does the fundamental light scalar really exist?

IL NUOVO CIMENTO VOL. 107A, N. 9 Settembre 1994

Does the Fundamental Light Scalar Really Exist?(*)

G. A. KOZLOV

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research 141980 Dubna, Moscow Region, Russia

(ricevuto 1'8 Marzo 1994; approvato il 29 Marzo 1994)

Summary. -- In a class of an extended low-dimensional scalar effective theory containing an additional light scalar, a standard Higgs-like particle with the mass below the top-quark mass and the bare mass parameter is presented. These spin-zero bosons may be invisible because they do not decay or they decay into light leptons (neutrinos). The Higgs-like particles may be light enough to be produced at the facilities with intermediate energies, but difficult to detect through their decay products.

PACS 12.90 - Miscellaneous theoretical ideas and models.

1 . - I n t r o d u c t i o n .

The discussion of a possible detection of Higgs particle observables has a long history and a lot of points of view [1-5]. The experimental search for the Higgs particle has been hampered by a wide range of its possible masses. The recent interest is provided by efforts both in experimental and theoretical areas of investigation of the Higgs-like particles in the light sector.

In a previous paper [6], we have presented the general aspects of an arbitrary dimensional system of two scalar neutral fields by means of the 1/N expansion method. It was shown that a hypothetical candidate for a (pseudo)scalar coupling between quarks would be a light Higgs-like exotic z-particle. The possible existence of a light boson (with mass mB which is smaller than the Higgs mass in the Standard Model (SM)) coupling to heavy quarks would influence the interaction between quarks. It would give rise to a force with a relatively long range (R} - 1~ms. The electron-positron colliders with an energy more than 100GeV are very useful physical instruments to search for the effects of Higgs-like exchange on both heavy quarkonia and heavy-light-quark bound systems.

Generally speaking, while the SM operates with the couplings of the Higgs boson to the gauge particles and to fermions (leptons and quarks) it leaves the Higgs mass

(*) The author of this paper has agreed to not receive the proofs for correction.

1739

1740 G.A. KOZLOV

mH absolutely unspecified. Nuclear physics leads to a lower bound m H > 13 MeV [7]. There is a low bound of mtt > 2 GeV coming from the heavy-quark level b--* s + + Higgs, i.e. from B-meson decays B --~ K § Higgs [8]. At present, the Higgs mass m n cannot be predicted in the SM, though there have been some restrictions on its bound 7 GeV < m H < 1 TeV [9-11]. We expect more complete searches for short distances of the Yukawa-type interactions mediated by exchange of the hypothetical light scalar ;<-bosom This exchange would generate an additional Yukawa-type potential to a heavy-quark interaction. Outside the quark bound system for the 1S-state, r > (R(1S)), the contribution via the ;<-particle exchange is given in the standard way

(1) V~ (r) = - ()~/r) exp [ - m~ r],

where mz is a z-boson mass and the Yukawa coupling gy of the z-particle to quarks is

g~ = 4r:Z = V~GF y2 Kmdmu .

Here Ge is the Fermi coupling constant, md and mu are the diagonal mass matrices for heavy quarks of charge - 1/3 (b-quark) and 2/3 (c, t quarks), respectively; K = kd ku, where the coupling constants kd and ku are equal to 1 for the SM only; y is the model-dependent constant (y = 1 and y > 1 in the SM of one Higgs doublet case and for several doublets of Higgs particles, respectively). Clearly, the z-particle contribution (1) is most important for large quark masses mQ and small enough m s.

Due to the fact, following from the SM, that the Yukawa coupling constant gv is proportional to the quark mass, the short-distance term potential must be taken into account for heavy-quark case calculations. Since the considered above model is instructive in the regime

m z ( R ( l s ) ) <~ 0(1) ,

we can estimate the ratio m~/mQ which due to the Coulomb contribution (R(1S)) = = 9/(4~smQ) looks like [12]

m~ 4~s(~)

mq 9

at the QCD mass scale t~. Taking into account the 1/(R(1S)) scale of the running coupling constant ~s((R(1S)) 1), we can restrict mq by means of

- i [ 4r~CF aS ((R( 1S))- 1) mQ t> Y X/2GF

that is for as = 0.3 and CF = 4/3 the decay heavy-quark mass should be more than 0.55TeV in the case of one Higgs doublet in the SM (~ = 1). Clearly, very heavy-quark bound systems of mass exceeding the top-quark mass m t - 140 GeV in the SM cannot exist if the Yukawa coupling scale ~, >> 1.

