributions? Peter Bamert Instttut de Phystque, Untversttk de Neuchdtel, CH-2000 Neuchdtel, Swuzerland
and Zoltan Kunszt Instttute of Theorettcal Phystcs, ETH, CH-8093 Zftrlch, Switzerland
Received 16 March 1993 Editor: R. Gatto
We discuss a model in which the Standard Model (SM) Higgssector has been extended by additional real and complextriplets. The p ~ 1 constraint is satisfied by restricting the potential to have an enlarged SU (2)L® SU (2)R global symmetry. This is fine tuning, which leads to a decreased predlctabthty in next-to-leadingorder. In this model, however, the triplet vacuum expectation values may give the dominating contribution to the gauge boson masses Using a renormalization group argument we constrain this region of the parameter space. Another mterestmg feature of this model is that one of the neutral scalars does not couple to the fermion sector at tree level and therefore could have a relatively largebranching ratio to 2y's It ~scoupled, however, to the Zboson and therefore it could be produced at LEP via the standard e +e---,Z~ mechanism with rates comparable to the ones of the Standard Model.
The Standard Model (SM) of electroweak and strong interactions successfully describes all elementary particle physics phenomena. Recently, it has been successfully confronted with the precision measurem e n t s at LEP [1 ]. The Higgs sector, a n d so the m e c h a n i s m of the electroweak symmetry breaking, however, is weakly constrained by the data. The most i m p o r t a n t constraint is given by the measured value of the rho parameter:
P=
eos2(0w)'
(1)
/gexp = 1.003 + 0.004, where Pexo is a fit of experimental data [2 ] with Mtop = 100 GeV. Models involving only SU (2)L doublets a n d singlets, satisfy in a natural way the p ~ 1 condition. This is due to the fact that in the SM the S U ( 2 ) L ® U ( 1 ) r gauge symmetry forces the scalar potential to exhibit a SU ( 2 ) L@ SU (2) R global symElsevier SciencePublishers B.V.
metry. Breaking the gauge symmetry the Higgs potential still exhibits an S U ( 2 ) c (custodial) symmetry which ensures p = 1 at tree level. Because of this, SU (2) L® SU (2) R violating radiative corrections to the Hlggs potential are finite and small. This also means that the rho parameter is close to one naturally, without fine tuning parameters. If we want to allow also for Higgs triplets or even higher multlplets, the value of the measured rho parameter becomes a strong constraint. One obvious posslbihty is that we d e m a n d the triplet v a c u u m expectation values (VEVs) to be small. This requires fine t u n i n g a n d an effective decoupling of the triplets from gauge bosons a n d fermions. Alternatively, we would require that the scalar potential, even with these higher multlplets, will remain i n v a r i a n t with respect to the enlarged S U ( 2 ) L N S U ( 2 ) R global symmetry [3,4]. For triplets, however, the S U ( 2 ) L @ S U ( 2 ) R symmetry of the Higgs potential will not be obtained automatically as a result of the gauge 33 5
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symmetry. It can only be achieved by fine-tuning the parameters of the Hlggs potential. Therefore, the pparameter in such models will suffer from a fine tuning problem and will depend "directly" on the fine tuned parameters of the scalar potential. In the SM model the p parameter is protected from this problem by the custodial symmetry. With fine tuned S U ( 2 ) L ® S U ( 2 ) ~ symmetry, however, the higher dimensional multiplets may generate a major part of the W and Z masses. Furthermore, in these models the doublet Yukawa couplings could be strongly enhanced with respect to the SM, indicating an interesting phenomenology. In this note we shall discuss the phenomenological viability of this scenario. The most straightforward extension of the SM Higgs sector, realizing the above situation, has been presented by Georgi and Machacek [ 3 ] who added a complex and a real triplet to the SM doublet. With this content of the scalar sector it is indeed possible to restrict the potential to a SU (2) e® SU (2) R symmetric version, as has been investigated in some detail by Chanowitz and Golden [ 4 ]. More recent papers [5-7] have taken a further look at the renormalization problems [ 7 ] and the phenomenology [5,6] of this model. From the tree level point of view the Higgs sector presents itself as follows [ 3 ]:
q~=_~-
4o
,
~-
zO* /
,
(2)
denoting a (½, ½) and a ( 1, 1 ) multiplet of SU (2) L ®SU(2)R, with hypercharge assignments of Y= 1, 0 and 2 for the doublet, the real and the complex triplet respectively and with the following phase conventions: q~-=--~+*, Z - = - Z +*, ~ - = - ~ + * , Z - - = Z + +* and C° = ~o.. They transform under SU (2) L® SU(2)R global transformations as CI)~ULq)U~ ( q ~ X ) , with UL ((JR) being the transformation matrices in the appropriate representation. The restricted Higgs potential takes the form [ 4 ]
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v(4~, x ) =21(Tr[~*~]-a2)2+22(Tr[X*X]-3b2)
2
+23 (Tr [ • *(/)] -aE+Tr[XtX] - 3 6 2 ) 2 +24(Tr[ rib¢~] Tr [X'X]
- 2 Yr[ q~*T'q~T j] Tr[ X*T'XTJ] ) +25(3 Tr[X*XXtX] - T r [X'X] 2) ,
(3)
where the sum is taken over z andj. The vacuum states [ q o > o = ( a / v / 2 ) l and [ X ) o = b l are defined by V( I q0)o, IX>o) = 0 together with the positivlty conditions [ 4 ] 21+22+223>/0 24)0
,
,
2122 "}-2123 "["2223 ) 0 ,
(4)
25)0.
The two VEVs a and b are related through the Wmass,
MZw=M~cos2(Ow)=¼g2(a2+8b2)=¼g2v 2 ,
(5)
with g being the SU(2)L coupling constant and v ( ~ 250 GeV) the VEV of the SM Higgs. This allows the definition of a mixing angle On denoted by its cosine c•, sine s/t (and tangent t ~ - t a n ( 0 n ) ) a c/~= - ,
/2
2v/2b sH=--
(6)
V
Within the Hlggs sector the gauged SU (2)L®U ( 1 ) r can be regarded as a subgroup of SU(2)LNSU(2)R with T 3 playing the role of the hypercharge. A diagonal subgroup of SU(2)LNSU(2)R survives the spontaneous symmetry breakdown. The physical Higgs bosons will form multiplets, which are degenerate in mass, under this custodml S U ( 2 ) c global symmetry. Expressed in terms of the fields defined in eq. (2) they are H++ =Z++, Z +Z_ ~- -'
H o= Z°+Z°*-2~ ° '
Ho,= )~°+Z°*+ ~° , H + = ci_iZ+ ~ - s+~ ~ r ~+
336
+,
(7)
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PHYSICS LETTERS B
Z ° _ Z °* 0o_0o. ~ +SH--
Hoif + - ~ M++ tH75 ,
~o+~o.
H?=
,/5
(7 c o n t ' d )
,
w i t h / / 5 , / / 3 , H1 and H1, denoting a S U ( 2 ) c fiveplet, triplet a n d 2 singlets respectively. The G o l d s t o n e triplet is orthogonal to the H3-plet
G~ = s Z- -+~+~+ - - +c++O+,
~,o_ ,,o.
0o_0o.
