Higgs and top mass from an extremality condition

Volume 238, n u m b e r 2,3,4

PHYSICS LETTERS B

5 April 1990

H I G G S AND T O P MASS F R O M AN EXTREMALITY C O N D I T I O N A.J. DAVIES School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia

S. MELJANAC and I. PICEK Ruder Bogkovi? Institute, P.O.Box 1016, YU-41001 Zagreb, Croatia, Yugoslavia Received 28 April 1989

We reconsider the vacuum stability analysis of the standard model with three generations and one Higgs doublet. By assuming the validity of this model up to the Planck scale, we impose extremality conditions at the Mw scale on the parameters of the H iggs potential. The conditions lead to the predictions mt ~ Mw and rnH ~ Mz. This result implies that the top might be discovered soon, while the discovery of the Higgs seems to be relegated to the SSC.

1. Introduction

8 ~ 2 ~-~ 32 92 34 d ~ = 6 2 2 --~g12--~gz2+~gl

There are sufficient reasons to view the standard model (SM) as an effective low-energy theory. Motivated by the forthcoming experiments at Fermilab and CERN, we take under scrutiny "the standard way beyond the SM", which relies on the notion of an elementary Higgs. After the discovery of the gauge bosons W ± and Z, both the top quark and the Higgs boson provide a crucial test of the SM and the window beyond the SM. The method adopted here represents a variant of the renormalization-group-equation ( R G E ) analysis of the scalar potential, providing a correlated constraint on the top-quark and Higgs masses. The usual analysis of that sort (for a review, see refs. [ 1,2] ) applies to the parameters 2 and ~2 of the classical potential of the SM. This potential of a single Higgs scalar doublet is represented at the tree level of perturbation theory by v(O)= -u~*O+~2(~*O) 2 .

(1)

The parameters 2 and Ft2, which are a priori unknown, depend on the logarithmic m o m e n t u m scale t = l n A through the RGEs in the MS scheme

+ 3~51s2 .2o2

9 4+6(2_g~)gt2 +~g2

(2a)

and 2 dfl 2

~=~

2

3 2 9 2 (32-~gl-~g2+3g2t).

(2b)

Here gl.2,3 are the gauge couplings for U( 1 ) v, SU(2)L and SU (3)c, respectively, while gt is the Higgs-top Yukawa coupling (other Yukawas and the quark mixing being neglected). The RGEs for the couplings g1,2,3,t a r e

8~ 2 d~t =aig 4 (no sum) ,

(3a)

where

(at,a2, a3)=(~, -~-, - 7 ) , and 892

~

=g2(9g2 -+~17g , 2 - z9g 22 - 8 g ~ ) .

(3b)

The system (2) and (3) was studied previously [ 3,4 ] with (2b) excluded, and more recently [ 5 ] with the RGE (2b) for/2 2 included.

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431

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PHYSICS LETTERS B

quire a mild flow of 2 and v up to the Planck scale by assuming u~ that

2. Extremality at the electroweak scale

In the following we find it more suitable to deal with the parameters 2 and v 2 of a shifted potential V(g~) = ½ 2 ( 0 * 0 - v 2) 2

M 2 =~(gl+g2)v 1 2 2 2

(5a)

and of the Higgs and the top quark: m2=22v2=2~t

2 ,

mt=gtv.

(5b)

At A=Mw, g2 =0.124, g22=0.424, g2 = 1.5, and the mass of the W boson implies V(Mw) = 174 GeV. We therefore study the complete system for g~,2,3,t. 2 and v 2, where the R G E for v 2 ( t ) = f 1 2 ( [ ) / 2 ( t ) obeys the equation related to (2): 87~2 d v 2 ( 1 2 d//2 " ~ - = 87~2V2 dt

~tMw ~ 0 '

dr2 dt Mw = 0 "

(7)

(4)

Such a classical potential is more convenient for two reasons: (i) it avoids a huge negative vacuum energy density provided by ( 1 ), and (ii) it is directly linked to the boundary conditions of the RGEs, i.e., the inputs at the scale A = M w : the VEV of ~, ( 0 ) = (o), generates masses of weak bosons: M2w = ~62 1.2,~ 2 ,

5 April 1990

Ida)

2

Let us stress the mathematical meaning of eq. (7): (i) First, it enables us to keep the couplings from exceeding certain limits up to the Planck scale and thus to keep the validity of perturbation theory. A similar requirement of perturbative validity has been used in ref. [ 9 ] in the context of the two-Higgs model. (ii) Eq. ( 7 ) may also be viewed as the generalizedstability-principle requirement: the content of an effective theory and the coupling constants at the characteristic energy scale should be such that//-functions of the couplings 2 and v, connected with the symmetry-breaking scale, are zero. This will soften the logarithmic divergences which are left over in the effective theory under consideration. This softening represents a generalization of cancelling quadratic divergences [ 10,2 ]. Having exposed the meaning ofeq. (7), let us turn to its predictions. For simplicity, by Mw we typify the electroweak scale at which the parameters of the Higgs potential are of zero slope. Then relations (7) can be written as conditions on mn and m t given by m 4 - m u2( M z2 + 2 M 2 - 2 m ~ ) +M4+2M 4-4m4=0,

