- Email: [email protected]
Bounds on the Higgs mass from cosmological constraints
Ast roparticle Physics AstroparticlePhysics 8 (1998) 309-315
EISEVIER
Bounds on the Higgs mass from cosmological constraints T. Ouali, J. Derkaoui Department of Physics, Mohamed 1st University*B.P 524, 6000, Oujaix, Morocco
Received 1 July 1997; revised 10 December 1997
Abstract We discuss lower and upper bounds on the Higgs mass using different values of the coupling constant A of the selfinteracting scalar field by considering a new inflationary scenario. We use the value of the amplitude of the primordial energy density fluctuations as inferred from the COBE results and the top quark mass as recently measured by the DO and CDF experiments. The upper bound obtained on the Higgs mass is compatible with the upper hound favoured by the
electroweak data. @ 1998 Elsevier Science B.V.
1. Introduction
One of the key problems in particle physics is the analysis of the electroweak symmetry breaking. The simplest mechanism for the breaking of the electroweak symmetry is real&d in the Standard Model (SM) . To accommodate all observed phenomena, an isodoublet field is introduced, which, after the spontaneous breaking of the symmetry, leaves us with one physical scalar boson. The present LEP experiments have set a lower bound of about 82 GeV on the mass of this particle (the so-called Higgs boson) [ 11. The mass squared term in the scalar potential depends quadratically on the cutoff that can be interpreted as the scale where new physics beyond the standard model sets in. For Higgs masses of the order of 100 GeV, the standard model can be extended to a supersymmetric theory where its simplest version is the Minimal Supersymmetric Standard Model (MSSM) [ 2-41. Lower bounds on the Higgs mass can be derived from the requirement of preserving the stability of the vacuum [5,6]. To illustrate our main idea, we consider only the case of the standard model. This limitation may be related to the discovery of the top quark by the CDF [ 71 and DO [ 81 experiments which constitutes a success for the standard model. This success in predicting the top quark mass has led many physicists to try the same game for the prediction of the Higgs mass [ 91. To investigate the Higgs mechanism, we consider its effects in cosmology. One of the effects of the electroweak phase transition is the expansion mode of the universe which may be exponential -but not inflationary[ 101. Another one is what we are interested in here, i.e. the effect on the Higgs mass. The interplay between particle physics and cosmology is given by the study of the high-temperature phase transitions in gauge theories [ 1 l-141. The one loop temperature-dependent effective potential V&( 4, T) is taken as the basic tool for the discussion of the electroweak phase transition [U-17]. This potential may be [ 17,181 0927-6505/98/$19.00 @ 1998 PflSO927-6505(97)00061-3
Elsevier Science B.V. ALIrights reserved.
I: Ouali, J. DerkaouilAstroparticle
310 %-dqkZ’>
= -(2A+
B)(qb)242
+ Aqb4
Physics 8 (1998) 309-315
+B#41n(42/(#)2)
+ &T2$2(3g2 +g’2 + 2hf) + K,
(1)
where B = 3/1024r2( 2g4 + (g’ + g’)2 - 16h:), g, g’ and h, are the coupling constants of the gauge boson particles and of the top quark which contribute to the one loop effective potential, K is an arbitrary constant, and (4) is the value of the field 4 that minimizes the effective potential. The energy density in the universe due to the existence of a vacuum expectation value for the Higgs field [ 191 contradicts the fact that the present energy density of the universe is less than 1O-29 g cm -3 [ 201. The presence of a large compensating cosmological term in Einstein’s equation may remove the effect of this vacuum energy density [ 191. At high temperature (i.e., in the early universe), symmetries which are presently spontaneously broken should have been exact [ 11,121. The authors of Ref. [ lo] give some details of the phase transition in the Weinberg-&lam model. They found that the condition to be satisfied by the Higgs mass when the vacuum energy density is greater than the radiation energy density is m(H) 5 11.5 GeV for T >> T, (T, is the critical temperature) ; the generalized energy density is dominated by the contribution of relativistic particles, and the scale factor varies as 4. This is the standard hot big bang expansion rate. For T 2 Tc and if m(H) 5 11.5 GeV, then the generalized energy density is dominated by the vacuum energy density and the scale factor has an exponential e’ form. While for T 2 T, the symmetry is broken and the vacuum energy density is compensated by the cosmological energy density [ 191 and once again the expansion rate is the standard one. If T, << mw, it is likely to be a first order phase transition, generating entropy and may be causing inhomogeneities to develop. If for some period the vacuum energy density is greater than the radiation energy density, then the effective pressure of the universe would be negative [ lo]. In this paper, we are not interested in the inflationary consequences of the phase transition of the WeinbergSalam model. But for a short time, the vacuum energy density of the universe due to the electroweak symmetry is much lower than the GUT one and gives cosmological effects. On the other hand, the cosmological perturbation is used to describe the growth of structures in the universe, to calculate the predicted microwave background radiation fluctuation, and in many other considerations [ 211. The inflationary universe in its improved scenario leads to the fact that the present state of our universe could have risen from a wide class of initial conditions, in contrast with the standard cosmology which assumes a particular initial state (nearly homogeneous and isotropic, i.e., a Friedman-Robertson-Walker (F.R.W.) one). And it offers also the possibility to explain a handful of very fundamental cosmological problems; among them the origin of density inhomogeneities. These density perturbations have been calculated by many authors [21-261 which found similar results. The main idea of this work is to find a relation between the cosmological perturbation and the Higgs mass. This relation explains the relation between the physics of the early universe and that of the present day universe. The phase transition, which we consider here, is based on Grand Unified Gauge Theories with a simple Lie group. To illustrate our purpose, we consider one of the viable candidates for GUT of rank higher than 5 and satisfying many of the desirable properties, the SO( 10) group [ 27,281. The same thing could be done for the E6 group by considering the chain of symmetry breaking E6--+SO(lO)-SU(5) x U( 1). 2. Spectrum of the density perturbation
in an inflationary
universe
We assume that the primordial energy density perturbations are due to quantum vacuum fluctuations. The method used to calculate the spectra of density perturbations produced in an inflationary universe with scalar matter field is the gauge invariant formalism developed by Bardeen [ 291. First, let us recall the main results from [ 211; the squared perturbation amplitude in a comoving scale k is s; = ( 1/47r2)((p’2/a)21U&7) 12k3’
(2)
where 77 is the conformal time, p is the homogeneous scalar field which breaks the grand unified symmetry, a(t) is the scale factor and Uk(v) is the mode function which satisfies the equation
T Ouali, J. Derkaoui/Astroparticle t&?7)
+ [k2 -
(l/zY(l/z)-‘luk(77)
Physics 8 (1998) 309-315
z = (a&H)
=o,
311
.
(3)
Here, the prime denotes the derivative with respect to the conformal time and H = a’/~. The solution of the mode equation for the short wavelength perturbation such as k* > ( l/z )“( 1/z ) -* is Q(V) =
4(77i)COS[k(T
-
%)I + [4(rli)/~lsin[~(v
For long wavelengths such as H(Qi)a(a)
- %)I .
< k < H( ~)a(~),
(4)
the solution is
(5) where We Let radius
fChis the crossing horizon time. can use the above results to find the power spectrum & by means of Eq. (2). us first consider the power spectrum during inflation. On scales which at time t are still inside the Hubble (kph(t) > H(t) ) , Uk (7) iS given by (4) and the power SpeCWLUn iS
Sk =
(1*/4g*) I@(t) 1,
(6)
where the dot indicates a derivative with respect to physical time t = Judr]; kph = k/u(v) is the wave number in physical coordinates, and 1*= 81r/3Mi. On scales which were inside the Hubble radius, at the beginning of inflation at time ti but which are outside at time f: H(t) > kph(?) > H(fi)U(tt)/U(t); We USC (5) t0 obtain
where H = b/u is the Hubble constant during inflation. The above equations (6) and (7) are valid for the respective wavelengths both during and after inflation. Here, we assume that after inflation the universe enters a period where the scale factor u(t) varies like Ji. On scales which are inside the Hubble radius at the beginning of inflation but outside at the end (at time t*), i.e., for which H(r*)a(r*)/a(t)
>
kph(t)
>
H(ti)a(ti)/a(f)
9
(8)
the power spectrum is obtained from (7), thus for any t > t*, one has & = (12/2,rr2)(g(t)H2/ji)t,h.
