The supersymmetric Singlet Majoron Model and the general upper bound on the lightest Higgs boson mass

29 June 1995 PHYSICS

LETTERS

6

Physics Letters B 353 (1995) 243-256

The supersymmetric Singlet Majoron Model and the general upper bound on the lightest Higgs boson mass J.R. Espinosa

’

Deutsches Elektronen Synchrotron DESK Notkestrasse 85. 22603 Hamburg, Germany

Received 13 March 1995; revised manuscript received 11 May 1995 Editor: R. Gatto

Abstract An upper bound on the tree-level mass of the lightest Higgs boson of the Supersymmetric Singlet Majoron Model is obtained. Contrary to some recent claims, it is shown to be of the same form as the general mass bound previously calculated for supersymmetric models with an extended Higgs sector. Soft-breaking masses or exotic vacuum expectation values do not enter in the tree-level bound [which is only controlled by the electroweak scale (Mz) ] and also decouple from the most important radiative corrections to the bound (the ones coming from the top-stop sector). The derivation of the upper bound for general Supersymmetric Models is reviewed in order to clarify its range of applicability.

1. The search for a Higgs boson is one of the most challenging goals for existing and planned accelerators. The discovery of such a fundamental scalar would be the first step in understanding the elusive mechanism of

electroweak symmetry breaking. In the framework of supersymmetric theories the Higgs sector is particularly constrained and provides a unique ground for checking whole classes of supersymmetric models. In contrast with the arbitrariness in the masses of most of the new particles predicted by supersymmetry (which are only weakly restricted by naturalness criteria) it seems to be a general feature of Supersymmetric Standard Models the presence of a light Higgs particle in the spectrum (with mass of order MZ even in the limit of unnaturally large supersymmetric masses). As is well known, in the Minimal Supersymmetric Standard Model the tree level mass mh of the lightest Higgs boson is bounded by MZ 1cos2/?]. Radiative corrections to the mass of this Higgs boson can be large if the mass of top and stops is large, and the tree level bound can be spoiled [ 11. After including next-to-leading log corrections [ 21 the numerical bound Mh < 140 GeV (for a top mass below 190 GeV and stops not heavier

than 1 TeV) is found, so that, even if the lightest Higgs can escape detection at LEP-200 its mass is always of the order of the electroweak scale (and the dependence on the soft breaking scale is only logarithmic). Similar bounds have been calculated for extended supersymmetric models. An analytical upper bound on the tree-level mass of the lightest Higgs boson (LHB) is known for very general Supersymmetric Standard Models with extended Higgs or gauge sectors [3,4]. This bound depends on the electroweak scale (given by Mz) and on ’ Supportedby the Alexander-von-HumboldtStiftung. 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00543-9

244

J.R. Espirwsa /Physics

Letters B 353 (1995) 243-256

the new Yukawa or gauge couplings that appear in the theory. Numerical bounds can be obtained by placing limits on these unknown parameters (e.g. assuming that the theory remains perturbative up to some high scale). These bounds are typically greater than the MSSM bound but still of order Mz. In a recent paper [ 51 a bound on the lightest Higgs boson mass was calculated in a particular supersymmetric extended model with spontaneous R-parity breaking, the supersymmetrized Singlet Majoron Model (SSMM). The obtained bound was found to be qualitatively different from the general bounds of [3,4]. In contrast with these general bounds it was found a dependence on exotic vacuum expectation values (VEVs) that are naturally of the order of the SUSY breaking scale so that the bound is no longer controlled by Mz, although it turned out to be numerically very similar to the MSSM bound due to the smallness of some Yukawa couplings. Theoretically this is a disturbing result, because it leaves open the possibility of finding similar models in which the mass of the LHEJ is much larger than MZ (evading the well behaved general bounds of [ 3,4] ) with important consequences for the Higgs phenomenology in such models. The purpose of this letter is twofold. First of all, to re-analyze the SSMM bound. In Section 2 we will show how the SSMM bound on the LHB tree-level mass can be improved, eliminating all the dangerous dependence on exotic VEVs [this will be true also after the inclusion of the most important one-loop radiative corrections (from top-stop and bottom-sbottom loops), see Section 31. And second of all, to re-derive in detail the general bound of Ref. [3] in order to clarify its applicability range. This will be done in Section 4. 2. The Supersymmetric Singlet Majoron Model [6] is the simplest viable extension of the MSSM which can accommodate spontaneous R-parity breaking (and so, lepton number breaking). The extra fields added to the MSSM are right-handed neutrino chiral superfields Ni (with L = - 1, and i is a family index, i = 1,2,3) and a singlet superfield @ (with L = 2). With this field content, the most general superpotential renormalizable and gauge invariant is f = C [h;Qi. HzUF + h$Hl . QiD; + h;Li ’ HzNj + h$HI . LiE; - AijNiNj@] + /.LH~. Hz, iJ