DOES THE FUNDAMENTAL LIGHT SCALAR REALLY EXIST? 1741

The interaction of the light z-boson with quarks is described by the Lagrangian density

L z Q = - Z_ (kddmdd + kuumuu), V

where, considering the neutral Z-scalar sector of the theory written in terms of real components

Z 9 = 2-U2(ZN + iz~) + vj 3

in the two-doublet Higgs case

X1 = , ~ ( 2 =

\ X l z ~

the vacuum expectation value v = ~ + v~ and

j = l , 2

Generally, the free parameter tg~ = v2/Vl is assumed to vary between 1 and the ratio mt/mb of top- and bottom-quark masses. It is very instructive to determine the

difference between the model-dependent couplings k d - ku ~ 0 [3] from the real physical processes to understand the deviation from the SM. We are going to focus our interest on the theoretical description of the light-~-boson (Higgs) phenomenon beyond the SM. One of the physically interesting schemes is the model with higher-derivative Lagrangian density. Such a model with calculable strongly interacting Higgs sector with heavy Higgs particles in the TeV region has been offered in[13]. The higher-derivative term in the kinetic energy of Higgs Lagrangian

(2M 4) - 1 [~]~ r ["]~,u r (2)

is keeping quantum fluctuation finite, while all the symmetries of the model are preserved. The term M 4 in (2) acts in the equation of motion as a Pauli-Villars regulator with the mass parameter M. The complex ghost pair with intrinsic 0(4) symmetry has been proposed with the mass scale M.

In this paper, we are going to introduce in the Lagrangian density an additional term

1 2 S ' r + r ,

describing the dipole-type field X(x) obeying the higher-order equation (written below for simplicity in the massless case)

(3) [:]2z(x) = 0.

1742 G.A. KOZLOV

The Lagrange-Euler equations for two scalar fields Z(x) and r

(4) [:]z(x) = r

(5) [:]r = o,

allow one to obtain (3). The non-trivial equal-time canonical commutator relation (CCR)

[ao r Z(y)]xO=yo = - i $ ( x - y ) = [~oZ(X), r

with (4) and (5) taken into account, leads to the CCR for an arbitrary field X on ~4 [14]:

[Z(x), Z(y)] = 2r: I d 4 p s ( P ~ ~,(p2) exp[ - i p ( x - y)] = 8r: e(zO ) O(z2 )

where ~'(p) is the well-defined generalized function ~'(p) = ~(p0)~,(p2) from S'(~4), ~ ' ( p ) = 0 a t p < 0 , z = x - y .

As has been noted in [15], the rote of a dipole field in 4-dimensions is held in 2-dimensions by the simple pole field. The reason for considering such a problem is that the analogy of behaviour between two and four dimensions has to be found at the level of Green's functions.

We present here the field •(x) as an observable object. This field transforms into the operator-generalized function ~(x) to arrive at the physical representation r%hys of algebra GH in the Hilbert space Hphys. We suppose the following point of view, that the additional r or e x p [ i r is physical, but non-observable to be compared with the fiction current j ~ C t ( x ) = - ~ , r This current could be considered as an observable transforming to the operator-generalized function J~Ct(x) = 0 in the physical representation.

The outline of the paper is as follows. In sect. 2, the lower-bound constraint for the z-boson is briefly explained. Section 3 reviews the necessary material of the effective model construction for the real scalar fields beyond the SM. The general case of massive relations between the physical quantities and bare parameters is presented. We give an explicit analysis both in the non-supersymmetric and supersymmetric case. The last section contains the conclusion.

2. - The lower-bound constraint.

The renormalization group flow to the unknown low z-boson masses, m z and a large enough top-quark mass m t = gt v is given by (v--- 175 GeV)

d~ 3 (6) dt -=/~ -= - - [(gt~ - n ) ~ + B - gt 4]

47~2

= V ~ v ,


where the coupling constants A and B are connected to each other and can be read in the standard form [16]:

1 2 A = ~ ( g l + 3g~), B = 1 (g~ + 2g~g~ + 394),

16

2 (mz2 _ m~v)

where mw and mz are the masses of W- and Z-bosons, respectively. The model considered here conserves the physical scalar ;~-bosons whose mases are surprisingly unconstrained.