(9)
To express this spectrum of perturbation as a function of the Hubble constant at the end of inflation (in which we are interested) and of other parameters which are model dependent, we use the slow roll approximation. From the equation of motion of the scalar field, one has ++33H+=-t$, where V is the effective potential in the grand unified symmetry and V+,= dV/dq
(10) We have in this approximation
(b=-!h_ 3H’
(11)
which is constant, i.e. @ is the same at the crossing horizon time and at the end of inflation. Thus,
(12)
312
T. Ouali, J. Derkaoui/Astroparticle
Physics 8 (1998) 309-315
We notice that this expression depends on the used model. In a recent work [21], the expression (12) has been used to compare the theoretical predictions with the observational constraints for two specifics cases. For a quadratic potential V(q) = ~m*q* (where the time evolution of the scalar factor after inflation varies like P with p = 3) the mass hierarchy is constrained by m/M, < 10m6, while for a quartic potential V(q) = Acp4 the coupling parameter A must be less than 10-14.
3. Higgs mass Now we need to relate this power spectrum to the Higgs mass by means of the Hubble constant at the end of inflation. For that purpose we assume that the universe undergoes different stages of evolution. Among them, the inflationary stage which after its rolling and its oscillations gives rise to particles which rapidly decay and reheat the universe up to the temperature of lOi GeV [30]. After reheating, the universe is in a radiation dominated phase where the scale factor varies as t ‘I* . During this phase, the universe enters the electroweak symmetry phase at a temperature which we approximate to be of the order of the vacuum expectation value (4). Let T be the temperature of the transition from the radiation dominated phase to the electroweak symmetry phase and H* and T* be respectively the Hubble constant and the temperature at the end of the inflationary stage. We use the relation at the beginning and at the end of the radiation dominated phase, i.e. UT = constant, where the scale factor a(t) goes like a*[2H*(t
- t*) + 1]“2,
(13)
where the star means that the values are taken at the beginning of this stage; H = b/u gives US the useful relation between the Hubble constant at the end of inflation and the one of the electroweak symmetry phase, (14) or approximately HzH*
(15)
This approximative result is compatible with what we can obtain by considering only the Stefan law and the relation UT = constant in the radiation dominated era. In field theory, this Hubble constant is related to the Higgs mass by assuming that the energy density of the electroweak symmetry dominates all other forms of energy density. For that, we use the mechanism which gives the effective potential V& From cosmological observations it follows that the vacuum energy density in the universe at present cannot be much greater than the critical density, then the K constant in Eq. ( 1) is chosen to keep negligible this energy density of the universe after symmetry breaking. Furthermore, in many realistic theories, the phase transition occurs even at T<(4) [ 161; then all high temperature corrections to the effective potential at 4 N (4) are negligible. Hence from Eq. (l), we obtain Veff(O) = K and V&( (4)) = 0, which means that
(16) where m(H) is the Higgs mass defined as m*(H) = d*V’&&)/d4* at q5 = (4). From the relation between the Hubble constant and the energy density (for sake of simplicity we take a flat universe),
b Ouali, J. Derkaoui/Astroparticle Physics 8 (1998) 309-315
313
(17)
where Mp is the Plank mass, and p = V&(O) in the electroweak phase, and using ( 15), ( 16) and ( 17), we find that
H*‘=[($+*)‘]
(g)‘[($)-;].
(18)
Finally, from ( 12) and ( 18) we obtain the relation between the spectrum of perturbation and the Higgs mass in terms of various parameters, t 19) To illustrate the behaviour of the spectrum of perturbation as a function of the Higgs mass, we consider an SO( 10) grand unified theory [ 3 1,281. Furthermore, to develop a new inflationary scenario, we are interested in a very flat potential near the origin, i.e. the spontaneous symmetry breaking arises from one loop radiative correction, hence the mass term is absent from the effective potential and the temperature transition from the inflationary phase to the radiation dominated era is given by T*’ =
&%* ,
(20)
where the one loop effective scalar potential is given by V(~,T~=~B*~4+B*~4[1.(~)2-~]+~/dir’ln{l-exp[-(x’+~)“2]}.
(21)
Here, q is the vacuum expectation value of the adjoint (45) Higgs field which breaks SO( 10) into SU( 5) [ 321, minimum (the GUT scale which can be greater than lOI GeV), B* = 9g4/576r2, and g is the gauge coupling constant. Now, if we suppose that the transition to the electroweak symmetry is done at T N (q5), then the spectrum of perturbations as a function of the Higgs mass becomes CTis the value of 4p at the SU(5)
& = 5324 where we have used A(p) =4B*(ln(g~/cr)*
(22) - $) and (19).