(1)

where the notation is self-explanatory. The new Yukawa couplings hb are responsible for the mass of neutrinos and therefore they have to be small. For the phenomenological restrictions used to put bounds in these couplings see 161. The tree-level scalar potential for neutral states, V( @, @, 9i”, g, @‘> can be readily derived (where the Ci” are the left-handed sneutrinos). It consists of three parts:

v = v, -4-VD+ vsoft,

(2)

with

(3)

V, = tG2 [IHe’

- @I2

+ c

,F?,z]i, i

where G2 f 2 + gf2, and

(4)

J.R. Espinosa / Physics Letters B 353 (1995) 243-256

245

R3

pi = iG2v2 cos 2/3 - rn: cot j3 + ~2 + $G2x2 -C(C”r~j)2-CJCh:xi)2, i

(9) j

j

i

where

c h:xiYj

u2 E

R3 E

9

p(+i

+

+ A:)Xiyj.

(10)

i,i

Due to the breaking are non-vanishing:

e,,=

hb(2Aj4

c

iJ

of lepton number,

Higgses and sneutrinos

h;yj(2/\j4

MS,2 = - C

iG2v1~i,

mix and the corresponding

+ AC) - iG2v2Xi + 2~2 C

i

matrix elements

hzhijxk,

(11)

j,k

where gi =_ Cj h$yj. And finally we also need the sneutrino

mass matrix

(12) For later use we also obtain from the condition

c

m2x? Y, 1 = v2R3 - iG2x2(x2

+ v2cos2~)

avaxi

= 0 the relation

- ~.LV,CJ~ - a4 - a;

C

from which sneutrino masses can be traded by other parameters Next we define the normalized field 3’ as Cixicy

p-Z

_A ’

i

of the potential.

-O Xivi 9

c x,X’

(13)

(ChjXi)‘, j

i

(14)

i

d--

such that (PO) = X, while any combination define the new field H{O as the combination /O

H,

=

u,g+XI0 = J$T7

-vi

L(vlHy

of the fields Ci” orthogonal

to fro has a vanishing

VEV. We also

+x60),

(15)

so that (Hi”) = v; s dv. This redefinition of fields amounts to a change of basis from to (Hi”, w, . . . ) where now the only doublet fields having a non-zero VEV are Hior and @. The important point is that, in this rotated basis, the 2 x 2 mass submatrix (for Hior, @>, Mij2, has a simpler form. Using (13), I 2 Ml1

=$I

v:M:~ -t 2~1 C

XiM:iI + C i

M’,22=M~2=-m~cot~+~G2u~i--,

i,j

R3

xiXjMi,c,

1=$-

vimi tan /? + $G2vi4 + R3v2] ,

(16)

(17)

u2

I

M,2

2

= L v;

XiMs,2

VIM~~ + C i

rn;vl - $G2v{2v2 - R3 + 2~2 C

(C j

h;ri) i

‘1 .

(18)

J.R. Espinoso /Physics Letters B 353 (I 995) 243-256

-

B,u@@

4j

+ C

A;hC@tii’q

Lj

- C

AiAij@‘qQ”

+ h.c.

1.

