Hierarchy of quark masses in the isotopic doublets in N=1 supergravity models

Volume 214, number 3

PHYSICS LETTERS B

24 November 1988

H I E R A R C H Y OF QUARK MASSES IN T H E I S O T O P I C DOUBLETS IN N = 1 SUPERGRAVITY M O D E L S Marek OLECHOWSKI and Stefan POKORSKI Institute of Theoretical Physics, University of Warsaw, H6za 69, PL-00-681 Warsaw, Poland Received 1 August 1988

In the framework of some generic softly broken SUSY models we study the SU (2) × U ( 1 ) breaking by radiative corrections, starting with the Yukawa couplings which at the Planck scale Mp satisfy h ' = h b. Physically acceptable solutions exist with the hierarchy of VEVs: vz/v~ ,,~mffmb provided m~> 50 GeV. When the SUSY breaking is driven only by the gaugino mass the solutions uniquely predict v2= O ( 10 ) v~ and the top mass in the range 50-65 GeV. Also, the mass ratio in the second quark generation can be accounted for.

The well known patterns of the quark mass matrix are the dramatic breaking of the generation symmetry and also very strong isotopic spin symmetry breaking within each generation. In the minimal version of the standard model both symmetries are explicitly broken by properly adjusted Yukawa couplings which in turn should be explainable by some deeper theory. In this paper we would like to discuss the possibility that the vertical symmetry is broken by the vacuum. For such a scenario a non-minimal Higgs sector is required but multi-doublet models have anyway been often invoked for various reasons and in particular at least two doublets are needed in the minimal SUSY models. In the latter models two different VEVs v~ and v2 are driving the down- and up-quark masses, respectively. It is then conceivable that the isotopic splitting within generations is mainly due to the hierarchy of scales v~ << v2 with the Yukawa coupling matrices h U~ h o. The question we address in this paper is whether such a vacuum can be generated dynamically in softly broken SUSY models. Several remarks are in order here. Firstly, such a scenario is perfectly consistent with present limits on the charged scalar Yukawa couplings [1 ]. Secondly, the mass patterns of the first generation are clearly different from the others and we shall discuss briefly this point at the end of the paper. And thirdly, strictly speaking we take h u = h D at the Planck scale M p and physics 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

at scale Mw is to be determined by the renormalization group ( R G ) evolution. In the supersymmetric extensions of the standard model the supersymmetry breaking drives SU (2) × U ( 1 ) breaking. Particularly attractive is the possibility [2] that the electroweak symmetry breaking is generated dynamically by radiative corrections. This mechanism has been studied in a large number of papers [3 ]. Here we adopt logically the most straightforward approach: take a concrete SUSY model with softly broken SUSY; assume that physics above Mp fixes the values of all the parameters of the lagrangian at the scale Mp and use those values as the boundary conditions for the R G evolution to low energies. In particular we take the dimensionful parameters of the SUSY breaking to be fixed by physics at and above Mp. This point of view has some support in the context of string induced supergravity models [4 ], although the exact origin of the SUSY breaking scale is far from being clear. Some further generalities of our procedure are the following. The R G evolution of the parameters of the lagrangian is performed by means of the supersymmetric one-loop R G equations (with soft terms breaking SUSY). At some scale Qo the minimum of the scalar potential begins to be at some non-zero VEV v= ~ v22signaling the breaking of SU (2) × U ( 1 ) and at some other scale Q~ < Qo one gets unstable solutions with v=oo (fig. 1 ). The region (Q~, Qo) de393

Volume 214, number 3

PHYSICS LETTERS B

~: [ ( ; , ~ * ~ ) / z ]