In terms o f those fields one sees that the quantity cu defined in eq. (6) i n d e e d denotes the cosine o f a S U ( 2 ) L multiplet rotation to S U ( 2 ) c triplets. The masses are ~1 [4] i~,
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H f ~e: Mf H+
1 + 75
tHVpn[
1 +Ys ( 10 c o n t ' d )
where the _+ in the second t e r m refers to up, down type fermions, p and n denote up and down type quarks a n d v and e stand for n e u t r m o type a n d electron type leptons respectively. Vp, is the C K M mixing matrix. In the following let V denote either a W or a Z boson. The VVH type vertices o f this m o d e l are ( o m i t t i n g an overall factor igg ~ ) [ 3 ]:
W + W + H ~ - : V/2 MwSH, W+ W_HO: _ M q ~w s . ,
= 3vZ(c~24 + s 2 2 5 ) ,
M ~ = v2~,.,
M2"H"=
W ~X/~CHSH23
~SS(2Z + 2 3 ) , / "
(9)
It is convenient to rotate the two S U ( 2 ) c slnglets to their mass eigenstates/7 o and/70,. The rotation angle shall be d e n o t e d b y its cosine cs a n d sine Ss./70 (/7o,) defines the mass eigenstates with the larger SU ( 2 ) c doublet ( H ° ) (triplet ( H ° , ) ) p o r t i o n ~2 Here we consider only Dirac type Yukawa couplings ~3. O f course, only those fields ofeq. (7) which have a nonzero doublet admixture will couple (at tree level) to the fermion sector. D u e to a 4 v the Yukawa couplings are enhanced c o m p a r e d with the SM. They are ~4 ( o m i t t i n g an overall factor ig).
HOlff.
My 1 2MwcH'
(10)
~1 Note that the pomtlvity of the squared masses is guaranteed byeq. (4). ~2 Note that by means of th~s definitmn ss takes values between - 1/,£5 and + l / q 2 ~3 Majorana type Yukawa couphngs induced by the SU(2)L triplets can be introduced but are unhkely to be phenomenologically important unless tH<< 1 [6]. ~4 For mmphcity we have assumed no smglet mixing (Cs=1) m the following expresmons.
+
o 2x/~ W-Hv: ~Mws,,
W + W - H °: MwCH,
ZZH°~
2 Mw
~ ~-,
zzH°,: 2,,/,%~ M ~w ~ , ZZH°I : ~ w c~, Mw W-ZH~: - --s,. Cw
(11)
Again the H toH t o, mixing has been taken to be zero. The absence o f H ° VV couplings is due to H ° being CP-odd. The 7 p a r a m e t e r s o f the restricted potential (eq. ( 3 ) ) can be reexpressed in terms o f the W-mass, the 4 Hlggs masses, the singlet mixing (cs) and t u
(=s,dcH). Let us first have a b n e f look at the limit "tu<< 1"
(b<
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has SM Hff and HVV and suppressed HHV couplings, whereas all other Higgs bosons have suppressed couplings to the fermion sector (eq. (10)) and to the HVVsector. The whole situation therefore resembles very much the SM with H ° playing the role of the SM Higgs boson. The main differences come from the non vanishing HHVtype couplings ~5, which can serve to bound the masses of the other (not H ° ) Higgs fields from below, and the fact that the other singlet (H°,) is hkely to be very light: M2,,..~
3U2S2(2122"~2223"~2123) / (21-'1-23). The opposite limit "t~>> 1" (a << b) is far more interesting, because here the SU (2)L triplet fields (X) are the source of the SSB, generating the W and Z masses. Again there is almost no singlet mixing: 8 e2=l--~[23/(22+23)]2tH2+O(tH4). Here we have the remarkable situation that S U ( 2 ) c multiplets either couple to the fermion sector (H1, H3), those couphngs being strongly enhanced with respect to the SM, or to the HVVsector (Hv, Hs), with couplings bemg roughly of the same order of magnitude as the SM ones, but not both. Again there is a field which is likely to be lighter than the others: M 2, 4c2(2 122 + 2 2 2 3 " } - 2 1 2 3 ) / ( 2 2 - ] - 2 3 ) . For medium values of tz4 ( ~ O ( 1 ) ) there can be significant HIHv o o mixing allowing the mass eigenstates/7 o and/7o, to have SM like couplings in both the fermmn and the HVV sector. As in the previous case H3 couples to the fermion sector while//5 has HVVcouphngs, but not the other way round. The restricted potential (eq. (3)) emerges after the additional parameters of the general SU (2) L® U ( 1 ) r invariant potential ~6 have been fine tuned to be zero ~7on a certain mass scale (e.g. on the Mm shell). However, due to the running of those parameters this fine tuned custodtal symmetry will not be exact anymore at another mass scale (e.g. the triplet mass shell Mm ). It is therefore reasonable to understand the term fine tuned custo&al symmetry as an approximate symmetry, which is valid only at tree level. Fine tuning in our case means that the model looses its #5 For a hst of the HHVtype couplings see e g. ref. [ 6 ]. #6 For the exphclt form of this potentml see e.g ref. [7 ]. #7 Note that one cannot simply ampose that those terms do not exist, because they are automatmally introduced by means of counterterms after the renormallzatlon procedure.
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predictive power at the next-to-leading order level #s At this point one might be tempted to supersymmetrize the model. In fact, supersymmetry provides a solution to those fine tuning problems since it ensures that the parameters of the superpotential, from which the scalar potential derives, do not receive radiative corrections. This is due to cancellations among various diagrams (Non-Renormahzatton Theorems). After supersymmetry has been softly broken those parameters will only receive logarithmic corrections. An extension of the supersymmetric SM containing one (necessarily complex) triplet with zero hypercharge ~ has been recently studied by Esplnosa and Quir6s [ 8 ]. In this model, however, the triplet VEV has to be kept small to fulfill the constramt p ~ 1. If we want to have large triplet VEVs, then at least one chlral triplet with nonzero hypercharge has to be added as becomes clear from the tree level relation P = ~{~ ] (q~)o ]2[TL(TL+ 1 ) _ (½Y)21 ~{q,) 2 ] (~/1)0 ] 2(½ y ) 2
,
(12)
where the sum goes over all (complex) multiplets, TL denotes the largest eigenvalue of the SU (2) e generator T3 in the appropriate representation and Y is the hypercharge. Adding e.g. a 2,.. (3, - 2) to {: ( 3, 0 ),/7: (2, 1) a n d H : (2, - 1 ) [(a, b ) ~ S U ( 2 ) L ® U ( 1 ) r ] would require ] (~)ol = ] ( Z ) o l / x / ~ for p = 1. But since Z misses its "counterpart" (3, 2) the Hlggsinos remain massless and give rise to a chiral anomaly. One then has to introduce a third chlral triplet 2: (3, 2). The conditions f o r p = 1 are now I (Z)o] 2+ ] (;~)o 12 = 2 ] ( ~ ) o] 2, resembling, in a special case, the non supersymmetrlc model: (Z) o= (;g) o = ( { ) o = b. In this model the scalar Higgs sector has 23 physical degrees of freedom corresponding to 2 doubly charged, 5 singly charged and 9 neutral scalars. The most general gauge invariant superpotential has 8 parameters (corresponding to the singlet combinations: H/7,/3~, ~, H(JI, HxH, H2H, ~ , Z~Z). The soft breaking terms add another 13 parameters, which are, however, not independent if one assumes that soft breaking is produced by spontaneous breakdown of local ~8 Note that custodial S U ( 2 ) c symmetry ts sufficmnt but not necessary f o r p = 1 One can, however, think of this model as an effective theory of a more fundamental structure from which thxs tuned property might emerge on some dynamacal reason