(5)

For simplicity, we will neglect any possible CP breaking effects in the following assuming that all the parameters in this potential are real. Then we will take A, = h$ij by an appropriate rotation of the Ni fields. AS was shown in Ref. [6], in a wide region of parameter space, not only I$’ and @ develop a VEV but also fii”, ii$’ and a0 do:

(6) with the hierarchy xi - h”u << u E dm < yj,d N O(Ms)* (MS supersymmetry breaking, e.g. MS 5 1 TeV) thus leading to the spontaneous lepton number L and R-parity. The implications of this interesting scenario are Following the general procedure, an upper bound on the tree level mass of obtained studying the scalar mass matrix for the fields m, @, fii”‘, q, Qor That matrix is now a 9 x 9 matrix but the analysis of the 2 x 2 submatrix bound. This upper bound was calculated in [5] and is

rni 5

.@2”2

cos2 2p

+ c

representing the scale of soft breaking of sum x U( 1 )Y, further studied in [ 6,7]. the LHB in the SSMM can be [with 4: = (&” + i@)/&].

for w,

@

XiXjhi”,h; (s,

xf

leads easily to such a

sin2 p + F

i

(7) with u2 = u: + uz and tan/3 = 02/01. As stressed in [5], there is an explicit dependence of the bound on the exotic VEVs xi and yk and on the soft breaking mass of sneutrinos, rnc;. In particular, the bound (7) is not finite in the formal limit xi, yk, rnfii --+ co, that is, there is no decoupling of the exotic VEVs and soft masses from the bound. The most important one loop corrections to this tree level bound, coming from the top-stop and bottom-sbottom sectors were also calculated in [5] and it was also found the same bad non-decoupling behaviour. Nevertheless, the smallness of h; and xi makes the bound numerically indistinguishable from the MSSM bound I@ cos2 2p. Anyhow, as we are about to see, the bound (7) can in fact be improved, that is, (7) is not saturated and a more stringent bound can be found. Moreover, this new bound will turn out to be independent of soft-breaking masses or exotic VEVs and is always 0( MS). In order to achieve this improvement we need to examine larger mass submatrices. In fact, we will need the 5 x 5 mass matrix for the fields @, Z!@ and cior. The elements of this matrix, derived from the effective potential (3)-(5) are

MT, =,u:

+ $G2(3v:

Mz2 = p; - ;G2(u;

- ~22)+ $G2x2 = -m;tanp+

- 34)

- iG2x2 + c i

(j c

iG2uf -pz,

,, ,)2+,(,h~~,)‘=-m:cot~+~c’uj+$. hY.y.

MT2 = rn: - 3G2qu2, with & = I,uU(~+ mf, x2 = xix? and rn$ = -BP_ The mass parameters VEVs UI and 02 using the minimization conditions avavl = JvJv2 = 0:

(8) ~1 and ,u:! have been traded by the

247

J.R. Espinma / PhysicsLettersB 353 (1995) 243-256

Or, writing 12 _ m3 = L(m$l 4

h;’ s + c

- R3),

tan@ 55 y, VI

h&, i

(note that this last definition

is the natural one in this model)

-mi2 tan p’ + iG2v{ 2

I2 m3 -

M2 = rni2 - ;G2v;v2 + 2 1

hj2v;v2

the 2 x 2 submatrix

(19) takes the form:

hi20’,02 i . -mi2 cot /I’ + iG2vi ;G2v’,v2 + 2 c

(20)

i As is well known, the full 9 x 9 scalar mass matrix must have an eigenvalue smaller than (or equal to) the smallest eigenvalue of this 2 x 2 submatrix. In this way the following upper limit for the LHB mass results: rni 5 $G2vt2 cos2 2p’ + vt2 c

h’f sin2 2/3’,

(21)

where v’~ z v: + U; + x2. Note that now

M; = 4 G2vf2 = ;(g*+gf2)

(22)

9

1 [ @) = x; breaks SU( 2)~ x U( 1)~ and then contributes 174 GeV. Remarkably, the bound (2 1) has the same form as the with an extended Higgs sector [3] and so it exhibits exclusively by the electroweak scale (note that the rotated even in the formal limit Xi + 03 because Xi/U: < 1) .

