Quark and lepton mass matrices in the extended standard model with discrete symmetry

Volume 255, number 1

PHYSICS LETTERS B

31 January 1991

Quark and lepton mass matrices in the extended standard model with discrete symmetry N.I. Polyakov ITEP, SU- 117 259 Moscow, USSR

Received 22 September 1990

We consider the standard model w~ththe addxttonal n = 512 heavy Hlggs doublets and spontaneously broken Zn-symmetry. The model correctly describes the quark and lepton masses and quark mixing angles without any substantlal hxerarchy of the parameters assumed Flavor changing neutral currents are suppressed for the masses of the addmonal Hlggs scalars greater than 5 TeV

1. Introduction The s t a n d a r d m o d e l ( S M ) d e v e l o p e d in the 1960's until now remains a viable theory. Most p a r a m e t e r s o f the m o d e l are m e a s u r e d except Higgs and t-quark masses. The observed spectrum o f the quark and lepton masses, s u p p l e m e n t e d with the lower b o u n d on t-quark mass [ 1 ], mt >/77 G e V ,

(1.1)

can find no reasonable explanation in the SM. Unnatural hierarchy o f Yukawa couplings ( Y C ) , 2~/2t < 10 -5 ,

(1.2)

m u s t be i n t r o d u c e d " b y h a n d " . M a n y models were p r o p o s e d to explain the quark a n d lepton mass spectrum [ 2 ]. In this p a p e r we consider a m o d e l w~th several Higgs doublets a n d discrete s y m m e t r y Z,. Z , s y m m e t r y fixes the Yukawa interactions ( Y I ) in such a way, that only the t-quark has a Yukawa coupling with the Z,-singlet Hlggs field, which develops a nonzero v a c u u m expectation value ( V E V ) . The masses o f all other fermions vanish due to Z,. After the spontaneous Z , - s y m m e t r y breaking (SSB, SB) it is possible to generate the realistic quark a n d lepton mass matrices ( Q L M M , M M ) . The variety o f the quark a n d lepton masses from me to mb originates in this a p p r o a c h from the structure o f the spontaneously b r o k e n Z , group. Thus " n e w physics" ~s responsible for the fermion mass spectrum.

In a previous p a p e r [ 3 ] we considered quark M M in the m o d e l with Z~2-symmetry. F o r the leptons to be also included it is necessary to extend the discrete symmetry. The m o r e subgroups the group Z , has, the m o r e mass scales can be generated after SB a n d the more opportunities appear for the Q L M M to be built. In this letter we describe Q L M M in the m o d e l with Z5 ~2 symmetry. The extended Higgs models are subjected to excessive processes with flavor changing neutral currents ( F C N C ) . If we want to realize " n e w physics" at the scale, experimentally accessible in the future it is necessarily to p r o v i d e some m e c h a m s m o f F C N C suppression.

2. F C N C in extended models It is well known that the quark diagonalization in the SM is that o f matrix o f YC, thus in the " m a s s " basis the nondlagonal vertices ~l,qj~0are absent. This is not the case for m a n y Higgs models. The experim e n t ( K - I ( - m i x i n g etc. ) imposes a severe constraint on the p a r a m e t e r s o f these models. I f all YC 2 v are o f the order o f unity the " n o n - d i a g o n a l " Higgses should be very heavy: m,d >/104 T e V . In spite o f this there exists a class o f n o n m l n i m a l models with F C N C p h e n o m e n a suppressed for

0370-2693/91/$ 03 50 © 1991 - Elsevier ScxencePubhshers B.V. ( North-Holland )

77

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rand -----5 TeV (also for 2 U= 1 ). Consider the FCNC of the d, s, b-quarks in the model with the following YIs: L~,uk = - 2 dvdL,dR/p v +h.c.

dR---~d'~- Vdg.

(3.2)

and assume that the YIs are invariant under this Z, symmetry. After the "standard Higgs" ~odevelops a VEV (~) =

(o)

m 2,

(d) the moduli of all YC are equal: 12~ [ =2a, the total contribution of the Hlggses mediated processes to the FCNC coefficient G,j in the effective lagranglan (2.3)

~s precisely zero for all z~j. This is shown in the extended version of this paper [ 4 ]. A discrete symmetry is the simplest way to fulfill the conditions ( a ) (c). The condition (d) should be imposed on the YC supplementarily. After the Z,-SB the validity of the conditions ( a ) - ( c ) is violated a little and as a result weak FCNC processes are generated. These processes do not contradict the experiment at m,d >/5 TeV.