24 November 1988

procedure, namely, we choose the r e n o r m a l i z a t i o n scale QR =MEXP and in specific softly b r o k e n SUSY models we look for the solutions (i.e. for the space o f the b o u n d a r y conditions at O ( M p ) ) which satisfy our constraints. These constraints are

z

c:/~ a

Mw =MEwxP ,

(2)

mb(Q=Mw) = (4+0.5) GeV,

(3)

and specifically for the present investigation hU_~h °

lnQa lnQo

lnQ

Fig. 1. Parameters entering the scalar potential in model I as functions of the renormalization scale Q. Also marked are the values of b=v/rho. For z3=~, CO=V2/VI=I and for b=0, w = co....... The scale QR= MZwx" is inbetween Q~ and Qo. pends only on the initial values o f the dimensionless parameters. On the way down from Qo to Q~ the mini m u m o f the tree level scalar potential becomes deeper a n d deeper and one m a y ask where is the true minim u m o f the potential (i.e. where to stop the R G evol u t i o n ) . However, the consistency o f this a p p r o a c h requires that one m u s t stop the evolution at Q>_-QR where QR is defined by the relation Mw = ½g(0R = M w ) v ( a R = M w ) .

( 1)

G o i n g below QR means eventually Q << Mw a n d we should then use a new set o f one-loop R G equations with all the massive particles (m>_-Mw), including Higgs scalars and massive gauge bosons, decoupled. Thus, for fixed initial values o f the p a r a m e t e r s (in particular for the fixed scale rho o f the soft SUSY breaking) and in the region o f applicability o f the used set o f R G equations the tree level scalar potential has a well-defined m i n i m u m at QR- It is natural to take Q = QR to m i n i m i z e higher o r d e r corrections which then become simply the s t a n d a r d SU ( 2 ) × U ( 1 ) corrections a n d there is no reason to expect t h e m to destabilize the m i n i m u m ~ In practice we do not know the actual values at O ( M p ) o f the SUSY breaking p a r a m e t e r s a n d o f the Yukawa couplings. Therefore we have to invert our ~ The sometimes used procedure of minimizing V,,cc+A V with V,ccalways obtained from the original set of the RG equations looks to us inconsistent for the same reason of using the RG equations beyond their region of applicability. 394

a t O ( M p ) ~2

(4)

Actually in the m a i n part o f this p a p e r we neglect the Yukawa couplings o f the first two generations and the last condition is replaced by h ' ~ h b at the Planck scale. Finally, let us m e n t i o n that instead o f Mp we start the evolution at Mx defined as the unification scale for a2 and a ~of, respectively, SU ( 2 ) and U ( 1 ) given the values o f the electric charge and sin20w=0.227 _+ 0.009 [6] at M w ~3. This way we circumvent the p r o b l e m o f possibly new thresholds a b o v e Mx. At Mx we arbitrarily take O<~ht-hb<~ O.2h t ,

(5)

as our b o u n d a r y condition. A n i m p o r t a n t r e m a r k is that for ht/hb<~0.96-0.98 the top quark becomes lighter than the b o t t o m quark. Thus, the hierarchy v2/Vl ~ m t / m b is the m a x i m a l one we can generate by radiative corrections in SUSY models. After those preliminaries let us specify our models. M o d e l I is the m i n i m a l SUSY m o d e l with the superpotential hu (~LORIg'I2 d-hDQLDRH~ q-hLLLERI2Ij q-g'I2Ii I212, (6) a n d with the supergravity induced soft SUSY breaking terms

~2 h U~hO e.g. in the minimal left-right symmetric supersymmetric model. In our RG evolution we neglect the scale Mw~ but the analysis of the neutral-current sector shows [ 5 ] that MwR must be close to the unification scale and therefore its inclusion would not alter much the present results. ~3The value of as of the SU (3) at Mw is then determined by the evolution and for the two models considered below a~=0.11 and as=0.10, respectively (for sin20w=0.227).

Volume 214, number 3

PHYSICS LETTERS B

-- E m~(~*Oi- ½ Z t~a~a~ka+AuhuCtLhRH2 i

a

+ A D h D q L d R n t +ALhL~'LeRHI

+B/t'HtH2. (7)

Model II is the superstring inspired S U ( 3 ) × SU(2) × U ( 1 ) × U ' (1) model with chiral superfields in 27 of E6. In this model # = 0 but there are extra terms in the superpotential [ 7 ]