to the gauge boson masses]

so that u’ is fixed to be

general bound calculated for supersymmetric models its same good properties, namely to be controlled Yukawa couplings h(i = xi hbxi/vi are well behaved

3. It is natural to ask whether the good effect of this field rotation also extends to radiative corrections. That is, do the one-loop radiative corrections to the bound (21) exhibit decoupling of the exotic VEVs Xi, yk and soft-breaking masses m;,? We will show in this section that this is actually what happens for the most important radiative corrections: the ones coming from the top-stop and bottom-sbottom sectors. The leading terms of the one-loop corrections to the Higgs mass can be easily calculated using the well known expression for the one-loop contribution to the effective potential (in the DR scheme) AV, = &StrMf(,ogg-l),

(23)

where Q is the renormalization scale and Mi are the field-dependent masses of the different species of particles. To fix the notation we list here the relevant masses for the top-stop sector, which are given by mf(v2)

= h:vl,

(24) rn$ + rnf -t M:LIicos2/?

II

Mf(ultU2~Xi~Yj) =

h,(&u:!

+ ,UVI + a21

with m$, m’$ the soft masses for left and right-handed h,Q3 . HzUg in the superpotential Di=$-$sin2f3w,

ht(A,uz + WI + (r21

(25)

,

rni + rn; + M$ Dk cos 2p /I stops, A, the trilinear coupling

associated

with the term

and

Dk = $ sin2 &,

(26)

248

J.R. Espitwsa /Physics

Letters B 353 (1995) 243-256

while g2 was defined in (10). We will call mi, rni the two eigenvalues of (25) with rni 2 rni. For the bottom-sbottom sector we have

m~+m~+M~Db,cos2P

M@l,

u2)

=

II

hd&Ul

hdA/a rni

+ P2)

+

~2)

(28)

rni + M$Di cos 2p /I ’

+

with rni the soft mass for right-handed sbottoms, & the trilinear coupling associated with the term hbHl. Q3 0: in the superpotential and D;=-$

0: = -f sin2 6’~.

+ i sin2 &,

(29)

2 As we are not considering the gauge We will call mi,, rniz the two eigenvalues of (28) with rnt, >_ mS-. contributions to (23), we will also neglect the D term contributions to (25) and (28) in the following. The one-loop corrected effective potential is a function of 4i = (@“, q, i;?, y, cp”‘) [or equivalently (~1, ~2, xi, yj, 4) ] and so, the squared-mass matrix Mi. will receive a one-loop correction J2A VI/i@id4j. After correcting also the minimization conditions (9) by including the one-loop contribution aAVt /&$i we obtain &r2(l Ul R3 --

M;,“’ =-(m~+6m:)cotp+$G2u~$

+Sf)

+A,,,

4 2 ;(T Sf +

A229

Y

(30)

MT,“’ = (mf + 8rn:) - $G2qu2 + A12, with m~‘~t~~ [f(mF,) - f(mk)] fl

-t- (t +

0)),

12

3h2 f(mi ) - .f(mi) rni -m?

Sf =j&

’

t2

3 AlI =8rr2

m? ml hfmfp2Ai2g( rn;,, mi) + him: log - b1 b2 -I-2AbAB log 2 4

, mt, > +

rni)

,

, .4fAk2g( mi, m:2) f? AbA*g(mi,, mt2)

2

+ 2A,Ab log 2

h:rnf log y

log ’ + AtAk(mf,, m?12

+ AiAig( rn$,m&)

m&

m?m? h~mh2&z(m~,

2

1

+ hirn$AB

+

2

log 2

+

b2

(31)

and f(m2)

A; =

(32)

= 2m2 [log$l],

1 rnt -m2-12

A,+pcotp+;

(+2

A = & +Ptanp B mi, m2b2

(33)