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supersymmetry. In this case the 13 parameters could, in principle, be computed by renormalizing the 4 independent parameters at the scale of supergravity breaking down to a low scale. In the end this leaves us with 12 parameters and 18 masses (including W and Z) indicating the existence of mass relations. Still there is quite a number of independent parameters. It is therefore clear that, although in the supersymmetrized model the parameters of the scalar potential can be restricted such that the theory has a vacuum which satisfies p = 1 at tree level, with stabilized fine tuning, such a model has difficulties with predictability due to the larger multiplet content and large number of parameters. Resuming the discussion of the non supersymmetric model we see that the Hlggs sector, as presented before, undergoes minor changes in terms of an approximate custodial symmetry. Note that the quantity t~r is only defined in the tree level approximation, smce the VEVs o f z and ~ are not identical in general. Also there will now be mixings between S U ( 2 ) c representations of different dlmensionalitles, leading to a breaking of the mass degeneracy of the S U ( 2 ) c multxplets. This means that p ¢ 1 in general, with ( p - 1 ) being small, of the order of l-loop corrections, but not computable, since in this order additional new parameters of the theory also contribute. Similarly, new parameters contribute also to mixing effects breaking the custodial SU (2) c. In fact, at l-loop level, all mixings, that respect charge and CP conservation, among Higgs and gauge boson fields are allowed now ~9. The fields H + and H ° for instance couple now to the fermion sector not only by means of triangle diagrams, but also by 0 o H5+ H3+ and HsH1 mixing terms. Decays of those fields to fermion pairs are therefore likely to be dominant below the W-threshold. Unfortunately the corresponding branching ratios cannot be computed due to the loss of nexMo-leading order predlctabihty. When trying to constrain the Hlggs sector of this model it seems reasonable first to concentrate on tm since it distinguishes between uninteresting (tH << 1 ) and interesting (t~>> 1 ) regions of the parameter t9 H o, consisting of the same xmaglnary field parts as G °, has the same CP asmgnment ( - 1 ), whereas the other neutral scalars 17o,/70, and H ° have CP ( + 1 ). It can be shown directly that the v i n o u s l-loop corrections to e g. H ° H ° m m n g cancel each other [ 7 ].
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space. Strict upper bounds on tn can only be obtained with the help ofunitarity arguments, because one can always render the enhanced Yukawa couplings phenomenologically negligible by demanding that the corresponding Higgs masses are big enough. Tree level partial wave unitarity constraints from H°tt and 0 0 scattering combined with the lower bound on H1H1 H m ( >~ ~ 5 GeV [2 ] ) coming from the Y"decay gives roughly 30 as an upper bound on tH [ 9 ]. Unitarity limits coming from longitudinal WLWL and WLZL scattering give upper bounds on H°t, and H ° masses (tH>> 1 ) that are similar to the SM bounds, roughly MI,, 1145~<1 TeV [ 6 ]. Some ad hoc arguments show what the general unitarity limit on tH might look like: Assuming for instance that the 2l parameters are about similar in magnitude would yield M3 ~< 1 TeV. Combining this with constraints coming from B/~ mixing [ 10] (see below) results in limits like the< ~ 10. The strongest bound on tH derives from a renorrealization group argument, based on two simple assumptions: "perturbative unification" and "desert". First one wants perturbation theory to be vahd up to a high scale (e.g. an unification scale Mu), furthermore one excludes new physics in between Mw and Mu, so that the renormalization group equations (RGE) evolve all the way up to Mu in an effective S U ( 3 ) c ® S U ( 2 ) L ® U ( 1 ) y theory. The point is that the RGE of the doublet Yukawa couplings have an infrared fixed point [ 11] and that those couplings inevitably blow up at some high scale if their value is close to or higher than this fixed point (at the Wscale). This upper bound on the Yukawa couplings gives upper bounds on the sum of the squared masses of quarks or leptons [ 11 ]. For the case of three families this results in an upper bound on the top mass [12] as Mtop ~<245 G e V ,
(13)
which translates in our case to Mtop~<245 CH (GeV) or
(14) With given lower bounds on the top mass one gets the following upper bounds on t~." t~<2.5
for Mtop > 92 G e V ,
the<4.4
for Mtop > 54 G e V .
(15) 339
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The first bound [ 13 ] is valid under the assumption that the decay t-~H+b and other non-SM decays do not occur. It therefore requires that both M m and M m are bigger than 92 GeV. M m or M m below 92 GeV asks for a decay mode independent lower bound on the top mass. Such a bound can be derived from the W width (Mtop>54 GeV) [14]. A low mass charged Higgs boson, therefore, strongly weakens the unitarity bound on tn. The result given by eq. ( 15 ) is rather significant: it tells us that if we demand to have a Higgs sector with Hlggs triplets such that the triplet contribution to the W-mass is as large as possible, the assumptions o f "perturbative unification" and "desert" then restrict the triplet contribution to less than 78% or 63%, respectively. The decay Z ° ~ H + H - provides a tn independent lower bound on the fiveplet and threeplet masses M m a n d M m [15]as MHs, MH3 >~41.7 G e V .
(16)
One might believe that, in addition to this high energy bound, there could also be a low energy bound coming from the mixing o f the B and/~ mesons, since the field H~- can give a dominant contribution, for high tH, to the flavor changing neutral currents responsible for this mixing. This bound depends strongly on tH (i.e. the Yukawa couplings) and on M~op, in the sense that a low Mtop weakens the constraint as does a low tH. It turns Out that the unitarity bound on tH is tOO strong for the BB bound to take effect. The threeplet m a s s MH3 is not constrained by B # mixing. Results for certain 2 Higgs doublet models [ 10,16] can be directly adapted to our case. Fortunately there is another low energy bound, that does constrain the triplet mass. It is coming from the bottom decay b---,sT, which is mediated by penguin diagrams involving the top, H + and W fields. Again this constraint is weakened for low Mtop and low tH. The results obtained in 2 Higgs doublet models [ 17 ] apply to our case directly. The combined constraints on MH3 a r e shown in fig. 1. The fiveplet m a s s MHs IS, in addition to eq. (16), indirectly constrained by the lower bounds o n MH3. Combining eqs. (4), (9) one gets a lower bound on
MHa a s MH5 >1_ / ~ 340
MH3.