,tfi, fi2~+kDISc~,

(8)

where N-(1,

l,O, ~ ) ,

D=(3, 1,-½,-]),

D¢-= (3, 1, ~, - 1 ) ,

(9)

and there is the respective modification of the SUSY breaking terms. The R G equations for both models are well known [ 8 ] and will not be repeated here (of course, we include h b # 0). In both models, we take as the origin of SUSY breaking at M~ only the gaugino masses r~o and we put m,2 = A~ = B = 0. An inspection of the R G equations shows that this case gives solutions with the maximal hierarchy v2/v~ ~ m,/rnb with the smallest possible value of m , In more general cases the solutions with the maximal hierarchy of VEVs also exist but they correspond to larger values of rn,. In model I the relevant potential for the breaking o f S U ( 2 ) × U ( 1 ) reads

V=-~(g]+ ~ g~j ) ( I H, i ~_ IH2 I~) 2 +/22 IH112+/222 [H2 12-/22(H~H2 + h . c . ) ,

(10)

where

u,~=mL+/2 ~, ~ = r n ~ , ~ + a 2, u ~ = - B a ,

(ll)

and in Model II

V=m 2, [H, 12+ t n ~ I//2 [2 + m 2 I N I 2 + (2AaH1H2N+h.c.) +).2( ]HL 12]N] 2+" [H2 [2[N12+ ]HIH212) + ~l ' 3~g j~( l H l

12_

]H21 2)2

1 2 + +~g2(Hl (½r)HI +HJ-(½r)H2)

+ 1.3.t2/5 ~s, t g lIlXI ~ , 1 2 - ] l H , I2 - ~ [ H 2 12)2 .

(12)

24 November 1988

ical formulae. The symmetry is broken at the tree level in the interval of Q such that the running parameters satisfy the inequality [ 9 ] /2~/2~
(13)

and, as before, rho is the universal gaugino mass at

Me. In the approximation /2~.22~ r n 2 _ (i.e. /2<< rn 2, ) this condition is depicted in fig. 1. The interval of Q in which the symmetry is broken depends merely on the initial values of the dimensionless parameters: Yukawa couplings, gauge couplings and/2/r~o. In our case we impose (5) and, according to ( 1 ) and (2), the region of symmetry breaking must include QR = M ExP. This restricts the space of the initial values of the parameters (h ~,/2/rho). Next, for each allowed h', the condition (3) constrains the ratio o9= v2/v~ at QR = MEwxeFor h t ~ h b the breaking of SU(2) X U ( 1 ) occurs at Q,~M~vxP only for such values of h t that m b ~ 4 GeV implies o9~O(10) and rnt ~ 50-65 GeV. More general boundary conditions at O(Mp) with non-vanishing scalar masses give solutions with larger mt. It is clear from fig. 1 that fixed h t and fixed o9(QR = MewxP) corresponds to a definite value of/2/ rho and also of 13(QR ) = V(QR)/ffZo which in turn implies a definite value of rho. Generically, co >> 1 corresponds to z3(QR) << 1 and consequently to rho >> Mw. The region in the space (h t,/2/r~o) which gives solutions satisfying our constraints is shown in fig. 2 and is compared with the analogous region for h t = ( 10-12) h b. We notice a somewhat different correlation of the two parameters in the two regions: in the former small changes in h t induce larger changes in/2/rho and in the latter the situation is reversed. Limits on the masses of the sparticles (in particular the chargino) put additional constraints on ffto which has to be O ( 1 TcV). The spectrum for a typical solution is collected in table 1 ~4. The solutions are automatically natural in the sense of ref. [ 10] due to the presence of only one scale (rho) at O(Mp). (As usual in those models, the strongest dependence of the scale of the SU (2) × U ( 1 ) symmetry breaking is on the Yukawa coupling: Mz/Me = exp [ -

O(1)/h2]. Results in Model II are similar and they are sum-

Let us now discuss in some detail S U ( 2 ) × U ( 1 ) breaking in model I. In model II the problem is similar but it is more difficult to get approximate analyt-

*$4The solutions satisfy constraints sufficient for the absence of the colour and the electromagnetic symmetry breaking.