4. Discussions and comments For the present horizon scales, the inflationary models require a very small perturbation amplitude (&N10-410v3) [25] and an extremely small coupling constant A~10-10-10-7 for galaxy formation. Letting & vary in the range [ 10s4, 10W3], we use Eq. (22) to derive limits on the Higgs mass as a function of A (see Fig. 1) for a top quark mass (mt,,r,) equal to 173.3 f 8.4 GeV [33] as given by the latest results of the CDF and DO experiments. In Fig. 1, solid lines are drawn for the central value of mk,r and the la effect on mtop is shown by the dashed lines. For a coupling constant A1lO-‘“, we obtain m(H) 532 GeY in this Higgs boson mass range, the symmetry restoring electroweak phase transition is a first order one [ 341. While for 10-10
314
‘I Ouali, J. Derkaoui/Astroparticle
Physics 8 (1998) 309-315
:; 0.09
: I
: : : : : : : : : :
0.06 0.07
:
:
:
:
:
0.06
6,
:
,s
0.05
:
0.04
: :
,’ 0.03
’
~ ;_;; ,,,, 0
/,,,/(,,,,,,, /I 40
60
120
160
200
240
260
m(H)(GeV)
Fig. 1. Spectrum of density petiations as a function of the Higgs mass. The solid line is for m, = 173.3 GeV, the band delimited by the dashed lines comes from AT,,,, around mt.
92 GeV For 10-910m8, this effect is not appreciable. On the other hand, the upper bound favoured by the electroweak data is m(H) < 300 GeV which, combined with our results for &?!10-3, suggests the A parameter to be less than 3 x lo-*. The method used here for a quartic potential which drives the new inflation may be applied to a quadratic potential which drives the chaotic inflation where the constraints on the amplitude of density perturbations requires constraints on the mass hierarchy. Other potentials (as those used in non-gauge theories) which describe the inflationary universe and are constrained by the same assumptions that are used here could help the discussion on the bounds on the Higgs mass.
5. Conclusions
In this work we have studied the behaviour of the Higgs mass (for a range of the coupling constant) constrained by the amplitude of density perturbations, i.e. we have tried to find a direct relation between the Higgs particle (for which detection is one of the main goals for future experiments in particle physics) and the spectrum of density perturbation, presently known indirectly by means of the microwave background radiation. The detection of the Higgs particle with a mass in the range mentioned in this paper (which is favoured by theory and by experimental data) would confirm the order of magnitude of the density perturbation amplitude. However, it should be emphasized that the results presented here rely strongly on the new inflation hypothesis.
T. Ouali, J. Derkaoui/Astroparticle Physics 8 (1998) 309-315
References [l] P. Janot, Nucl. Phys. B 38 (1995) 264; OPAL ColIaboration, special Physics note for the LEPC presentations of 11-11-1997. [2] HP. Nilles, Phys. Rep. C 110 (1994) 1. [ 31 H.E. Haber, G.K. Kane, Phys. Rep. C 117 (1985) 75. [4] R. Barbieri, Riv. Nuovo Cimento 11 ( 1988) 1. [5] M. Sher, Phys. Len. B 317 (1993) 159; G. Altarelli, I. Isidori, Phys. Lett. B 337 (1994) 141. [6] J.A. Casas, J.R. Espinosa, M. Quiros, Phys. I&t. B 342 (1995) 171; J.R. Espinosa, M. Quiros, Phys. L&t. B 353 (1995) 257. [ 71 Abe et al., CDF collaboration, Phys. Rev. Lett. 74 (1995) 2676. [ 81 S. Abachi et al., DO collaboration, Phys. Rev. Lett. 74 (1995) 2632. [9] J. Ellis et al., Phys. Lett. B 333 (1994) 118. [lo] E.W. Kolb, S. Wolfram, Astrophys. J. 239 (1980) 428. [ 1l] D.A. Kirzhnits, JETP L&t. 15 (1972) 529. [ 121 D.A. Kits, A.D. Linde, Phys. L&t. 42B ( 1972) 471. [ 131 L. Dolan, R. Jackiw, Phys. Rev. D 9 (1974) 3320. [ 141 S. Weinberg, Phys. Rev. D 9 (1974) 3357. [ 151 D.A. Kirzhnits, A.D. Linde, Ann. Phys. 101 (1976) 195. [ 161 A.D. Linde, Rep. Prog. Phys. 47 (1984) 925. [ 171 M. Quiros, J.R. Espinosa, F. Zwhner, Preprint CERN-TH.6577/92 (1992). [ 181 M. Shaposhniiov, Preprint CERN-TH. 6918/93 (1993). [ 191 A.D. Linde, JETP Lett. 19 (1974) 183. [20] S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1973); C.W. Misner, K.S. Thome, J.A. Wheeler, Gravitation (Wiley, New York, 1973). [21] V.F. Mukhanov, H.A. Feldman, R.H. Brandenbetger, Phys. Rep 215 (1992) 203-333. [22] S. Hawking, Phys. Lett. B 115 (1982) 295. [23] A.A. Starobinsky, Phys. Lett. 117 B (1982) 175. [24] A. Guth, S.Y. Pi, Phys. Rev. I_&. 