(17)
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Bounds
MH3
on
M~3(GeV) 250 200
150
i00
50
0
0
5
10
15
20
25 tan2(OH)
M m (GeV) 600 5O0 4O0 300 200 i00 0 0
2
4
6
8
tan2(0H)
Fig 1. Combined bounds on M m , for M m being smaller (a) (resp higher ( b ) ) than 92 GeV. " t a n ( 0 n ) -= tH" denotes the tangent of the SU (2)L doublet-triplet mixing angle as defined in eq (6). In both diagrams the high energy bound of eq. ( 16 ) xs represented by the dashed hne, the sohd hne refers to the u m t a n t y bound of eq. (15) and the dash-dotted line denotes the constramt coming from b~sy decay [ 17 ] In both figures the shaded regton is excluded. The dotted hne in (b) and the upper dotted hne m (a) represent the B # mlxmg b o u n d for " n o m i n a l " values of the hadromc m a t n x elements, that enter the c o m p u t a h o n of
B/~ mixing, as used mref. [ 10] For companson the B# bound for "pessimistic" values of the parameters (which would correspond to an actual bound) has been plotted m (a) (lower dotted hne) It obwouslydoes not cover any new parameter space. Constraints on MH~ are shown m fig. 2. "Z°--,Z°*/7 ° , / 7 ° ~ h a d r o n s '' bremsstrahlung can be used to bound the n o mass from below. Results for the SM Higgs boson [15] translate to our case with the help of the ratio
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Bounds
on
MH~
Bounds MH
M ~ (GeV)
on
MH1
(G~v)
70 (for t a n ( 0 H ) < 2 5)
80
65 60
60
55
q
50
40
45 40
20
35 30 0
5
i0
15
20
tan2(0H)
Fig. 2. Combined bounds on Mm. Again the horizontal dashed line denotes the high energybound of eq ( 16), whereas the vertmal solid hne marks the umtarity hmat on tan (0H) ( --=tn) The dotted line represents the bound coming from eq (17) and eq. ( 16). The b~sy constraint as plotted in fig. 1 ymldstogether wxth eq. (17) the dash-dotted lane. The shaded regmn is excluded. Again the strict bound from B/~ mixing ("pesstmlstlc" values) covers no new parameter space For tan(0n) =0 the hmlt reads MH~> 72 GeV.
B= r( Z°-~ Z°*B°) F( Z°-~ Z°*H°M) 8 2 2 = c2c 2 + gSsSH + 2X/~g8 CsSsCHSH.
( 18 )
Those b o u n d s are displayed in fig. 3. The I" decay yxelds the additional low energy b o u n d [ 2 ] Mno ~> ~ 5 GeV
(19)
as long as the Yukawa couplings of this field are enhanced with respect to the SM, or in other words as long as the condition Cs/CH>~ 2 2 1 is met. For the case of zero mixing this is always the case• This b o u n d is also valid for the f i e l d / 7 °, i f s ~ / c ~ >>.1. I n the limit of tH<< 1 and c s ~ 1 (no H ° H °, mixing) the field H °, is likely to be lighter than the other scalar Higgs fields, but it can not be constrained due to its strongly suppressed H V V a n d H f f t y p e vertices. In this model, therefore, it is possible to have a very light Higgs boson, that escapes detection. Let us now concentrate for a m o m e n t on the properties of H °. Assuming that MHO ~<92 GeV weakens, as m e n t i o n e d before, the unitarity b o u n d on tH (~<4.4). A n H ° below the Z threshold, therefore, could be produced by bremsstrahlung ( Z ° - - , Z ° * H° )
i
2
3
4
5
tan 2
oH)
Fig. 3. Bremsstrahlungbound on M,q~for the case of zero slnglet mixing (Ss=0 (dashed)) and maxamal smglet mixing (Ss= 1/ xf2 (dotted), Ss= - 1/x/2 (dot-dashed)) and for tan (0H) < 2.5, whmh corresponds to the stronger bound on tan(0n) as given in eq. (15) This expenmental bound represents the combined results for the four LEP experiments for the process Z--*Z*H° with H ° decayingto hadrons [ 15] In each case the region below the corresponchngline is excluded. at LEP with rates comparable to the SM. The relative production rates are equal or enhanced, compared to the SM, for tH being in the interval [ 1.8, 4.4 ] with a maximal value of 1.27 for t/_/= 4.4. Such a Higgs could also be produced through W W fusion at h a d r o n colliders, but there the production rate is suppressed by a factor of three or more compared to the case of SM Hlggs production. The decay channels of H ° within the Higgs sector are H s0- - + H0v H v0 and H s0- + H 30H 30 or H ~ H ; ~1o. A "close to the unitarity limit" (tH> 1.8 ) H ° below the W, H °, and H3 threshold could therefore only decay through ,1~,** l,,rr°*ra°* V ' V * , H 3 H 3 , 1-100p and S U ( 2 ) c violating mixing diagrams, since it has no tree level couplings to the fermion sector (eq. ( 1 0 ) ) . Decays to f f p a l r s are mediated by triangle diagrams involving gauge bosons a n d members of the H3-plet and by the H soH ~o mixing term, which is taken to be of the same order of magnitude as other l-loop diagrams. Such a H ° could therefore have a relatively large ( ~ 10%) branching ratio to 2y's ~1, given by l-loop ~l°Agam we consider only the "'no smglet H~Hv° o mixing'" subllmlt ~1~Compare wath the 2y branching ratios of the SM Hlggsboson (see e.g. ref. [ 18] ) 341
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diagrams, with all the charged particles, that couple to H °, running a r o u n d the loop. Again it should be emphasized that, due to the fine tuning in the model, the corresponding branching ratio cannot be computed. In summary, we p o i n t e d out that in Higgs triplet models where the p ~ 1 constraint is satisfied by imposing a custodial S U ( 2 ) c s y m m e t r y on the potential, the gauge boson masses still can be d o m i n a t e d by triplet contributions. U n f o r t u n a t e l y (or fortunately?) this idea can only be i m p l e m e n t e d invoking fine tuning o f some o f the p a r a m e t e r s o f the general SU ( 2 ) L ® U ( 1 ) y invariant potential, which leads to a loss o f the next-to-leading order predictability. The assumptions o f " p e r t u r b a t l v e unification" a n d "desert" then lead to a r e n o r m a l i z a t i o n group argument which requires that at least ~ 20% o f the W-mass must come from doublet contributions. Close to this limit a neutral scalar with a mass below the W-threshold could have a large B ( H ° ~ 2 7 ) branching ratio, a n d could be p r o d u c e d with rates c o m p a r a b l e to the SM at LEP through z°---,Z°*H° bremsstrahlung. In the other limit, with the gauge b o s o n masses p r o d u c e d by the S U ( 2 ) L doublet field, the m o d e l resembles very much the SM. Although here one o f the neutral scalars could be very light and still escape detection. We thank J.P. Derendinger for illuminating discussions a n d for calling our attention to the result o f ref. [111.
References [ 1] L. Rolandl, preprlnt CERN-PPE/92-175, talk given at the XXVI ICHEP 1992 (Dallas, TX, USA, 1992)
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[2] Particle Data Group, Review of particle properties, Phys Rev. D 45 (1992) 1; and Phys. Lett. B 239 (1990) 1 [3] H Georgl and M. Machacek, Nucl. Phys. B 262 (1985) 463. [ 4] M.S. Chanowitz and M. Golden, Phys. Lett. B 165 (1985 ) 105. [51J.F Gunlon, H.E. Haber, G.L Kane and S. Dawson, The Higgs hunters guide (Addison-Wesley, Redwood City, CA, 1990). [ 6 ] J.F Gumon, R Vegaand J. Wudka, Phys. Rev. D 42 (1990) 1673 [ 7 ] J.F. Gunlon, R Vegaand J. Wudka, Phys. Rev. D 43 ( 1991 ) 2322. [8] J.R Espinosa and M Qulrds, Nucl. Phys. B 384 (1992) 113. [9] P. Bamert, Diploma Thesis (ETH Zurich, Switzerland, 1990). [10] G Athanaslv, P. Franzlnl and F. Gilman, Phys. Rev. D 32 (1985) 3010. [ 11 ] J. Bagger, S. Dlmopoulos and E Mass6, Nuel. Phys. B 253 (1985) 397. [12] J Bagger, S Damopoulos and E Mass6, Phys. Lett. B 156 (1985) 357. [ 13 ] F. Abe et al, Phys. Rev D 45 (1992) 3921 [ 14 ] H. Plothow-Besch,Proe. Joint Intern. Lepton-photon Symp. & Surophyslcs Conf. on High energy physics (Geneva, Switzerland, July-August 1991 ), Vol. l, p. 36. [ 15 ] M. Davier, Proe. Joint Intern Lepton-photon Symp. & Europhysacs Conf. on High energy physics (Geneva, Switzerland, July-August 1991 ), Vol. 2, p. 153. [ 16] L.F. Abbott, P. Silovie and M.B. Wise, Phys. Rev. D 21 (1980) 1393. [ 17 ] J.L. HeweR, Phys. Rev. Lett. 70 ( 1993) 1045; prepnnt ANLHEP-CP-92-125 (Argonne National Laboratory, 1992) [BulletinBoard hep-ph 9212260 ] [ 18] Z. Kunszt, preprlnt ETH-TH92-20 (ETH Ziirlch, Switzerland, 1992).