J.R. Espinosa / Physics Letters B 353 (1995) 243-256

All of the matrix elements M$(‘) (i,j

= 1,2) diverge in the decoupling SUSY limit ,%yk,rni + 8m:

249 cxx

In

fact, writing p2 - y: - rn: + am: - CY(Mi), all the MS(‘) grow like Mi: M:,(l) = -(mz + ami) tanp - --&lfSf) ’ M&(l) = -(m: Mi2(‘)

=

+ am:> cot/I + $

f...,

- %*8f+ 02

... ,

(m: + c?rni) + . . . ,

(34)

where the dots stand for contributions that are finite in the decoupling limit. The eigenvalues of this matrix will grow also like Mz if Det M*(‘) grows like M$ but, if due to some cancellation, Det M*(t) grows only like Mi one of the eigenvalues (the lightest) will remain finite in the decoupling limit [ 81. In fact one can see that this cancellation takes place for the (rng + am:)* contribution to Det M *(I) but not for the rest of the terms. Then, as was found in [ 51, the bound derived from this unrotated submatrix receives one-loop corrections which are not controlled only by the electroweak scale. To study the one-loop rotated matrix we need also

(35) with the tree level matrix elements on the right hand side as given in ( 11) and ( 12) and 3h2 - ‘m*A’* f Tg( m*fiT rni) v-g&

+Sf.

(36)

Note that ci,j MZ,fjXiX,i contains the term Ci rni,xp which receives also the one-loop correction SC&+

= -i

i

xxi%

= I

i

-$$2&7*

[f(mi)

-

f(m?L,)] .

(37)

The rotated 2 x 2 mass matrix then takes the one-loop form: M’* =

-(mk*

+Smi2)

tanp’+A{,

(mi2 + am;*)

+ Ai

(m;* +Smi*)

+A;*

-(m~2+6m;2)cotp’+A;211’

1) (38)

with 6rng2 = f

[6m3*vl + A,(r26f]

.

(39)

1

The terms A~j are finite in the decoupling limit and are given by Ai, = $G*v’,~+ $h:m:a;“(cr*

+2pvl)g(m~,m~) Vi”

A;* = ;G2v; + A22,

u: + ?Alt, 1

J.R. Espimsa / Physics Letters B 353 (I 995) 243-256

A look at (38) shows immediately that there is decoupling, that is, Det M'2= M{,2Mi22 - M',24 N O( Mt). In fact, the one-loop corrected version of the tree-level bound (21) takes the simple form rni5 A{, cos2 p’ + Ai sin2 p’ + Ai sin 2p’ =

MS cos2 2p’ + 2M& F

$

sin2 2/?’

(41) where 82 = (At

+ pcotfl+

I

~~1~2)~

9

rni -mi

x2 = (Ab b

+ ittaW2 2 2

“6, - mi&

(42)

’

Note that the one-loop corrections have exactly the same form as in the MSSM, the only difference arising in 2: which now includes the term cr2/v2. 4. In this section we will re-derive the bound on the lightest Higgs boson mass for general Supersymmetric Standard Models under very general assumptions. The particular form of the bound for some special cases of interest [ 31 will be presented, and at the end we will show how to apply the general bound to the SSMM case finding the same result obtained in the direct calculation of Section 2. Let us first assume that the Higgs sector of the general Supersymmetric model under consideration contains at least two SU( 2) L doublets (HI, H2 with hypercharges f l/2) taking vacuum expectation values

(HI) = ~1 (H2)

v2 z

= ~2 >

V: + vi

5

(

174GeV)2,

(43)

with the equality holding when only these two doublets drive electroweak breaking. In the general case other fields may contribute to gauge boson masses. When there are extra doublets (d doublets with hypercharge -l/2 and d with hypercharge l/2) is well known that a field rotation can be made such that only one doublet of each type takes a non-zero VEV [ 91 (that can be taken real and positive if electric charge is conserved). In that case these two rotated fields will be the ones we are calling HI and H2 (as we did for the SSMM in the previous sections). In general other fields in higher SU( 2) L representations can participate in the electroweak breaking (but satisfying the constraints from Ap) . In the following we will make the reasonable assumption that no fields in representations higher than triplets take non-zero VEVs2 (models with unsuppressed triplet VEVs have been studied in the literature). In general the LHB has the same quantum numbers as the Standard Model Higgs so that we will concentrate in the study of the CP even Higgs sector 3 . Let us consider the most general tree-level potential for the fields w, I$’ (coming from a renormalizable full scalar potential): ’ The vast majority

of the models of interest satisfy this requirement

although

a bound that applies even without this restriction

can be

derived (see Ref. [IO]).