49 (1982) 1110. [25] J. Bardeen, P. Steinhard, M.S. Turner, Phys. Rev. D 28 (1983) 679. [26] R. Brandenberger, R. Kahn, W. Press, Phys. Rev. D 28 (1983) 1809. [27] R.W. Robinett, Jonathan L. Rosner, Phys. Rev. D 26 (1982) 2396. [28] Ma&i Yasu, Phys. Rev. D 24 (1981) 1005. [29] J. Bardeen, Phys. Rev. D 22 (1980) 1882. [30] A.D. Linde, Phys. Lett. B 108 (1982) 389-393. [31] D. Bailin, A. Love, Nucl. Phys. B 233 (1984)204, A. Love, Nucl. Phys. B 243 (1984) 537; F. Wilczek, A. Zee, Phys. Rev. D 25 (1982) 553; A. Zee, Unity of Force in the Universe (World Scientific, Singapore, 1982). [32] H. Georgi, D.V. Nanopoulos, Nucl. Phys. B 155 (1979)52. [33] S. Abachi et al., DO collaboration, Fermilab-pub-97/059-E. [34] 1. Montvay, preprint CERN-TH/95-204, hep-lat/9507024.
315
EISEVIER
Bounds on the Higgs mass from cosmological constraints T. Ouali, J. Derkaoui Department of Physics, Mohamed 1st University*B.P 524, 6000, Oujaix, Morocco
Received 1 July 1997; revised 10 December 1997
Abstract We discuss lower and upper bounds on the Higgs mass using different values of the coupling constant A of the selfinteracting scalar field by considering a new inflationary scenario. We use the value of the amplitude of the primordial energy density fluctuations as inferred from the COBE results and the top quark mass as recently measured by the DO and CDF experiments. The upper bound obtained on the Higgs mass is compatible with the upper hound favoured by the
electroweak data. @ 1998 Elsevier Science B.V.
1. Introduction
One of the key problems in particle physics is the analysis of the electroweak symmetry breaking. The simplest mechanism for the breaking of the electroweak symmetry is real&d in the Standard Model (SM) . To accommodate all observed phenomena, an isodoublet field is introduced, which, after the spontaneous breaking of the symmetry, leaves us with one physical scalar boson. The present LEP experiments have set a lower bound of about 82 GeV on the mass of this particle (the so-called Higgs boson) [ 11. The mass squared term in the scalar potential depends quadratically on the cutoff that can be interpreted as the scale where new physics beyond the standard model sets in. For Higgs masses of the order of 100 GeV, the standard model can be extended to a supersymmetric theory where its simplest version is the Minimal Supersymmetric Standard Model (MSSM) [ 2-41. Lower bounds on the Higgs mass can be derived from the requirement of preserving the stability of the vacuum [5,6]. To illustrate our main idea, we consider only the case of the standard model. This limitation may be related to the discovery of the top quark by the CDF [ 71 and DO [ 81 experiments which constitutes a success for the standard model. This success in predicting the top quark mass has led many physicists to try the same game for the prediction of the Higgs mass [ 91. To investigate the Higgs mechanism, we consider its effects in cosmology. One of the effects of the electroweak phase transition is the expansion mode of the universe which may be exponential -but not inflationary[ 101. Another one is what we are interested in here, i.e. the effect on the Higgs mass. The interplay between particle physics and cosmology is given by the study of the high-temperature phase transitions in gauge theories [ 1 l-141. The one loop temperature-dependent effective potential V&( 4, T) is taken as the basic tool for the discussion of the electroweak phase transition [U-17]. This potential may be [ 17,181 0927-6505/98/$19.00 @ 1998 PflSO927-6505(97)00061-3
Elsevier Science B.V. ALIrights reserved.