The solution

(7) ,~(~) = R[1 - (~o/~) a] + 2(/Zo)(/Zo/~) a

obeys eq. (6) at m t ;~ (m~/2 + m2) 1/2, where

g -B 8 - - , a = - - ( A - gt2).

R - gt2 _ A 4= 2

2/(2v 2) leads to a lower bound of Solution (7) together with 2(/x) > 0 and 2(/z 0) = mz the ;~-boson mass for large m t . Here t~ is an arbitrary model scale and due to a natural physical restriction t~ < 2, on the high scale 2, of validity in the SM (2 ̀is supposed to be of an order of a few TeV). Thus, the above-mentioned approximation for the lower bound of the z-boson mass, mz, leads to the following relation:

m 1 mz < 2, a 2v2R

which shows how the ;~-boson bound starts at a certain value of m t and A, and how it grows or decreases with both the value and sign of the coupling-constant parameter a.

In case one could make A within the SM scheme very large (up to - 1015 GeV), the lower bound of mz is defined by the top-quark mass, mt > 78.5 GeV, at f'Lxed masses of gauge bosons contained in the coupling constants A and B :

3 (8) m~ 2r~v 2

[mt 4 - l(mz4 + 2m~v)] ln( A / - 4 ~ m z /

Note that 2(~) in (7) can become negative immediately for mz < A, if mt 2 > (m~/2 + + m~v) and ~(mz) goes to zero. Even though 2, is a free parameter, it is clear that in the framework of SM 2, should be large enough and the condition that the top-quark mass is large enough leads to the 2,-independence of the lower bound of the z-bosom From the additional point of view [17] it is known that unless some unusual mechanism solves the hierarchy problem within the SM, 2, should not be very large, most likely only of a few TeV.

1744 G.A. KOZLOV

3. - The model and the technique.

We consider the real scalar field Z(x) as a multiplet Z~(x) with N components Z ~ (x) = {X 1 , ..., Z N} belonging to the regular representation of O(N). The essence of the model can be found without inclusion of quark fields.

The Lagrangian density looks like

1 1 2 2 (9) L(x) = -~ 8,,~8~' Z + -~ mo Z - - -

where

1 2 4~N go (Z2) 2 + L~ho~t(x),

1 Lghost(X ) = a~z(x)a~r + 2 r

N N 8~XS'Z = ~ a~Z~8"X~, Z 2= ~ Z~Z ~, mo and go 2 are the bare mass and coupling

a = l a = l

constant, respectively. For the Lagrangian density (9) the equations of motion are -

E]~Z = r

where

go m3- )z,

~2 N - 1 ~2 E]~- + ~

= 1 S x y

is the Laplace-Beltrami operator in the Minkovski space M(N - 1, 1) if ~ --- - 1, or in the Euclidean space E(N) if s = + 1. We consider Z as a physical Higgs-like field and r as an additional ghost field.

Minimization of the classical energy density, obtained by neglecting the zero-point energy in the fields, leads to the possible equation

(10) R~Z(x) = 0, k = 1, 2, . . . ,

for the weak limit [3~ r = 0. Anticipating the spontaneous breaking of the global U(1) symmetry, we shift the

Z(x)-field by

z ( x ) = + Z o ,

where we stipulate that (~(x)} = 0. Formally, let us consider a vector r belonging to the pseudo-Hilbert space hi. In

the pseudo-Hilbert-Fock space h =Fv(hl) of the Bose particles, we define the creation a*(r) and annihilation a(r) operators by means of

IA* (p ) f (p )d4p = a*(r), ~A(p) f (p)d4p : a(r),

where f ( p ) e S(9tN). The scalar field ~ looks like

~(p) = A(p) + A * ( - p ) .


Considering the standard composition

C = a(r) + a*(r)

and the shifting procedure for the Z(x)-field we have used before, one can find that at any function f ~ S(~ N) in the space of S ~-real functions on ~N and for Zo e ~N we have the following clear representation:

exp [iCzo] exp[i f~(x) f (x)d4x] e x p [ - i C z o ] - - e x p [ i f x(x)f(x)d~x]

Using the mean-field expansion procedure, we define the total Lagrangian density (9) a s

L(x) = - V(Zo) + Lo (x) + L1 (x),

where the mean-field potential energy V(xo) is

(11) V(Zo)- g~z~ (Z~ 12Nm~ ) 4!N go 2 '

1 _ _ 1 2-2 go 2 ( Z O O ) 2 (12) Lo(X) = -~a~Z(x)&~ X(x) + ~g. X 3 !N '

(13) L l ( x ) - 4!~g~ 2 (~z)2 _ 3!Ng---~--~ (Z~ _ 21(/z 2 _ m~ + z~176 )~2+

) 1 r + m~176 Z~ (ZOO) + a ~ a , r + ~ .