3 This terminology to be correct.

is only valid when CP is a good symmetry

(as in the MSSM),

but we do not need this assumption

for the derivation

J.R. Espinosa / Physics Letters B 353 (1995) 243-256

V=v,+M;fl+M;t$+t~&fl)~+

251

2i~&fl@+rn;(H3~

+ MII(Z$')~+ M,2(H32@+M2,(H32H7+M22(H33

+Al(Hyy4+

h12(~)2(@92+A2(kq)4+A~2(Hy)3Hp+A;,fp(Hy)3,

(44)

appearing in this potential are some function of other scalar fields, e.g. MI z o will indicate that such functions have to be evaluated with these extra scalar fields at their vacuum expectation values: MT F Ml (,yor = (x0'), .$@' = (9'))...). By symmetry considerations some of the terms in this potential can be forbidden. For example, imposing many couplings in (44) should be invariance of the potential under the symmetry transformation v ---f -@ set to zero. As we will see later, we do not need to impose this symmetry by hand but it will arise automatically for the quartic couplings of the supersymmetric theories in which we are interested. As @, @ are the neutral components (more precisely the real part) of two S(1(2)L doublets and the full potential has to be gauge invariant, it can be immediately deduced that MT,2must come from a field or combination of fields transforming as a sum doublet, that is In general,

the parameters

Ml (,yor, lo', ...) . The superindex

(M7,2)3 - m2(4,

(45)

with m2 some gauge invariant squared mass and (d) representing the VEV of some S(1(2),_ doublet (fundamental or not, but different from q and q by construction). Now, as HI and H2 are the only doublets with a non-vanishing VEV (and no fields in representations higher than triplets take a non-zero VEV) we get the restriction

OQ3 = (M;)3 =O. Using similar arguments,

(46) SCJ(2)t

x U( 1)~ invariance

of the full potential

implies4

Mb - (d),

(47)

so that one can also deduce the relations

MC, = My, = M& = M;2 = 0.

(48)

Next, taking into account that the scalar potentials we are considering come from a (softly broken) supersymmetric theory, we will get some restrictions on the quartic couplings in V.As is well known, these quartic couplings are of fundamental importance for the tree-level upper bound we want to derive. Let us consider separately the two different types of supersymmetric contributions to the effective potential. i) F terms These are given by the well known formula

*=y$

(49) I

with f(k) = Wtfi -+ cbi) [WC&)is the superpotential, ii the chiral superfields and & their scalar components]. The superpotential W is at most cubic in the superfields implying that the quartic F-terms in the potential must come from the cubic terms in W. It is easy to see that quartic terms like (q)3H!=$ or (@)3w can only arise if both a superfield ii and its hermitian conjugate ii* appear in W, which is not possible ( W being an analytic function of the chiral superfields). This proves that F-terms do not contribute to the “ii couplings. 4 By assumption

we discard the possibility

of M$ transforming

as a W(2)

quadruplet

and taking a VEV.

252

J.R. Espinosa / Physics Letters B 353 (1995) 243-256

ii) D terms The contribution of D terms to the scalar potential takes the form

where a runs over the gauge groups (with coupling constant go) and Ta are the generating matrices of group a in the representation of the fields +i. According to (50) the only quartic terms obtained are of the form HI HI H; H;, HI H2H; Hz or H2H2Hz Hz. So, neither D terms nor F terms contribute to the A’ couplings and then

Ai = A;, = 0.