I: Ouali, J. DerkaouilAstroparticle
310 %-dqkZ’>
= -(2A+
B)(qb)242
+ Aqb4
Physics 8 (1998) 309-315
+B#41n(42/(#)2)
+ &T2$2(3g2 +g’2 + 2hf) + K,
(1)
where B = 3/1024r2( 2g4 + (g’ + g’)2 - 16h:), g, g’ and h, are the coupling constants of the gauge boson particles and of the top quark which contribute to the one loop effective potential, K is an arbitrary constant, and (4) is the value of the field 4 that minimizes the effective potential. The energy density in the universe due to the existence of a vacuum expectation value for the Higgs field [ 191 contradicts the fact that the present energy density of the universe is less than 1O-29 g cm -3 [ 201. The presence of a large compensating cosmological term in Einstein’s equation may remove the effect of this vacuum energy density [ 191. At high temperature (i.e., in the early universe), symmetries which are presently spontaneously broken should have been exact [ 11,121. The authors of Ref. [ lo] give some details of the phase transition in the Weinberg-&lam model. They found that the condition to be satisfied by the Higgs mass when the vacuum energy density is greater than the radiation energy density is m(H) 5 11.5 GeV for T >> T, (T, is the critical temperature) ; the generalized energy density is dominated by the contribution of relativistic particles, and the scale factor varies as 4. This is the standard hot big bang expansion rate. For T 2 Tc and if m(H) 5 11.5 GeV, then the generalized energy density is dominated by the vacuum energy density and the scale factor has an exponential e’ form. While for T 2 T, the symmetry is broken and the vacuum energy density is compensated by the cosmological energy density [ 191 and once again the expansion rate is the standard one. If T, << mw, it is likely to be a first order phase transition, generating entropy and may be causing inhomogeneities to develop. If for some period the vacuum energy density is greater than the radiation energy density, then the effective pressure of the universe would be negative [ lo]. In this paper, we are not interested in the inflationary consequences of the phase transition of the WeinbergSalam model. But for a short time, the vacuum energy density of the universe due to the electroweak symmetry is much lower than the GUT one and gives cosmological effects. On the other hand, the cosmological perturbation is used to describe the growth of structures in the universe, to calculate the predicted microwave background radiation fluctuation, and in many other considerations [ 211. The inflationary universe in its improved scenario leads to the fact that the present state of our universe could have risen from a wide class of initial conditions, in contrast with the standard cosmology which assumes a particular initial state (nearly homogeneous and isotropic, i.e., a Friedman-Robertson-Walker (F.R.W.) one). And it offers also the possibility to explain a handful of very fundamental cosmological problems; among them the origin of density inhomogeneities. These density perturbations have been calculated by many authors [21-261 which found similar results. The main idea of this work is to find a relation between the cosmological perturbation and the Higgs mass. This relation explains the relation between the physics of the early universe and that of the present day universe. The phase transition, which we consider here, is based on Grand Unified Gauge Theories with a simple Lie group. To illustrate our purpose, we consider one of the viable candidates for GUT of rank higher than 5 and satisfying many of the desirable properties, the SO( 10) group [ 27,281. The same thing could be done for the E6 group by considering the chain of symmetry breaking E6--+SO(lO)-SU(5) x U( 1). 2. Spectrum of the density perturbation
in an inflationary
universe
We assume that the primordial energy density perturbations are due to quantum vacuum fluctuations. The method used to calculate the spectra of density perturbations produced in an inflationary universe with scalar matter field is the gauge invariant formalism developed by Bardeen [ 291. First, let us recall the main results from [ 211; the squared perturbation amplitude in a comoving scale k is s; = ( 1/47r2)((p’2/a)21U&7) 12k3’
(2)
where 77 is the conformal time, p is the homogeneous scalar field which breaks the grand unified symmetry, a(t) is the scale factor and Uk(v) is the mode function which satisfies the equation
T Ouali, J. Derkaoui/Astroparticle t&?7)
+ [k2 -
(l/zY(l/z)-‘luk(77)
Physics 8 (1998) 309-315
z = (a&H)
=o,
311
.