P H YSIC S I_ETT ER S B
Gauge boson masses dominantly generated by Higgs-triplet contributions? Peter Bamert Instttut de Phystque, Untversttk de Neuchdtel, CH-2000 Neuchdtel, Swuzerland
and Zoltan Kunszt Instttute of Theorettcal Phystcs, ETH, CH-8093 Zftrlch, Switzerland
Received 16 March 1993 Editor: R. Gatto
We discuss a model in which the Standard Model (SM) Higgssector has been extended by additional real and complextriplets. The p ~ 1 constraint is satisfied by restricting the potential to have an enlarged SU (2)L® SU (2)R global symmetry. This is fine tuning, which leads to a decreased predlctabthty in next-to-leadingorder. In this model, however, the triplet vacuum expectation values may give the dominating contribution to the gauge boson masses Using a renormalization group argument we constrain this region of the parameter space. Another mterestmg feature of this model is that one of the neutral scalars does not couple to the fermion sector at tree level and therefore could have a relatively largebranching ratio to 2y's It ~scoupled, however, to the Zboson and therefore it could be produced at LEP via the standard e +e---,Z~ mechanism with rates comparable to the ones of the Standard Model.
The Standard Model (SM) of electroweak and strong interactions successfully describes all elementary particle physics phenomena. Recently, it has been successfully confronted with the precision measurem e n t s at LEP [1 ]. The Higgs sector, a n d so the m e c h a n i s m of the electroweak symmetry breaking, however, is weakly constrained by the data. The most i m p o r t a n t constraint is given by the measured value of the rho parameter:
P=
eos2(0w)'
(1)
/gexp = 1.003 + 0.004, where Pexo is a fit of experimental data [2 ] with Mtop = 100 GeV. Models involving only SU (2)L doublets a n d singlets, satisfy in a natural way the p ~ 1 condition. This is due to the fact that in the SM the S U ( 2 ) L ® U ( 1 ) r gauge symmetry forces the scalar potential to exhibit a SU ( 2 ) L@ SU (2) R global symElsevier SciencePublishers B.V.
metry. Breaking the gauge symmetry the Higgs potential still exhibits an S U ( 2 ) c (custodial) symmetry which ensures p = 1 at tree level. Because of this, SU (2) L® SU (2) R violating radiative corrections to the Hlggs potential are finite and small. This also means that the rho parameter is close to one naturally, without fine tuning parameters. If we want to allow also for Higgs triplets or even higher multlplets, the value of the measured rho parameter becomes a strong constraint. One obvious posslbihty is that we d e m a n d the triplet v a c u u m expectation values (VEVs) to be small. This requires fine t u n i n g a n d an effective decoupling of the triplets from gauge bosons a n d fermions. Alternatively, we would require that the scalar potential, even with these higher multlplets, will remain i n v a r i a n t with respect to the enlarged S U ( 2 ) L N S U ( 2 ) R global symmetry [3,4]. For triplets, however, the S U ( 2 ) L @ S U ( 2 ) R symmetry of the Higgs potential will not be obtained automatically as a result of the gauge 33 5
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symmetry. It can only be achieved by fine-tuning the parameters of the Hlggs potential. Therefore, the pparameter in such models will suffer from a fine tuning problem and will depend "directly" on the fine tuned parameters of the scalar potential. In the SM model the p parameter is protected from this problem by the custodial symmetry. With fine tuned S U ( 2 ) L ® S U ( 2 ) ~ symmetry, however, the higher dimensional multiplets may generate a major part of the W and Z masses. Furthermore, in these models the doublet Yukawa couplings could be strongly enhanced with respect to the SM, indicating an interesting phenomenology. In this note we shall discuss the phenomenological viability of this scenario. The most straightforward extension of the SM Higgs sector, realizing the above situation, has been presented by Georgi and Machacek [ 3 ] who added a complex and a real triplet to the SM doublet. With this content of the scalar sector it is indeed possible to restrict the potential to a SU (2) e® SU (2) R symmetric version, as has been investigated in some detail by Chanowitz and Golden [ 4 ]. More recent papers [5-7] have taken a further look at the renormalization problems [ 7 ] and the phenomenology [5,6] of this model. From the tree level point of view the Higgs sector presents itself as follows [ 3 ]:
q~=_~-
4o
,
~-
zO* /
,
(2)
denoting a (½, ½) and a ( 1, 1 ) multiplet of SU (2) L ®SU(2)R, with hypercharge assignments of Y= 1, 0 and 2 for the doublet, the real and the complex triplet respectively and with the following phase conventions: q~-=--~+*, Z - = - Z +*, ~ - = - ~ + * , Z - - = Z + +* and C° = ~o.. They transform under SU (2) L® SU(2)R global transformations as CI)~ULq)U~ ( q ~ X ) , with UL ((JR) being the transformation matrices in the appropriate representation. The restricted Higgs potential takes the form [ 4 ]
3 June 1993
v(4~, x ) =21(Tr[~*~]-a2)2+22(Tr[X*X]-3b2)
2
+23 (Tr [ • *(/)] -aE+Tr[XtX] - 3 6 2 ) 2 +24(Tr[ rib¢~] Tr [X'X]
- 2 Yr[ q~*T'q~T j] Tr[ X*T'XTJ] ) +25(3 Tr[X*XXtX] - T r [X'X] 2) ,
(3)
where the sum is taken over z andj. The vacuum states [ q o > o = ( a / v / 2 ) l and [ X ) o = b l are defined by V( I q0)o, IX>o) = 0 together with the positivlty conditions [ 4 ] 21+22+223>/0 24)0
,
,
2122 "}-2123 "["2223 ) 0 ,
(4)
25)0.
The two VEVs a and b are related through the Wmass,
MZw=M~cos2(Ow)=¼g2(a2+8b2)=¼g2v 2 ,
(5)
with g being the SU(2)L coupling constant and v ( ~ 250 GeV) the VEV of the SM Higgs. This allows the definition of a mixing angle On denoted by its cosine c•, sine s/t (and tangent t ~ - t a n ( 0 n ) ) a c/~= - ,
/2
2v/2b sH=--
(6)
V
Within the Hlggs sector the gauged SU (2)L®U ( 1 ) r can be regarded as a subgroup of SU(2)LNSU(2)R with T 3 playing the role of the hypercharge. A diagonal subgroup of SU(2)LNSU(2)R survives the spontaneous symmetry breakdown. The physical Higgs bosons will form multiplets, which are degenerate in mass, under this custodml S U ( 2 ) c global symmetry. Expressed in terms of the fields defined in eq. (2) they are H++ =Z++, Z +Z_ ~- -'
H o= Z°+Z°*-2~ ° '
Ho,= )~°+Z°*+ ~° , H + = ci_iZ+ ~ - s+~ ~ r ~+
336
+,
(7)
Volume 306, number 3,4
H°=CH
PHYSICS LETTERS B
Z ° _ Z °* 0o_0o. ~ +SH--
Hoif + - ~ M++ tH75 ,
~o+~o.