Here ~(x) is the elementary excitation instead of X = ~ + Xo. The potential (11) has the minimum at

(14) ~ =- v 2 = 6Ngo2m~ .

The minimization condition for mo 2 = mo2(Z~) in (14) fLxes the boundary condition in the (Zo, g~, t~o) space of the integral curve for the Calan-Symanzyk/~-function/3(g~) = = ~o (3go 2/3fZo) at which the energy is minimized (~o is an arbitrary subtraction point at which one can define the coupling constant go2(tZo) and the field Zo(t~o))-

For further development of perturbative expansion we will use the partial normal ordering (PNO) procedure for the quadratic field operators ~2 (x). All the dynamical variables dynamically dependent on the operators containing equal arguments, we are rewriting in the following form by definition [14]:

(15) ~(x) 2 = :~(x)2 : + (~(x)2>o,

(16) (~(x)2)o = NA(x; Z~, ~2) ,

where d(x; X~, t z2) is the two-point Wightman function. In fact, our estimations will be formal since d(x; Xo 2,/z 2) includes the divergence from the short-distance limit. Substituting (15) into (13) with (16) taken into account, we obtain the following expression for the interaction Lagrangian density in the framework of the PNO

1746 G.A. KOZLOV

procedure scheme:

(17) L l(x) = - g~ [:Z(x) 2 :]2 _ _ _ 4 !N

1 [ - ~ $ t ~ 2 : ~ ( x ) 2: + mo 2 - -

where

and /I _= A(x; Z~, t~2) �9

3 !N [Z0~(x)]:~(x) 2 : -

g~ (Z~ + AN)] [ZoZ(x)] - 3 !N

g~ N z l 2 - 2 g~x~ ) - t~ 2 - m~ + (?CA) 3 I N

+ a . ~ ( x ) a . r + 1 r

2 go , 2

~/~2 - /z2 _ m~ + 3.---~(Z0 + NA)

To investigate the scalar boson-,,quark, interplay, let us consider the gradient model characterized by the ,,classical- variant of the Lagrangian density:

- - 1 (18) LBQ (x) = Q[ia - m - gSz(x)] Q + 8 ,z (x )8" r + ~ r (x).

The solution to the -quark- field Q(x) is as follows:

Q(x) = exp [ - igy.(x)] Q(O) (x) ,

where Z(x) is realized on the pseudo-Hilbert space hi and obeys eq. (10), but Q(0)(x) is the solution of the free Dirac equation. We imply the dipole-type character of the r field in the Lagrangian density (18) (see the introduction). The , ,quark, field Q(z) obeys the renormalized quantum field equation

{ i~ - g].~ N[a~Z(z)] } Q(z) = 0,

which is an analog of the classical equation. Here N denotes normal ordering defined as a limit of

N{[a,Z(z)] Q(z')} = a, [z(z) + igw(z - z')] Q(z ' ) , z----) z ' ,

where the two-point Wightman function w(x) is introduced in ~4 as

1 In - + iO(x ~ w(x) = (0 IX(x)z(0)I0 ) - (4=) 2 -~- ,

formed in the time-ordered wC(x)- funct ion

wC(x - y) = (O I T z ( x ) z ( y ) I O ) = O(x ~ - y ~ - y) + O(y ~ - x ~ - x ) ,

obeying the equation

[3~wC(z) = ~2~(z), v = 1, 2, . . . ,

in 2v-dimensions with an arbitrary scale l. Note that the global gauge transformations


on the Z(x)-field with the pseudounitarity operators exp [iCx0] lead to the fact that the Wightman functions for the elementary excitations should be non-invariant at Zo ;~ 0 [14]

(0 ly.(x)z(y)IO) = (O 17,(x)~(y)lO} + Z~ .

Due to the restriction that the model must not depend on /~ , we fLX it SO that ~t~ ~ = 0, i.e.

2 go [ 2 (19) ~2 _ m~ + - - ~ Z 0 + Nd) = 0.