(51)

Inserting Eqs. (46), (48) and (51) in (44), the tree-level scalar potential for w, form I+(*,@) +

= Vt + i(mp>2(e)2

$,2(*)2w32

+

+ (mP2)2*@

+ i(mg)2(*)2

@

takes the following

+ ~AI,(*)~

(52)

$22(@74.

After writing I$‘r = hy + &ui [which corresponds to (@) E (( @ + i@) /2) = ui], rni’ and rnq can be expressed as functions of the other parameters of the potential just imposing that Vs has its minimum at ( UI ,u2 >, that is, using the minimization conditions -%

=o

ah!

(i= 1,2).

(53)

In that way we are led to rni = -mf2 cot /? - 2Az& - A12u:,

my = -my2 tan/3 - 2Attuf - At2v,2,

(54)

where tanp = UZ/U~and the superindex o is omitted for simplicity. The analysis of the mass submatrix (M2) for the fields h y, hy will give us the mass bound on the LHB mass. This submatrix is now M2=

=

rn: + 6At1v: + A&

II

mi2 +

mT2+ 2A12~1~2 rni + 6A22ui + A~V: II

2A120u2

-mt2 tan /I + 4At I uf mi2

mi2 +

Cotp +

-m;2

+ 2h2~1~2

2A12~1~2

(55)

4A22V;

where the minimization conditions (54) have been used to write the second expression. It is simple to show that the eigenvalues of this matrix can be written as

mi = f

-i-iif2 + 4AllV; + 4A&

f

(?$, + 4vf, - ~U:Z)~COS22p + (ET2 + 2A12u~)~sin2 2p] 1’2} , (56)

with Eif2 E 2mT,/ sin 2p and

V: 3

Aiiuf/

cos

2p. Using the inequality

[a2 COST 2p + b2 sin2 2p] “2 1 a cos2 2p + b sin2 2/3,

(57)

J.R. Espimsa / Physics Letters B 353 (1995) 243-256

the following

253

bound results:

rni 5 m2 I (4A,,

COS4 p

+

sin4 p + A12sin2 2p) v2,

‘i&2

(58)

which is the central formula we were looking for. Note that the bound is determined by the quartic couplings, as anticipated. It implies that the bound is not sensitive to the details of the soft-breaking. Then the only scale that enters the bound is u, the electroweak scale and so, even if the soft breaking scale gets large there is always a light scalar Higgs [that is, of mass 0( Mz) ] in the spectrum. One can go beyond (58) and obtain the particular form of the bound in some classes of models: 4.1. Models with an extended Higgs sector Having obtained the general formula (SS), we can now find the particular form of the Higgs mass bound in a class of non-minimal Supersymmetric models in which the Higgs sector is extended and contains, apart from the two MSSM Higgs doublets, HI, HZ, other extra fields (singlets, more doublets without VEVs, triplets, etc. See Refs. [ 11-151). In that class of models the Aij couplings in (58) come from: i) F terms The only cubic terms in the superpotential that can give a contribution are of the form AWN

A$Aifij,

(59)

with i, j = 1,2, and 4 an extra chiral superfield (note that there are no such terms in the MSSM). specifically with a superpotential containing a part like f =Ao~o~~+~h,~,~~+~A_,~_,~~+ the Aij couplings SA,l =$A:,

....

(60)

in (52) receive the contributions 8A22 = $A:,

,

6At2 = A;.

(61)

On top of that, note that sum x U( 1 )r invariance of the superpotential or neutral components of triplets with hypercharge Y = 0, fl. ii) D terms The contributions are exactly the same as in the MSSM: SA,t = SA22 = f (g2 + g’2) , so

More

SA12 = -+

(g2 + d2)

that the bound on the LHB in these non-minimal

,

requires the C#Qscalars to be singlets

(62)

models is

-m’h “2 I $ (g2 + gr2) cos2 2/3 + Ai sin2 2p + ATcos4 /3 + A?, sin4 /I,

(63)

with v2 5 v: + ui as usual. This bound was first obtained in [ 31. Of course, if Ak = 0 the MSSM bound is recovered. The non-minimal correction, dependent on the new Yukawa couplings, Ak, is positive definite so that the bound is weaker than in the MSSM. Also note that it is necessary to write the bound with an explicit v2 dependence instead of using the formula (g’ -I- g’2)v2/2 = M;. This relation between v: + IJ; and Mi will no longer be valid in models with an extended Higgs sector if other fields apart from the two doublets HI and HZ contribute with their VEVs to M: (e.g. triplets). If this happens, then (g2 + gf2)v2/2 < Mi and the bound will be more restrictive.