H?=
,/5
(7 c o n t ' d )
,
w i t h / / 5 , / / 3 , H1 and H1, denoting a S U ( 2 ) c fiveplet, triplet a n d 2 singlets respectively. The G o l d s t o n e triplet is orthogonal to the H3-plet
G~ = s Z- -+~+~+ - - +c++O+,
~,o_ ,,o.
0o_0o.
In terms o f those fields one sees that the quantity cu defined in eq. (6) i n d e e d denotes the cosine o f a S U ( 2 ) L multiplet rotation to S U ( 2 ) c triplets. The masses are ~1 [4] i~,
3 June 1993
H f ~e: Mf H+
1 + 75
tHVpn[
1 +Ys ( 10 c o n t ' d )
where the _+ in the second t e r m refers to up, down type fermions, p and n denote up and down type quarks a n d v and e stand for n e u t r m o type a n d electron type leptons respectively. Vp, is the C K M mixing matrix. In the following let V denote either a W or a Z boson. The VVH type vertices o f this m o d e l are ( o m i t t i n g an overall factor igg ~ ) [ 3 ]:
W + W + H ~ - : V/2 MwSH, W+ W_HO: _ M q ~w s . ,
= 3vZ(c~24 + s 2 2 5 ) ,
M ~ = v2~,.,
M2"H"=
W ~X/~CHSH23
~SS(2Z + 2 3 ) , / "
(9)
It is convenient to rotate the two S U ( 2 ) c slnglets to their mass eigenstates/7 o and/70,. The rotation angle shall be d e n o t e d b y its cosine cs a n d sine Ss./70 (/7o,) defines the mass eigenstates with the larger SU ( 2 ) c doublet ( H ° ) (triplet ( H ° , ) ) p o r t i o n ~2 Here we consider only Dirac type Yukawa couplings ~3. O f course, only those fields ofeq. (7) which have a nonzero doublet admixture will couple (at tree level) to the fermion sector. D u e to a 4 v the Yukawa couplings are enhanced c o m p a r e d with the SM. They are ~4 ( o m i t t i n g an overall factor ig).
HOlff.
My 1 2MwcH'
(10)
~1 Note that the pomtlvity of the squared masses is guaranteed byeq. (4). ~2 Note that by means of th~s definitmn ss takes values between - 1/,£5 and + l / q 2 ~3 Majorana type Yukawa couphngs induced by the SU(2)L triplets can be introduced but are unhkely to be phenomenologically important unless tH<< 1 [6]. ~4 For mmphcity we have assumed no smglet mixing (Cs=1) m the following expresmons.
+
o 2x/~ W-Hv: ~Mws,,
W + W - H °: MwCH,
ZZH°~
2 Mw
~ ~-,
zzH°,: 2,,/,%~ M ~w ~ , ZZH°I : ~ w c~, Mw W-ZH~: - --s,. Cw
(11)
Again the H toH t o, mixing has been taken to be zero. The absence o f H ° VV couplings is due to H ° being CP-odd. The 7 p a r a m e t e r s o f the restricted potential (eq. ( 3 ) ) can be reexpressed in terms o f the W-mass, the 4 Hlggs masses, the singlet mixing (cs) and t u
(=s,dcH). Let us first have a b n e f look at the limit "tu<< 1"
(b<
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PHYSICS LETTERS B
has SM Hff and HVV and suppressed HHV couplings, whereas all other Higgs bosons have suppressed couplings to the fermion sector (eq. (10)) and to the HVVsector. The whole situation therefore resembles very much the SM with H ° playing the role of the SM Higgs boson. The main differences come from the non vanishing HHVtype couplings ~5, which can serve to bound the masses of the other (not H ° ) Higgs fields from below, and the fact that the other singlet (H°,) is hkely to be very light: M2,,..~
3U2S2(2122"~2223"~2123) / (21-'1-23). The opposite limit "t~>> 1" (a << b) is far more interesting, because here the SU (2)L triplet fields (X) are the source of the SSB, generating the W and Z masses. Again there is almost no singlet mixing: 8 e2=l--~[23/(22+23)]2tH2+O(tH4). Here we have the remarkable situation that S U ( 2 ) c multiplets either couple to the fermion sector (H1, H3), those couphngs being strongly enhanced with respect to the SM, or to the HVVsector (Hv, Hs), with couplings bemg roughly of the same order of magnitude as the SM ones, but not both. Again there is a field which is likely to be lighter than the others: M 2, 4c2(2 122 + 2 2 2 3 " } - 2 1 2 3 ) / ( 2 2 - ] - 2 3 ) . For medium values of tz4 ( ~ O ( 1 ) ) there can be significant HIHv o o mixing allowing the mass eigenstates/7 o and/7o, to have SM like couplings in both the fermmn and the HVV sector. As in the previous case H3 couples to the fermion sector while//5 has HVVcouphngs, but not the other way round. The restricted potential (eq. (3)) emerges after the additional parameters of the general SU (2) L® U ( 1 ) r invariant potential ~6 have been fine tuned to be zero ~7on a certain mass scale (e.g. on the Mm shell). However, due to the running of those parameters this fine tuned custodtal symmetry will not be exact anymore at another mass scale (e.g. the triplet mass shell Mm ). It is therefore reasonable to understand the term fine tuned custo&al symmetry as an approximate symmetry, which is valid only at tree level. Fine tuning in our case means that the model looses its #5 For a hst of the HHVtype couplings see e g. ref. [ 6 ]. #6 For the exphclt form of this potentml see e.g ref. [7 ]. #7 Note that one cannot simply ampose that those terms do not exist, because they are automatmally introduced by means of counterterms after the renormallzatlon procedure.