3!N

In this scheme, we neglect loop corrections to the vacuum expectation value, therefore, from (17) we can obtain

[ g~ (~+N3)] ~~ m~ 3! N

from which two possibilities can be extracted

~ 6Ym~ yd. Xo = 0, ~o = go2

Taking into account the condition ~t~ 2 = 0 at the present stage of the perturbation order (the two-point function does not acquire the loop corrections), the physical masses can be extracted directly from (12) if we rewrite the last one in the following form:

1 1 2_a ( O~aOtb)~b(x)+ Lo(x) = a.~(x)a~(x) + ~,~ ~ (x) ~ab ~2

1( + 2 ~2 - 3N Z~ ~a(X)--7~b(X).

Since ~(x) is a scalar multiplet of O(N) one can find ( N - 1) Goldstone-like particles and one massive boson with the following corresponding masses:

m~ = _~2 = 0,

m ~ = - / z 2 + g~ Z~" 3N

Using the mass relation (19) by taking into account

A(//. 2 , ,ts 2) ---- ([A2)I+k-D/2A'([A 2 ' ~t2),

where the generalized function A'(...) has the form [6]

exp[(rr/e)(D-1)i] ( t~ ) (20) A'(/z2' t~~ = ( - 1)D/2-1i 4k(k - D/2)!F(k) (~2 + io)D/2-k In + io ,

1748 G.A. KOZLOV

we can get the following equation to obtain z = ln(tz2/tz~ + io):

(21) exp [z(~ + 1)] + a~o 2 exp [z~] + f l z ( / s 1+~ ~- 0.

In formulae (20) and (21) k is the rank of the Laplace-Beltrami operator D~= -1 in the even D-dimensional space-time, ~ = k - D / 2 >1 0,

_- _ , n o + 3IN

2 / 2",1+ ~ go ~, g-o)

fl - A ( k , D ) , 3]

A ( k , D) = ( - 1 ) D / 2 - 1 i exp [(=/2)(D - 1)i] 4 k (k - D /2 ) ! F (k )

Let us consider the simple and instructive example due to restriction (/z0//z) 2 > ~ on small enough g~. In the case one could make tZo within the SM scenario very large, i.e. it0 - A ---- few TeV, we can read the following expression for the mass of new scalar field, omitting the origin where go 2 = 0, since it should hold that /z 2= mo 2 in the non-interacting case (~--+ 0, i.e. k ---~D/2):

(22) t~2=mo 2 - 3T ~ Z~ 1 ~ - .

The prediction that the vacuum of the theory can be unstable is connected with the possibility that in the running coupling-constant scheme the Higgs self-coupling g~(tZo) becomes negative, such that the Higgs potential is unbounded from below [18]. One of the recent physical issues is how to solve the hierarchy problem between the vacuum expectation value v - 175 GeV and a very high scale of cut-off A in the SM. At an arbitrary large number n in (22) in order to be physically relevant one has to require that g~ < 0 at the fLxed range of validity of the SM scale A. The low limit of ~ 2 can be obtained for negative values of g~ at a very high scale of new physics A:

fZ~A2>>z] ---- m~ A 2 A ( D ) . 3!

The A scale should probably not exceed a few TeV at low N, if we suppose that g~ > 0. The mass bound for X~ = 3N~2/g2o,

[ ( tz 2 _ 2 1 g~ A 2 A ( D ) 1 + (23) mo 2 3 31 mo 2 A 2 '

shows how the bound starts at a certain value of mo 2 and how it decreases with A taking into account the N-independence of this bound.


In the case of the supersymmetric system described by a super-Lagrangian density

1_ 1 g~

4!N

the field ~(~) is a superfield, where ~ =_(z, 0, 0) and (0, 0) are (2G)-dimensional Grassman coordinates and ~2=z2+4i00 is invariant under the OSp(D/2G) transformation. The vacuum expectation value 3(/z 2) of the quadratic operator (0 I~(z) 210> in the D-dimensional space is calculated as [19]

A( t~ 2) = F(1 - D/2 + G)

(47c)D/2 (~ 2)l-D/2 +G "

The condition to adjust the physical mass is given by the ratio t~2/m~ depending on the dimensionless parameter - g ~ / m ~ (g2--->g~2A4) if we specify that D = 2G:

t ~2 I [ _ I T ~ I + 2 g~ ] (24) mo 2 = 2 3 (4rz)D/2m4 "

Due to reality of the physical mass, this supersymmetric scheme is valid if the bare parameters go 2 and mo 2 are connected by means of

for an arbitrary D. We will omit the case go'2 = 0 since it should hold that [z2/m~ is equal to - 1 or 0 in the non-interacting case.