,

V

(2.2)

there are many nondiagonal vertices, but when (a) all the fields ~0Uare orthogonal (are not mixing ), (b) the masses of the fields R v - Re 9,j, I v - I m ~0v, t, j = 1, 3 are equal, (c) all the masses of the fields ~0v are equal: m 2 =

~ f f = Go (d, y5dj) 2

S / 123...n~ =~234...1)

(2.1)

where the ~0,jare the neutral components of the different Higgs doublets. Let's assume, that these scalars develop some appropriate VEVs. After MM dlagonallzatlon

dL ~dM--UdL,

31 January 1991

(3.3)

the fields ~o,also get nonzero VEV: (~0,) = a ( ~ ) .

(3.4,

In this vacuum SU (2) ®U ( 1 ) is broken to U ( 1 ), and the W, Z-bosons get masses

rnZw=½g~v2(1 +na2), m~ = ½(g2 +g,2)V2 ( 1 + na 2) .

(3.5)

From eqs. (3.1), (3.5) we see that when xvZ<< m 2, b2=na2<< 1 low-energy sector of the theory is precisely the CM. The field Re ~ - v with the mass (3.6)

m 2 =4KV 2

is the standard Hlggs. All other scalars (n charged and 2n neutral) have the large mass m. We introduce "Fourier transformed" fields Ok, 1

~ exp(i 27~k(m- 1 ) ) tpm,

(3.7)

~k= ~m=l

which are transformed with definite charges under Zn: 3. General description of the m o d e l

[ 2~tk'X S: 0k ~exp~i --~-) 0k.

We reahze discrete symmetry in the simplest possible way. Let us introduce the standard Higgs doublet ~ breaking SU (2) ® U ( 1 ) spontaneously to U ( 1 ) and n additional heavy H~ggs doublets with the potential

(3.8)

Since all VEVs (~o,) are equal, it is clear that only the doublet ~o has a nonzero VEV ( ¢ k ) = b(0v)(~k,O

(3.9)

•

V(¢, ~o,)= x ( 1012- v2) 2 Let us assume that the quark and lepton fields +m 2

~ I~o,--aOIz,

(3.1)

(u)

QLt ~-'~ d L ~ ,

where ot is some numerical parameter. This potential is symmetric under all possible permutations of the fields ~o,.We choose some Z,-subgroup, e.g. one generated by the cyclic permutation 78

URt,

d~,, (3.10)

e Lt

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PHYSICS LETTERS B

(where I is the generation index) nontrivially transform under Z~:

[2hi )

S: ~dL,~ e x p t ~ q,

and with the transformation law under Zn, S: )fi --'X2, ZE~Zt

(3.18)

It is clear that the vacuum

QL,,

u m - - * e x p ( ~ u,)ld R . . . . .

31 January 1991

(3.11)

=lz,

=o,

< ~ 1 > "~--0 ,

= f l -

(3.19)

is not Zn invariant. The doublets 0k "feel" this Zn-SB due to Zn-invariant ~-z-interaction:

with Z~ charges

q,,u,,d,,l,,e,.

(3.12)

V~x= fl[Z2( I¢0, 12+ 1~312+ 1~512+...)

Z~-symmetry fixes the YIs:

+g2( 1~'212+ 1~412+...)1 • - &u~ = 0~, u~.(.L30~,_ ~,~ + ,ga..u.0 ~)

The vacuum (3.19) is invarlant under the S z transformation. This means that Zn is broken to the group Zn/2 which is generated by the S 2 transformation. The doublet 0.12 which is an S 2 singlet acquires the VEV

. 4 .QL, - dR,(a,jO(q,_aj) d ~- ~ d aq,,dj0 )

+&,eR.('t,30~t,-e,~ + ~/,3&,.~,0) +h.c.

(3.13)

The three second terms in the round brackets in (3.13) are nonzero only when the following equations hold:

q,=ua, q,=4,

l,=ej.

(3.14a,b,c)

We choose the Z~ charges so, that only eq. (3.14a) is vahd (say for t=j= 3). This means that the doublets 0, 0o with nonzero VEV have YCs with the t-quark only, -- '~Yuk = 0L3

(3.20)

UR3 ('~ 7308 "JFl~t~30c )

+ L/%uk{0k, k # 0 } +h.c.

(3.15)

and as a result the t-quark gets the mass mt = (b2~3 +q~3)v.