338
3 June 1993
predictive power at the next-to-leading order level #s At this point one might be tempted to supersymmetrize the model. In fact, supersymmetry provides a solution to those fine tuning problems since it ensures that the parameters of the superpotential, from which the scalar potential derives, do not receive radiative corrections. This is due to cancellations among various diagrams (Non-Renormahzatton Theorems). After supersymmetry has been softly broken those parameters will only receive logarithmic corrections. An extension of the supersymmetric SM containing one (necessarily complex) triplet with zero hypercharge ~ has been recently studied by Esplnosa and Quir6s [ 8 ]. In this model, however, the triplet VEV has to be kept small to fulfill the constramt p ~ 1. If we want to have large triplet VEVs, then at least one chlral triplet with nonzero hypercharge has to be added as becomes clear from the tree level relation P = ~{~ ] (q~)o ]2[TL(TL+ 1 ) _ (½Y)21 ~{q,) 2 ] (~/1)0 ] 2(½ y ) 2
,
(12)
where the sum goes over all (complex) multiplets, TL denotes the largest eigenvalue of the SU (2) e generator T3 in the appropriate representation and Y is the hypercharge. Adding e.g. a 2,.. (3, - 2) to {: ( 3, 0 ),/7: (2, 1) a n d H : (2, - 1 ) [(a, b ) ~ S U ( 2 ) L ® U ( 1 ) r ] would require ] (~)ol = ] ( Z ) o l / x / ~ for p = 1. But since Z misses its "counterpart" (3, 2) the Hlggsinos remain massless and give rise to a chiral anomaly. One then has to introduce a third chlral triplet 2: (3, 2). The conditions f o r p = 1 are now I (Z)o] 2+ ] (;~)o 12 = 2 ] ( ~ ) o] 2, resembling, in a special case, the non supersymmetrlc model: (Z) o= (;g) o = ( { ) o = b. In this model the scalar Higgs sector has 23 physical degrees of freedom corresponding to 2 doubly charged, 5 singly charged and 9 neutral scalars. The most general gauge invariant superpotential has 8 parameters (corresponding to the singlet combinations: H/7,/3~, ~, H(JI, HxH, H2H, ~ , Z~Z). The soft breaking terms add another 13 parameters, which are, however, not independent if one assumes that soft breaking is produced by spontaneous breakdown of local ~8 Note that custodial S U ( 2 ) c symmetry ts sufficmnt but not necessary f o r p = 1 One can, however, think of this model as an effective theory of a more fundamental structure from which thxs tuned property might emerge on some dynamacal reason
Volume 306, n u m b e r 3,4
P H Y S I C S LETTERS B
supersymmetry. In this case the 13 parameters could, in principle, be computed by renormalizing the 4 independent parameters at the scale of supergravity breaking down to a low scale. In the end this leaves us with 12 parameters and 18 masses (including W and Z) indicating the existence of mass relations. Still there is quite a number of independent parameters. It is therefore clear that, although in the supersymmetrized model the parameters of the scalar potential can be restricted such that the theory has a vacuum which satisfies p = 1 at tree level, with stabilized fine tuning, such a model has difficulties with predictability due to the larger multiplet content and large number of parameters. Resuming the discussion of the non supersymmetric model we see that the Hlggs sector, as presented before, undergoes minor changes in terms of an approximate custodial symmetry. Note that the quantity t~r is only defined in the tree level approximation, smce the VEVs o f z and ~ are not identical in general. Also there will now be mixings between S U ( 2 ) c representations of different dlmensionalitles, leading to a breaking of the mass degeneracy of the S U ( 2 ) c multxplets. This means that p ¢ 1 in general, with ( p - 1 ) being small, of the order of l-loop corrections, but not computable, since in this order additional new parameters of the theory also contribute. Similarly, new parameters contribute also to mixing effects breaking the custodial SU (2) c. In fact, at l-loop level, all mixings, that respect charge and CP conservation, among Higgs and gauge boson fields are allowed now ~9. The fields H + and H ° for instance couple now to the fermion sector not only by means of triangle diagrams, but also by 0 o H5+ H3+ and HsH1 mixing terms. Decays of those fields to fermion pairs are therefore likely to be dominant below the W-threshold. Unfortunately the corresponding branching ratios cannot be computed due to the loss of nexMo-leading order predlctabihty. When trying to constrain the Hlggs sector of this model it seems reasonable first to concentrate on tm since it distinguishes between uninteresting (tH << 1 ) and interesting (t~>> 1 ) regions of the parameter t9 H o, consisting of the same xmaglnary field parts as G °, has the same CP asmgnment ( - 1 ), whereas the other neutral scalars 17o,/70, and H ° have CP ( + 1 ). It can be shown directly that the v i n o u s l-loop corrections to e g. H ° H ° m m n g cancel each other [ 7 ].
3 June 1993
space. Strict upper bounds on tn can only be obtained with the help ofunitarity arguments, because one can always render the enhanced Yukawa couplings phenomenologically negligible by demanding that the corresponding Higgs masses are big enough. Tree level partial wave unitarity constraints from H°tt and 0 0 scattering combined with the lower bound on H1H1 H m ( >~ ~ 5 GeV [2 ] ) coming from the Y"decay gives roughly 30 as an upper bound on tH [ 9 ]. Unitarity limits coming from longitudinal WLWL and WLZL scattering give upper bounds on H°t, and H ° masses (tH>> 1 ) that are similar to the SM bounds, roughly MI,, 1145~<1 TeV [ 6 ]. Some ad hoc arguments show what the general unitarity limit on tH might look like: Assuming for instance that the 2l parameters are about similar in magnitude would yield M3 ~< 1 TeV. Combining this with constraints coming from B/~ mixing [ 10] (see below) results in limits like the< ~ 10. The strongest bound on tH derives from a renorrealization group argument, based on two simple assumptions: "perturbative unification" and "desert". First one wants perturbation theory to be vahd up to a high scale (e.g. an unification scale Mu), furthermore one excludes new physics in between Mw and Mu, so that the renormalization group equations (RGE) evolve all the way up to Mu in an effective S U ( 3 ) c ® S U ( 2 ) L ® U ( 1 ) y theory. The point is that the RGE of the doublet Yukawa couplings have an infrared fixed point [ 11] and that those couplings inevitably blow up at some high scale if their value is close to or higher than this fixed point (at the Wscale). This upper bound on the Yukawa couplings gives upper bounds on the sum of the squared masses of quarks or leptons [ 11 ]. For the case of three families this results in an upper bound on the top mass [12] as Mtop ~<245 G e V ,
(13)
which translates in our case to Mtop~<245 CH (GeV) or
(14) With given lower bounds on the top mass one gets the following upper bounds on t~." t~<2.5
for Mtop > 92 G e V ,
the<4.4
for Mtop > 54 G e V .
(15) 339
Volume 306, number 3,4
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The first bound [ 13 ] is valid under the assumption that the decay t-~H+b and other non-SM decays do not occur. It therefore requires that both M m and M m are bigger than 92 GeV. M m or M m below 92 GeV asks for a decay mode independent lower bound on the top mass. Such a bound can be derived from the W width (Mtop>54 GeV) [14]. A low mass charged Higgs boson, therefore, strongly weakens the unitarity bound on tn. The result given by eq. ( 15 ) is rather significant: it tells us that if we demand to have a Higgs sector with Hlggs triplets such that the triplet contribution to the W-mass is as large as possible, the assumptions o f "perturbative unification" and "desert" then restrict the triplet contribution to less than 78% or 63%, respectively. The decay Z ° ~ H + H - provides a tn independent lower bound on the fiveplet and threeplet masses M m a n d M m [15]as MHs, MH3 >~41.7 G e V .
(16)
One might believe that, in addition to this high energy bound, there could also be a low energy bound coming from the mixing o f the B and/~ mesons, since the field H~- can give a dominant contribution, for high tH, to the flavor changing neutral currents responsible for this mixing. This bound depends strongly on tH (i.e. the Yukawa couplings) and on M~op, in the sense that a low Mtop weakens the constraint as does a low tH. It turns Out that the unitarity bound on tH is tOO strong for the BB bound to take effect. The threeplet m a s s MH3 is not constrained by B # mixing. Results for certain 2 Higgs doublet models [ 10,16] can be directly adapted to our case. Fortunately there is another low energy bound, that does constrain the triplet mass. It is coming from the bottom decay b---,sT, which is mediated by penguin diagrams involving the top, H + and W fields. Again this constraint is weakened for low Mtop and low tH. The results obtained in 2 Higgs doublet models [ 17 ] apply to our case directly. The combined constraints on MH3 a r e shown in fig. 1. The fiveplet m a s s MHs IS, in addition to eq. (16), indirectly constrained by the lower bounds o n MH3. Combining eqs. (4), (9) one gets a lower bound on
MHa a s MH5 >1_ / ~ 340
MH3.