395

Volume 214, number 3

PHYSICS LETTERS B

24 November 1988

0.5-

0.02-

ho =0.07 COmia =B.3

0.0 -0.5 ¸

=

~

~

h

~

1D" 12 0.01-

o

G

-1.0 -1.5 0.00

I

-2.5

0.00

I

I

o.15

o.~o

ho=O.09 ¢~mla =iO.O ~om~, =14.1

-2.0

o.6s

o.io

o.is

o.~o

o.zs

h~

0.01-

Fig. 2. We plot for two different ranges of values of the ratio ht/ h t"the space of parameter values at Mp for which SU(2) × U ( 1) is broken and the conditions ( 1)- ( 3 ) satisfied in ( model I ). The small shaded areas correspond to fixed values of sin20~ (from the left to the right sin20~=0.218, 0.227, 0.236, respectively) and the overall region corresponds to the experimentally allowed range sin20w=0.227 _+0.009. m a r i z e d in fig. 3 a n d table 1. O n e p o i n t w o r t h stressing is t h a t for small 2 (see eq. ( 8 ) ) the global m i n i m u m o f the p o t e n t i a l Vu~ggs= Wneutral -[" Wcharged ( t h e p o t e n t i a l is explicitly w r i t t e n d o w n in ref. [7 ], eqs. ( 4 . 2 9 ) a n d ( 4 . 3 0 ) ) c o r r e s p o n d s to Vcharg~d= 0 . I n d e e d , o n e gets

0.00" 0.05

o.io

o.zs

ko Fig. 3. We show for two different values ofh'=hb=ho the space of values at Me of the other two Yukawa couplings (in model II) for which SU(2) × U ( 1 ) is broken and the constraints ( 1)-(3) satisfied. Also given are the corresponding values of (o= v2/v~. The small shaded areas correspond to fixed values ofsin20~ (from the left to the right sina0~=0,227, 0.236, respectively) and the overall region corresponds to the range sin20~.=0.227_+0.009. For h t = h b= ho SU (2) × U ( 1 ) is broken for ho in the range (0.04, 0.12).

Table 1 Model I

h'/h b h~j )-o k. Po (GeV) ~ho (GeV)

v2/v, Mz (GeV) mr, (GeV) mt (GeV) q¢+-i71+ (GeV) (GeV) lightest neutralino (GeV) ~1(GeV) (GeV) H + (GeV) neutral scalars (GeV)

396

1.2 0.08

16 980 9.8 4.5 53.9 22, 1000 2150 18 1900-2060 420-680 415, 415 90, 405, 405

Model II 1.0 0.08 0.009 0.117 1500 11.5 910 4.4 57.2 34, 715 1500 23 1520-2570 680-930 480-930 90-1290

VHiggs =2.2.AaVlV2X+J. 2VlV2"}-ag2v2(IVl 2 2 1 2 2 - ]2--V2)

+Jtv2,x, (Iv712+v~)l,

(14)

w h e r e (after a p p r o p r i a t e gauge t r a n s f o r m a t i o n s ) (H2)=

(0)
Vi-

'

(N)mX,

(15)

w i t h v2, v, a n d x real. M i n i m i z i n g the p o t e n t i a l ( 1 4 ) w i t h respect to a a n d ¢~ w h e r e v, = a cos ¢~, I v i- I = a sin ¢~o n e gets the m i n i m u m for 0 = nn p r o v i d e d k4ax 222-2

v2x/v2+lv?~

[ -g2<0"

(16)

( T h i s c o n d i t i o n is surely satisfied for 2 2 < ½g2.) I n this case it is sufficient to f i n d the global m i n i m u m (x, vi, v2) o f Vneutral g i v e n in eq. ( 4 . 2 9 ) o f r e f . [ 7 ] .