2,

/m

2

(0"~

"V~ l,},

(3.21)

which is much mess than the electroweak scale v if m >>/L. When the group order n has a divisor q, the formulae ( 3.17 ) - (3.20) can obviously be generalized. For the SB Z ~ Z q to take place we must introduce n/q additional scalars. As a result of this SB the doublets 0q, 02q, 03q,... get nonzero VEVs. The masses of the additional scalars will be several hundreds of GeV. These scalars do not interact with the vector and spinor fields and thus are practically invisible in experiment.

(3.16)

All other quarks (and leptons) have YCs with the fields 0k, k # 0 and thus are massless. Zn-SB is needed for these fermions to become massive. We shall break Z. spontaneously. For this purpose we introduce the additional scalar fields - total electroweak slnglets which are nontrivially transformed under Z~. Due to their quantum numbers these fields do not have YCs with the fermlons and their VEVs do not contribute to the W, Z-boson masses. We consider Zn-SB in a simple example. Let n be an even number. To break Z~ to Z~/2 we introduce two additional scalars Z~, )(.2with the potential: V= a (,~12-~-,~2- ,u2) 2-~-oL1,~2,~2

(0,/2>=/z

(3.17)

4. The choice of a discrete group In the following we accept the standard set of the quark [5 ] and lepton masses and recalculate all the masses to the scale #,~ I00 GeV: mu = 3 MeV,

mc = 1 G e V ,

m d = 5 MeV,

ms=0.1 G e V ,

me=0.5MeV,

mr=?,

mo=0.1GeV,

mb=3.8 GeV, m,=l.8GeV. (4.1)

Since we have already assumed that the moduh of all YC's are equal, it is necessary to attribute some VEV < 0k) to each mass scale in (4.1). Moreover, to generate the appropriate Kobayashi-Maskawa ( K M ) 79

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matrix element values V~s, V~b we must also include two other mass scales:

Vtot=/c(l¢12-v2)2+m 2 ~ I~,-a¢l 2 z=l

mo-~22 M e V ~ I V,~I = m o / m ~ = 0 . 2 2 , mr = 180 M e V ~ I V~bl = m v / m b = O . 0 4 5 .

31 January 1991

p--I

(4.2)

Suppose that the group Z, contains p different subgroups Zq, q~ = 1, q2, ..., qp, than after Z~-SB p + 1 different mass scales appear: - 1 scale-VEV < ¢o ) = b~, p scales correspond to the breakings Zn--, Zoo, ..., Zqp. Thus to get ten different mass scales in (4.1), (4.1) we must choose a group containing nine subgroups. The most simple group o f such a type is Z , with n=29=512. -

+ E (vq+Vq),

YI (3.13 ) are written via the "Fourier transformed" doublets Ck (3.7). After SSB the fields ¢,, k # 0 get small VEVs which can be expressed via the small parameters

z q - 2q-Pfldt21m2

(5.6)

o f the breakings Z(2~) to Z(2~): V o - ( ¢ , ) = -VgZo,

v, - ( ¢2 ) = -Vg(Z, + Zo) , V2 ~ ( ¢ 4 ) U3 ~ < ¢ 8

5. Model with Zs~2-symmetry

(5.5)

q=O

"-~ -- V9(Z2 "[- Z1 "{- ZO) '

>=

-- U9(Z3 JI-Z2 "1- ZI "[--70) '

/34 ~ <¢16 > = -- U9(Z4 J-Z3 -I-Z2 +-71 + ' 7 0 ) '

We generalize the Z,-SB mechanism for the case when n = 2 p, p = 9. For every q = 0, p - 1 we introduce l = 2 p- q additional electroweak singlets - scalars Zq.~, Zq,2..... Zq.t which form an/-dimensional representation o f Zn: S : )~q.l ---~')~q,2, ..., Zq, l----~)~q,l •

(5.1)

/)5 ~ ( ¢ 3 2 > : -- U9(Z5 "l-'"nt-Zo) ' V6 ~ <064 > "~- - - U 9 ( Z 6 "~-"'"[- ZO) ' U7 ~ ( 0 1 2 8 > = - - U 9 ( Z 7 "[-"'"[- ZO) ' US ~- <¢256 > = -- U 9 ( Z s ' ~ - ' " - I - Z o ) '

Z,-invariant potential o f these fields

v9 -= <¢512 > - <0o > =b~.