(17)
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Bounds
MH3
on
M~3(GeV) 250 200
150
i00
50
0
0
5
10
15
20
25 tan2(OH)
M m (GeV) 600 5O0 4O0 300 200 i00 0 0
2
4
6
8
tan2(0H)
Fig 1. Combined bounds on M m , for M m being smaller (a) (resp higher ( b ) ) than 92 GeV. " t a n ( 0 n ) -= tH" denotes the tangent of the SU (2)L doublet-triplet mixing angle as defined in eq (6). In both diagrams the high energy bound of eq. ( 16 ) xs represented by the dashed hne, the sohd hne refers to the u m t a n t y bound of eq. (15) and the dash-dotted line denotes the constramt coming from b~sy decay [ 17 ] In both figures the shaded regton is excluded. The dotted hne in (b) and the upper dotted hne m (a) represent the B # mlxmg b o u n d for " n o m i n a l " values of the hadromc m a t n x elements, that enter the c o m p u t a h o n of
B/~ mixing, as used mref. [ 10] For companson the B# bound for "pessimistic" values of the parameters (which would correspond to an actual bound) has been plotted m (a) (lower dotted hne) It obwouslydoes not cover any new parameter space. Constraints on MH~ are shown m fig. 2. "Z°--,Z°*/7 ° , / 7 ° ~ h a d r o n s '' bremsstrahlung can be used to bound the n o mass from below. Results for the SM Higgs boson [15] translate to our case with the help of the ratio
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PHYSICS LETTERS B
Volume 306, number 3,4
Bounds
on
MH~
Bounds MH
M ~ (GeV)
on
MH1
(G~v)
70 (for t a n ( 0 H ) < 2 5)
80
65 60
60
55
q
50
40
45 40
20
35 30 0
5
i0
15
20
tan2(0H)
Fig. 2. Combined bounds on Mm. Again the horizontal dashed line denotes the high energybound of eq ( 16), whereas the vertmal solid hne marks the umtarity hmat on tan (0H) ( --=tn) The dotted line represents the bound coming from eq (17) and eq. ( 16). The b~sy constraint as plotted in fig. 1 ymldstogether wxth eq. (17) the dash-dotted lane. The shaded regmn is excluded. Again the strict bound from B/~ mixing ("pesstmlstlc" values) covers no new parameter space For tan(0n) =0 the hmlt reads MH~> 72 GeV.
B= r( Z°-~ Z°*B°) F( Z°-~ Z°*H°M) 8 2 2 = c2c 2 + gSsSH + 2X/~g8 CsSsCHSH.
( 18 )
Those b o u n d s are displayed in fig. 3. The I" decay yxelds the additional low energy b o u n d [ 2 ] Mno ~> ~ 5 GeV
(19)
as long as the Yukawa couplings of this field are enhanced with respect to the SM, or in other words as long as the condition Cs/CH>~ 2 2 1 is met. For the case of zero mixing this is always the case• This b o u n d is also valid for the f i e l d / 7 °, i f s ~ / c ~ >>.1. I n the limit of tH<< 1 and c s ~ 1 (no H ° H °, mixing) the field H °, is likely to be lighter than the other scalar Higgs fields, but it can not be constrained due to its strongly suppressed H V V a n d H f f t y p e vertices. In this model, therefore, it is possible to have a very light Higgs boson, that escapes detection. Let us now concentrate for a m o m e n t on the properties of H °. Assuming that MHO ~<92 GeV weakens, as m e n t i o n e d before, the unitarity b o u n d on tH (~<4.4). A n H ° below the Z threshold, therefore, could be produced by bremsstrahlung ( Z ° - - , Z ° * H° )
i
2
3
4
5
tan 2
oH)
Fig. 3. Bremsstrahlungbound on M,q~for the case of zero slnglet mixing (Ss=0 (dashed)) and maxamal smglet mixing (Ss= 1/ xf2 (dotted), Ss= - 1/x/2 (dot-dashed)) and for tan (0H) < 2.5, whmh corresponds to the stronger bound on tan(0n) as given in eq. (15) This expenmental bound represents the combined results for the four LEP experiments for the process Z--*Z*H° with H ° decayingto hadrons [ 15] In each case the region below the corresponchngline is excluded. at LEP with rates comparable to the SM. The relative production rates are equal or enhanced, compared to the SM, for tH being in the interval [ 1.8, 4.4 ] with a maximal value of 1.27 for t/_/= 4.4. Such a Higgs could also be produced through W W fusion at h a d r o n colliders, but there the production rate is suppressed by a factor of three or more compared to the case of SM Hlggs production. The decay channels of H ° within the Higgs sector are H s0- - + H0v H v0 and H s0- + H 30H 30 or H ~ H ; ~1o. A "close to the unitarity limit" (tH> 1.8 ) H ° below the W, H °, and H3 threshold could therefore only decay through ,1~,** l,,rr°*ra°* V ' V * , H 3 H 3 , 1-100p and S U ( 2 ) c violating mixing diagrams, since it has no tree level couplings to the fermion sector (eq. ( 1 0 ) ) . Decays to f f p a l r s are mediated by triangle diagrams involving gauge bosons a n d members of the H3-plet and by the H soH ~o mixing term, which is taken to be of the same order of magnitude as other l-loop diagrams. Such a H ° could therefore have a relatively large ( ~ 10%) branching ratio to 2y's ~1, given by l-loop ~l°Agam we consider only the "'no smglet H~Hv° o mixing'" subllmlt ~1~Compare wath the 2y branching ratios of the SM Hlggsboson (see e.g. ref. [ 18] ) 341
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diagrams, with all the charged particles, that couple to H °, running a r o u n d the loop. Again it should be emphasized that, due to the fine tuning in the model, the corresponding branching ratio cannot be computed. In summary, we p o i n t e d out that in Higgs triplet models where the p ~ 1 constraint is satisfied by imposing a custodial S U ( 2 ) c s y m m e t r y on the potential, the gauge boson masses still can be d o m i n a t e d by triplet contributions. U n f o r t u n a t e l y (or fortunately?) this idea can only be i m p l e m e n t e d invoking fine tuning o f some o f the p a r a m e t e r s o f the general SU ( 2 ) L ® U ( 1 ) y invariant potential, which leads to a loss o f the next-to-leading order predictability. The assumptions o f " p e r t u r b a t l v e unification" a n d "desert" then lead to a r e n o r m a l i z a t i o n group argument which requires that at least ~ 20% o f the W-mass must come from doublet contributions. Close to this limit a neutral scalar with a mass below the W-threshold could have a large B ( H ° ~ 2 7 ) branching ratio, a n d could be p r o d u c e d with rates c o m p a r a b l e to the SM at LEP through z°---,Z°*H° bremsstrahlung. In the other limit, with the gauge b o s o n masses p r o d u c e d by the S U ( 2 ) L doublet field, the m o d e l resembles very much the SM. Although here one o f the neutral scalars could be very light and still escape detection. We thank J.P. Derendinger for illuminating discussions a n d for calling our attention to the result o f ref. [111.
References [ 1] L. Rolandl, preprlnt CERN-PPE/92-175, talk given at the XXVI ICHEP 1992 (Dallas, TX, USA, 1992)
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