= v 2 ( _ 3 2 + 3 zg~2 +zg2 9 2 + 3g~

1 ) - - -2- [~g4+3-'2"2"2"4a-6(2--g2)g2] 4~51,52 ~8,52 ~

m 4

(6)

The running of the non-asymptotically free Higgs selfcoupling has already been given by (2a). If we knew the principle by which nature selects values of v and 2, we would be able to predict the Higgs and top masses in (5b). We observe that some insight into v and 2 might come from the physical picture of a "great desert" up to the Planck scale, which enters both the familiar grand unification [6 ] and a somewhat less orthodox antiunification [7] programme. In the latter approach, the running of gauge coupling constants leads to the prediction of the three fermion families [7 ] which is also built into our picture and into our eqs. ( 2 ) - ( 6 ) . In order to ensure such a picture, we re432

(8a)

- m H2 ( M z2 + 2M2w-2m 2)

+2(M 4+2M 4-4m

4)=0.

(8b)

The corresponding two curves have two intersections (full and empty dot in fig. 1 ). The result acceptable in the present framework is represented by the full dot: 1A//'4 ~1/4 -~.o "/Q G e V , m , = ( ~1 M z4 + ~,~w~

(9a)

mu = [M~ + 2M~v - (M~: + 2 M 4 ) '/2] ,/2 -~ 94 G e V .

(b)

In order to check the stability of this prediction, let us look for a change in it when we impose (7) at ~ 3 *a An explicit check on similar conditions has been presented in

ref. [81.

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PHYSICS LETTERS B

mH(GeV]

/

//

150

// i / / I I I

Mz -

I

Mw -

C=2Ma+M4=

5 April 1990 ~m.4 _ 4 m 4 >~0 ,

(12)

obtained by requiring that the Coleman-Weinberg potential [ 12 ] be bounded from below. However, the conditions (7) allow us to go beyond the correlated mH-m¢ constraints in the form o f bounds, which typically have the shape of a dashed curve ~3, as shown in fig. 1.

I I I I

SO

3. Discussion and conclusion

I I I

/ 210

410

,

60

/

I

~,Mz, 80

100

120

1~0 1150 180 m t (GeV}

Fig. 1. Our prediction of m, and mr~ (full dot), Veltman's values (solid curve ), and constraint from triviality (dashed curve). Mw (the naive value of the Fermi scale, 2 - ~/4G ~- 1/2 . . . 250 GeV ). This shift is expressed by d~ 3Mw , ~ ~d2M w + d~Z2 dt2 Mw = 0 ,

dv ~ dt

d~ .w 3Mw

--~

d~v + dt 2

MW=0,

(10)

and leads to a slight modification o f eqs. (8), having as solutions m~=85 G e V ,

(lla)

mH = 9 2 G e V .

(1 lb)

A change o f the top mass within 10%, and the Higgs mass by 2%, can be considered as the sufficient stability of the prediction. We observe an interesting piling-up of the spectrum ~2 around the values close to Mw-~ 80 GeV and Mz - 91 geV. Relations (7) can be viewed as saturated bounds (equality ~ replacing inequality < ) of the inequalities which are somewhat stronger than the corresponding condition ~2 We thank Professor V.D. Shirkov for bringing to our attention a physical scenario of dimensional reduction of multidimensional gauge theories [ 11 ], where mH~_Mz is automatic.

Let us first relate our results (9) or ( l 1 ) to some other correlated Higgs-top mass bounds, which already exist on the market: (i) The conclusions deduced from the triviality o f the Higgs self-coupling are represented by the dashed line in fig. 1 corresponding to Beg et al. in ref. [3 ]. This has recently been established independently of perturbation theory [ 1 ]. In particular, the triviality bound supplement with the desert up to Mp~anckgives mn < 140 GeV, in agreement with the value obtained in the present paper. (ii) The reduction of the coupling method [ 14] (assuming that the RGEs for all couplings are driven by the largest gauge coupling and keeping only the tquark Yukawa coupling) offers two options, either m , = 61 G e V ,

m t = 81 G e V ,

or

mH>40GeV,

mt<81GeV.