(3)
Here, the prime denotes the derivative with respect to the conformal time and H = a’/~. The solution of the mode equation for the short wavelength perturbation such as k* > ( l/z )“( 1/z ) -* is Q(V) =
4(77i)COS[k(T
-
%)I + [4(rli)/~lsin[~(v
For long wavelengths such as H(Qi)a(a)
- %)I .
< k < H( ~)a(~),
(4)
the solution is
(5) where We Let radius
fChis the crossing horizon time. can use the above results to find the power spectrum & by means of Eq. (2). us first consider the power spectrum during inflation. On scales which at time t are still inside the Hubble (kph(t) > H(t) ) , Uk (7) iS given by (4) and the power SpeCWLUn iS
Sk =
(1*/4g*) I@(t) 1,
(6)
where the dot indicates a derivative with respect to physical time t = Judr]; kph = k/u(v) is the wave number in physical coordinates, and 1*= 81r/3Mi. On scales which were inside the Hubble radius, at the beginning of inflation at time ti but which are outside at time f: H(t) > kph(?) > H(fi)U(tt)/U(t); We USC (5) t0 obtain
where H = b/u is the Hubble constant during inflation. The above equations (6) and (7) are valid for the respective wavelengths both during and after inflation. Here, we assume that after inflation the universe enters a period where the scale factor u(t) varies like Ji. On scales which are inside the Hubble radius at the beginning of inflation but outside at the end (at time t*), i.e., for which H(r*)a(r*)/a(t)
>
kph(t)
>
H(ti)a(ti)/a(f)
9
(8)
the power spectrum is obtained from (7), thus for any t > t*, one has & = (12/2,rr2)(g(t)H2/ji)t,h.
(9)
To express this spectrum of perturbation as a function of the Hubble constant at the end of inflation (in which we are interested) and of other parameters which are model dependent, we use the slow roll approximation. From the equation of motion of the scalar field, one has ++33H+=-t$, where V is the effective potential in the grand unified symmetry and V+,= dV/dq
(10) We have in this approximation
(b=-!h_ 3H’
(11)
which is constant, i.e. @ is the same at the crossing horizon time and at the end of inflation. Thus,
(12)
312
T. Ouali, J. Derkaoui/Astroparticle
Physics 8 (1998) 309-315
We notice that this expression depends on the used model. In a recent work [21], the expression (12) has been used to compare the theoretical predictions with the observational constraints for two specifics cases. For a quadratic potential V(q) = ~m*q* (where the time evolution of the scalar factor after inflation varies like P with p = 3) the mass hierarchy is constrained by m/M, < 10m6, while for a quartic potential V(q) = Acp4 the coupling parameter A must be less than 10-14.
3. Higgs mass Now we need to relate this power spectrum to the Higgs mass by means of the Hubble constant at the end of inflation. For that purpose we assume that the universe undergoes different stages of evolution. Among them, the inflationary stage which after its rolling and its oscillations gives rise to particles which rapidly decay and reheat the universe up to the temperature of lOi GeV [30]. After reheating, the universe is in a radiation dominated phase where the scale factor varies as t ‘I* . During this phase, the universe enters the electroweak symmetry phase at a temperature which we approximate to be of the order of the vacuum expectation value (4). Let T be the temperature of the transition from the radiation dominated phase to the electroweak symmetry phase and H* and T* be respectively the Hubble constant and the temperature at the end of the inflationary stage. We use the relation at the beginning and at the end of the radiation dominated phase, i.e. UT = constant, where the scale factor a(t) goes like a*[2H*(t
- t*) + 1]“2,
(13)
where the star means that the values are taken at the beginning of this stage; H = b/u gives US the useful relation between the Hubble constant at the end of inflation and the one of the electroweak symmetry phase, (14) or approximately HzH*
(15)
This approximative result is compatible with what we can obtain by considering only the Stefan law and the relation UT = constant in the radiation dominated era. In field theory, this Hubble constant is related to the Higgs mass by assuming that the energy density of the electroweak symmetry dominates all other forms of energy density. For that, we use the mechanism which gives the effective potential V& From cosmological observations it follows that the vacuum energy density in the universe at present cannot be much greater than the critical density, then the K constant in Eq. ( 1) is chosen to keep negligible this energy density of the universe after symmetry breaking. Furthermore, in many realistic theories, the phase transition occurs even at T<(4) [ 161; then all high temperature corrections to the effective potential at 4 N (4) are negligible. Hence from Eq. (l), we obtain Veff(O) = K and V&( (4)) = 0, which means that
(16) where m(H) is the Higgs mass defined as m*(H) = d*V’&&)/d4* at q5 = (4). From the relation between the Hubble constant and the energy density (for sake of simplicity we take a flat universe),
b Ouali, J. Derkaoui/Astroparticle Physics 8 (1998) 309-315
313
(17)
where Mp is the Plank mass, and p = V&(O) in the electroweak phase, and using ( 15), ( 16) and ( 17), we find that
H*‘=[($+*)‘]
(g)‘[($)-;].