J.R. Espinosa /Physics

254

Letters B 353 (1995) 243-256

4.2. Models with an extended gauge sector In a similar manner, an upper bound on the LHB mass can be obtained in Supersymmetric models which gauge group (at low energy, that is N 1 TeV) is different from the Standard one [3,8,16,17]. Usually, this type of models require the introduction of extra representations in the Higgs sector to give a correct gauge symmetry breaking and of extra fermions to cancel anomalies. The influence of the extra Higgs representations on the lightest Higgs bound has been considered in the previous subsection while the presence of extra exotic fermions can affect the bound through radiative corrections [ 181. In particular models all these effects have to be combined. In this subsection we concentrate in the modifications of the tree level bound when the two doublets Ht , H2 transform non-trivially under the extra gauge groups. Examining the D terms the following contributions to the Aij couplings in (52) are obtained: (64) with (i= 1,2). This translates

in the following

(65)

modification

of the bound on rni [ 31:

[TF cos2 j3 - TF sin2 /3]” .

Ami = 2u2 cgz ‘I

(66)

As a check, for SU( 2) L T: =T;=

3,

(67)

gives Am; = (g2v2/2) TF =T;=

cos2 2p. And for U( 1) r

;,

(68)

so that Am; = (g’2u2/2) cos2 2p, in agreement with the MSSM bound. After the general discussion in this section it should be clear that one can apply the bound (63) SSMM. Making use of the field rotation ( 1.5) directly in the superpotential ( 1) we get the term h$Li ’ H2Nj = hy’$N,/H;“@$ I i.e. a &-type

coupling

Am; = ut2c

also to the

(69)

+ .. . ,

which, according

to (60) and (63) contributes

to the bound with (70)

(~hC~)2sin22~f. j

i

This is the same result that was obtained by the direct calculation. bound, note that the effect of the field rotation (15) on V, is

= $G* [jH;‘12 -

1@1’12 +...,

Concerning

the D term contribution

to the

(71)

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255

which is formally equivalent to the MSSM result with the replacement Ht” -+ Hi0 and so gives the contribution Ami = ~G*(u{* + v;) cos* 2p’, in agreement with (21). 5. In conclusion, we have improved the tree-level upper bound on the mass of the lightest Higgs boson in the Supersymmetric Singlet Majoron Model, finding a new bound which is controlled by the electroweak scale and remains light in the limit of heavy exotic VEVs or soft breaking masses that decouple from the bound. We have also proved (computing the most important one-loop corrections to this bound) that this decoupling is not spoiled by radiative corrections. The similarity of the improved bound calculated in this paper for the lightest Higgs boson mass in the Super-symmetric Singlet Majoron Model with previous bounds derived for general Supersymmetric models with an extended Higgs sector has motivated the re-analysis of the derivation of these general bounds in order to clarify its range of applicability. We have shown that those general bounds are in fact based on very general assumptions, namely that the model contains a pair of doublets participating in the electroweak breaking and no fields in Sum representations higher than triplets take a VEV With this simple starting input and using gauge symmetry and supersymmetry to constrain the effective potential one is able to obtain a bound on the mass of the lightest Higgs boson of the theory. As a particular example we have re-derived the bound in the Supersymmetric Singlet Majoron Model using these general results. It would be interesting to study whether the decoupling of exotic scales in one-loop radiative corrections to this tree-level bound is general, that is, to see if it is automatic for the class of models in which the tree level bound applies or if further assumptions are required. This subject is currently under investigation [ 191.

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