(13)

(iii) However, our prediction seems to be closer to that proposed by Veltman [ 10,2 ]. That prediction is given by the sum rule relying on the absence of quadratic divergences, 4 ~ m~=6M~v+3MZz+3m 2 . f

Refs. [ 10,2] report rnt = 69 GeV

for rnH << M w ,

mt=77.5GeV

forma=Mw,

(14)

up to a modification of a few GeV due to radiative corrections. This prediction with up-to-date values for Mw and Mz is also displayed in fig. 1. ~3 Or some modification [ 13] for a heavy top (m~>Mw). 433

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As a s e c o n d type o f a r g u m e n t in f a v o u r of the cond i t i o n s ( 7 ) let us refer to the r a d i a t i v e corrections in the SM which are sensitive to rnt a n d m n which are considered here. T h e analysis by Ellis a n d Fogli [ 15 ], based o n precision electroweak m e a s u r e m e n t s , is o f interest for us. T h e y d e t e r m i n e d sin20w by c o m b i n ing the four different n e u t r a l - c u r r e n t sectors using the least-squares m e t h o d . U s i n g rnH=Mz as a n i n p u t , they o b t a i n e d the overall p r o b a b i l i t y d i s t r i b u t i o n m a x i m i z e d for r n c ~ 1.45 G e V a n d rnt ~ 7 5 - 8 5 G e V .

(15)

Such surprising a g r e e m e n t with o u r p r e d i c t i o n s m a y suggest that the e x t r e m a l i t y c o n d i t i o n s ( 7 ) are at work. However, the final c o n f i r m a t i o n c a n c o m e o n l y f r o m e x p e r i m e n t s . B ° - B ° m i x i n g [ 16 ] a n d the m o s t recent results f r o m a T e v a t r o n already i n d i c a t e that the t o p - q u a r k mass a p p r o a c h e s the M w region. To conclude, we have presented the following chain of arguments: Physical scenario ~ e x t r e m a l i t y c o n d i t i o n s ( 7 ) ~ p r e d i c t i o n s ( 9 ) or ( 11 ) , w h e r e b y the last step has b e e n checked against the p r e d i c t i o n o b t a i n e d using different m e t h o d s w i t h i n the s a m e physical picture. A c c o r d i n g to o u r p r e d i c t i o n , the t o p - q u a r k discovery is to be expected already d u r i n g the c u r r e n t Ferm i l a b collider run, b u t a n e l e m e n t a r y Higgs stays elusive. T h u s , the next g e n e r a t i o n of accelerators (such as the SSC) is expected to be crucial in clarifying the m e c h a n i s m o f electroweak s y m m e t r y breaking.

Acknowledgement O n e o f us ( I . P . ) acknowledges partial s u p p o r t by

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5 April 1990

the N S F / J F 899-31. S.M. wishes to t h a n k the U n i versity of M e l b o u r n e for hospitality.

References [ 1 ] P. Langacker, in: Proc. XXIV Intern. Conf. on High energy physics (Munich 1988), eds. R. Kotthaus and J.H. Kiihn (Springer, Berlin, 1989) p. 190; see also A. Billoire, in: Proc. XXIII Rencontre de Moriond '88, Electroweak interactions and unified theories, ed. J. Tran Thanh Van (Editions Fronti6res, Dreux, 1988) p. 147; P. Franzini, in: Proc. XXIII Rencontre de Moriond '88, Electroweak interactions and unified theories, ed. J. Tran Thanh Van (Editions Fronti6res, Dreux, 1988 ) p. 159. [2] M. Veltman, in: Proc. XXIII Rencontre de Moriond '88, Electroweak interactions and unified theories, ed. J. Tran Thanh Van (Editions Fronti6res, Dreux, 1988 ) p. 133. [ 3 ] M.A.B. Beg, C. Panagiotakopoulos and A. Sirlin, Phys. Rev. Lett. 52 (1984) 883; D.J. Callaway, Nucl. Phys. B 233 (1984) 189; E. Ma, Phys. Rev. D 31 (1985) 322. [4] M. Lindner, Z. Phys. C 31 (1986) 295. [ 5 ] C. Wetterich, preprint DESY 87-154. [6] A. Zee, Unity of forces in the universe (World Scientific, Singapore, 1982). [7] D.L. Bennett, H.B. Nielsen and I. Picek, Phys. Lett. B 208 (1988) 275; H.B. Nielsen, preprint NBI-HE-89-01 (1989). [8] A.J. Davies and S. Meljanac, Mod. Phys. Lett. A 4 (1989) 137; J. Basecq, S. Meljanac and L. O'Raifeartaigh, Phys. Rev. D 39 (1989) 3110. [9] G. Kreyerhoffand R. Rodenberg, Phys. Lett. B 226 (1989) 323. [ 10] M. Veltman, Acta Phys. Pol. BI2 ( 1981 ) 437. [ 11 ] I.P. Volobujev and Yu.A. Kubyshin, Teor. Mat. Phys. 68 (1986) 368. [ 12] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888. [ 13] E. Gross and E. Duchovni, Phys. Rev. D 38 (1988) 2308. [ 14] J. Kubo, K. Sibold and W. Zimmermann, Nucl. Phys. B 259 (1985) 331. [15] J. Ellis and G. Fogli, Phys. Lett. B 213 (1988) 526. [ 16 ] H. Albrecht et al., Phys. Lett. B 192 (1987) 245.