4. - C o n c l u s i o n .

In this paper, we have examined the lower-dimensional system of two scalar neutral fields by means of a 1/N expansion at zero temperature. Using the partial normal-ordering procedure, an application to solve the equation to determine the physical mass of a ,,light, scalar )~-boson is proposed. From explicit calculations at the zero temperature we have obtained the relation between the physical quantities and the bare parameters. The comparison between the O(N) symmetric Lagrangian theory and the supersymmetric case in deriving some relations to calculate the physical mass of (dight, Higgs-like exotic z-particle is illustrated.

The function [z2(go2) in (24) looks like a parabola in the {g.2/m~, z= = gg2A4N/[3(4:r)D/2m3]}-plane, where ([z2/m~) < 1 in the region z < 4, but the range - 1 / 2 < z < 0 provides the ratio restriction ([z2/m~)< - 1 in the negative domain. Here A represents the range of validity of the SM and should not probably exceed a few TeV. However, the bare parameter mo and the self-coupling constant gg2, generally depending on m0, are essentially unspecified. For instance, suppose we choose gg2 = 0.01 and m0 -= 63.5 GeV, defined as a lower bound of experiments at LEP1 and SLC to detect the decay Z--~ff + Higgs [20]. Then, the calculation in four-dimensional space-time leads to the A restriction: A < 642N 1/4 GeV. Note

1750 G.A. KOZLOV

that the ratio ~2/m~ in (23) can become smaller than 1 immediately for the restriction on go ~

go 2 > - 48(1 + A2/m2) -1 ,

that is true for the arbitrary values of mo and A, otherwise, the negative values go 2 allow us to get an expected fact of (t~2/m~) > 1, if

A 2 < 48(lg31) m3.

Thus, in the non-supersymmetric scenario we show the restriction of the bound corresponding to (23) in the few TeV's model scale region, i.e. A < 4.4 TeV for (t~2/m~) > 1, if we choose I g ~ l - 0.01 and mo= 63.5 GeV. Finally, there possibly remain the physical real scalar Higgs-like z-particles whose masses are surprisingly unconstrained. In the minimal version of the SM there is only one such scalar boson, but the extensions beyond the SM often have more. For a -light- %-boson with the mass m~ < 1 GeV the direct decay channels are pure hadronic in the light sector (~=) and leptonic (e + e - , ~+ ~-) if m x is above twice the hadron (::)-mass.

According to the idea of Kirzhnitz and Linde [21], the vacuum expectation value of the Higgs doublets vanishes at sufficiently high temperatures and the electroweak symmetry should be restored. It is supposed that a high enough temperature, T, is presumably realized in the very early Universe. Following Landau [22], in the simple version the temperature effects are provided by

m02 = const(T - To)

in the Lagrangian density (9), where Tc is the critical temperature. Thus, as a next step we need a more precise analysis by taking into account the temperature effects with the first (go 2 < 0)- and second (g~ > 0)-order transitions. Finally, to search for the exotic z-boson accompanied with spontaneous breaking of chiral symmetry, the temperature dependence of the z-boson mass needs to be calculated.

R E F E R E N C E S

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BNL-42853 (1989). [3] A. PICH: ~-Decays and chiral Lagrangians, FTUV/90-17, IFIC/90-14 (1990); J. PRADES and

A. PICH: The decay ~ -o u~176 in two-Higgs-doublet models with a light scalar, FTUV/90-13, IFIC/90-10 (1990).

[4] M. S. ALAM et al.: Phys. Rev. D, 40, 712 (1989). [5] R. [6] G. [7] R. [8] R. [9] S.

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[14] N. N. BOGOLIUBOV, A. A. LOGUNOV, A. I. OKSAK and I. T. TODOROV: The General Principles of Quantum Field Theory (Nauka, Moscow, 1987).

[15] E. D'EMILIO and M. MINTCHEV: A gauge model with perturbative confinement in four dimensions, IFUP/Sept . Pisa (1979).

[16] M. LINDER: Z. Phys. C, 31, 295 (1986). [17] U. ELLWANGER and M. LINDNER: Constraints on new physics from the Higgs and top masses,

HD-THEP-92-48 (1992). [18] A. D. LINDE: Particles Physics and Inflationary Cosmology (Harwood Academic

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does the fundamental light scalar really exist?

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