(18)
Finally, from ( 12) and ( 18) we obtain the relation between the spectrum of perturbation and the Higgs mass in terms of various parameters, t 19) To illustrate the behaviour of the spectrum of perturbation as a function of the Higgs mass, we consider an SO( 10) grand unified theory [ 3 1,281. Furthermore, to develop a new inflationary scenario, we are interested in a very flat potential near the origin, i.e. the spontaneous symmetry breaking arises from one loop radiative correction, hence the mass term is absent from the effective potential and the temperature transition from the inflationary phase to the radiation dominated era is given by T*’ =
&%* ,
(20)
where the one loop effective scalar potential is given by V(~,T~=~B*~4+B*~4[1.(~)2-~]+~/dir’ln{l-exp[-(x’+~)“2]}.
(21)
Here, q is the vacuum expectation value of the adjoint (45) Higgs field which breaks SO( 10) into SU( 5) [ 321, minimum (the GUT scale which can be greater than lOI GeV), B* = 9g4/576r2, and g is the gauge coupling constant. Now, if we suppose that the transition to the electroweak symmetry is done at T N (q5), then the spectrum of perturbations as a function of the Higgs mass becomes CTis the value of 4p at the SU(5)
& = 5324 where we have used A(p) =4B*(ln(g~/cr)*
(22) - $) and (19).
4. Discussions and comments For the present horizon scales, the inflationary models require a very small perturbation amplitude (&N10-410v3) [25] and an extremely small coupling constant A~10-10-10-7 for galaxy formation. Letting & vary in the range [ 10s4, 10W3], we use Eq. (22) to derive limits on the Higgs mass as a function of A (see Fig. 1) for a top quark mass (mt,,r,) equal to 173.3 f 8.4 GeV [33] as given by the latest results of the CDF and DO experiments. In Fig. 1, solid lines are drawn for the central value of mk,r and the la effect on mtop is shown by the dashed lines. For a coupling constant A1lO-‘“, we obtain m(H) 532 GeY in this Higgs boson mass range, the symmetry restoring electroweak phase transition is a first order one [ 341. While for 10-10
314
‘I Ouali, J. Derkaoui/Astroparticle
Physics 8 (1998) 309-315
:; 0.09
: I
: : : : : : : : : :
0.06 0.07
:
:
:
:
:
0.06
6,
:
,s
0.05
:
0.04
: :
,’ 0.03
’
~ ;_;; ,,,, 0
/,,,/(,,,,,,, /I 40
60
120
160
200
240
260
m(H)(GeV)
Fig. 1. Spectrum of density petiations as a function of the Higgs mass. The solid line is for m, = 173.3 GeV, the band delimited by the dashed lines comes from AT,,,, around mt.
92 GeV For 10-9
5. Conclusions
In this work we have studied the behaviour of the Higgs mass (for a range of the coupling constant) constrained by the amplitude of density perturbations, i.e. we have tried to find a direct relation between the Higgs particle (for which detection is one of the main goals for future experiments in particle physics) and the spectrum of density perturbation, presently known indirectly by means of the microwave background radiation. The detection of the Higgs particle with a mass in the range mentioned in this paper (which is favoured by theory and by experimental data) would confirm the order of magnitude of the density perturbation amplitude. However, it should be emphasized that the results presented here rely strongly on the new inflation hypothesis.
T. Ouali, J. Derkaoui/Astroparticle Physics 8 (1998) 309-315
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