Volume 214, number 3

PHYSICS LETTERS B

Let us also m e n t i o n that in b o t h m o d e l s the isotopic s y m m e t r y breaking by the v a c u u m is essentially encoded in the initial values o f the p a r a m e t e r s at Mp. In particular rho ~ O ( 1 T e V ) >> Mw is relevant for this result. One more r e m a r k is that on the way down to the scale QR ~ O ( M w ) one is passing several thresholds o f different sparticles (since in general the effective SUSY breaking scale is s o m e w h a t larger than M w ) and possibly extra gauge bosons a n d in principle each t i m e the R G equations should be modified. This effect partially cancels against two-loop effects [ 7 ] b u t the systematic study o f both would be an obvious imp r o v e m e n t to the present calculation. Finally, let us briefly extend our considerations to other quark generations. At least two possibilities are open. One is that the horizontal hierarchy o f Yukawa couplings is related to the f u n d a m e n t a l p r o b l e m o f generations (as e.g. in string theories compactified on orbifolds). One thus takes some specific matrices h u ~ h D at O ( M p ) and at the same t i m e assumes that only two Higgs multiplets H~ a n d H2 develop nonzero VEVs. In the other approach, a d v o c a t e d by Ib~ifiez [ 11 ], also the horizontal hierarchy o f masses is supposed to be due to the hierarchy o f VEVs. In this case each fermion m u s t couple d o m i n a n t l y to its own Higgs multiplet. Let us a d o p t the first attitude and take the Fritzseh form for the Yukawa coupling matrices at Mx for the second and the third generation: hU~hD=(~

h)"

(17,

It turns out that the structure o f the R G equations is such that at Mw one gets h t > h b and at the same time t u < e d. A small difference in the diagonal elements drives the off-diagonal elements in the opposite directions. F o r instance, for the set o f parameters: h ~ = 0 . 0 8 , h V h b = l . 1 5 , # o = 1 6 GeV, r ~ o = l TeV, e = 0 . 0 1 2 one gets m~=55.1 G e V ,

mb=4.25 GeV,

mc=l.23GeV,

ms=160MeV,

sin 02_3 = 0 . 0 4 4 .

24 November 1988 (18)

A similar effect also operates for the first generation but is much too weak to invert the ratio o f the masses. However, one m a y expect that for the lighter quarks other tiny effects (like two-loop corrections or non-renormalizable interactions) are also important.

Note added. After c o m p l e t i o n o f this work we became aware o f a p a p e r by G i u d i c e a n d Ridolfi [ 12 ] in which the possibility o f generating the hierarchy v2/v~ ~ mt/rnb in softly broken SUSY models is also mentioned. References [ 1] P. Krawczyk and S. Pokorski, Phys. Rev. Lett. 60 (1988) 182. [2] L.E. Ib~ifiez,Phys. Lett. B 118 (1982) 73; H.P. Nilles, Phys. Lett. B 115 (1982) 193. [3] See, e.g., L. Alvarez-Gaumr, J. Polchinski and M.B. Wise, Nucl. Phys. B 221 (1983) 495; L.I. Ibfifiez,Nucl. Phys. B 218 (1983) 514; J. Ellis, D. Nanopoulos and K. Tamvakis, Phys. Len. B 121 (1983) 123; L.E. Ib~ifiezand C. L6pez, Nucl. Phys. B 233 (1984) 511; J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B 276 (1986) 14. [4] See, e.g., P. Binetruy, S. Dawson and I. Hinchliffe, Phys. Lett. B 179 (1986) 262; J. Ellis, D.V. Nanopoulos, M. Quiros and F. Zwirner, Phys. Lett. B 180 (1986) 83. [5] N.G. Deshpande and R.J. Johnson, Phys. Rev. D 27 (1983) 1165; M. Olechowski and S. Pokorski, Z. Phys. C 23 (1984) 349. [6] G. Altarelli, in: Proc. 23rd Intern. Conf. on High energy physics (Berkeley, CA, 1986), S.C. Loken, ed. (World Scientific, Singapore, 1978 ). [ 7 ] J. Ellis, K. Enqvist, D.N. Nanopoulos and F. Zwirner, Nucl. Phys. B 276 (1986) 14. [8] B. Gato, J. Leon, J. Perez-Mercader and M. Quiros, Nucl. Phys. B 253 (1985) 285; N.K. Falck, Z. Phys. C 30 (1986) 247. [9] K. lnoue, A. Kakuto, H. Komatsu and S. Takeshita, Prog. Theor. Phys. 67 (1982) 1889. [ 10] R. Barbieri and G.F. Giudice, preprint CERN-TH.4825/ 87. [ 11 ] L.E. Ib~ifiez,Phys. Lett. B 139 (1984) 363. [ 12 ] G.F. Giudice and G. Ridolfi, prepfint SISSAE.P. 18 (1988).

397