(5.7)

V q = a ( ) ~ 2 q , l "~Zq,2"~l-..."~)~q,l--]'~2) 2 2 2

The VEVs o f all other doublets Ck can be reduced to (5.7) with the aid o f the equation

2 2 2 2 2 2 "~ a q , l (Zq, lZq,2 dI-Zq,2)~q,3 + "" dI-Xq, l)~q,l ) (Ok)=(¢k)

2 2 2 2 2 2 "JI-O/q,2 (Xq, IZq,3 ']-Zq,2Xq,4 '1- ... dl-~q,l)~q, 2 )

+...

if (5.2)

breaks Z~2,) spontaneously to Z(2~). The additional terms are introduced in (5.2) for all the fields Xq.,, z= 1, l to get nonzero masses in the v a c u u m (Zq,,) = #0~,,o •

k'=k(2t-1),l=l,2,....

(¢(2q+2,)> = (¢(2q)> ,

q
= (0(2,)) ,

q>t,

= (0(2~+,)) ,

q=t.

The Z,-invariant interaction potential l

2q-- 1

t=l

(1~°,+~,1z - a z v 2 )

(5.4)

j=0

is necessary to transfer this Z~-SB to the sector of Higgs doublets. The total potential o f the scalars in the model takes the form 80

(5.8)

For example, (¢24) = ( ¢ 8 ) , as 24 = 8.3. It is also useful to write down the equation

(5.3)

V'~=fl~ Z Z2,, Z

,

(5.9)

We attribute the following VEV to each mass scale in (4.1,2): /~V0 = m e ,

Aul:mu

Avg=ms=m~,

,

~v2=md,

),vs=mv,

,~,V3= m o ,

Av6=mc,

,~.v7=m~, (5.10)

Volume 255, number 1 2l)8 = m b ,

PHYSICS LETTERS B

2"I)9+q1)=mt.

(5.10 cont'd)

YIs in the model can easily be constructed if we assume all the quark and lepton Z, charges to be some powers of two. In this case eq. (5.9) simplifies the calculation of the VEV of the doublets q~ which have a YC with given fermions. We choose all YC matrices /~u,d,e equal and hermitean: 1 exp (iOl) X~,d.e=2 exp(--iOl) 1 \ e x p ( -- iO2) exp ( -- iO3)

exp(iO2)~ exp(iO3)].

1

Only the combination d = 01 - 02 + 193 of three phases O, is physical - it is the source of CP-violation. The conditions (5.10) uniquely fix the quark Zn charges:

(23]

Ul ~

_

e, =

\28]

_

24

.

Mu=

Md=

mu \mu

Me =

mo

md

ms e-'arn s

e l a m ,t

me mo ms

\m e

-

so

so

m,l/mb

sOsvlR*)

1

sosv(1-R)

,

(5.14)

-sv

,

(5.13a)

v = v 9 = 125 G e V .

(5.13b)

GeV,

b - (n)-l/2a=v9/v= 1 , a = 2 -9/2=0.044 .

,

(5.13c)

Let us show how these matrices develop from the Zn charges (5.12). For example, the 1-3 element of the matrix Md appears from the VEV (Ok) with k=q~-d3=23+2 s. Using eq. (5.9) and taking into account eqs. (5.7), (5.10) we find /~(~(23+28))= )~ ( ~ ( 2 3 ) ) =,,~v3=mo. For simplicity we express all the matrix elements in ( 5.13 ) through the masses mu, .... The true masses differ from those in ( 5.13 ) by small mixing factors.

(6.1)

This corresponds to m t = , ~ , v 9 + q v ,'-' 125

mb

mo e'am e - ' a m s mx

(5.15)

To estimate the typical values of the parameters we assume 2=q=-0.5,

me e 'amv e -16my m t / mo

me

1 =

--

6. Numerical estimates

O'/0

rod md

\ m o / m b ( m3,/ms -- e ta )

e-lamolmb~ m, l l m b )

(5.12)

27

The charges (5.12) correspond to the following QLMM: W/u

molms 1

R = e'ams/mr = e'a/sr, ms/mb ~--e la/2.

29

,

1 -mo/ms

2 5

\29/

24

VKM

where

q,=l,= 25 ,

d,=-

This effect is essential only for md and me which become about 1 +So" 1.2 times less after mixing. The d-type quark MM is diagonalized (see (2.2) ) by two unitary matrices U, Vwritten in ref. [4]. Neglecting the small u-type quark mixing angles we find the KM matrix Vvoa= U +, which by the phase rotation can be reduced to the Maiani parameterization

/ (5.11)

31 January 1991

(6.2)

Using eqs. (5.9), (5.11 ) we find the SB parameters zq and the corresponding VEVs/~q:

Zo = mdmt=4

× 10-6, #o = ( 2 9 z o m 2 ) 1 / 2 = 0 . 0 5 m ,

zl = 2 . 0 × 10 -5

#~=0.08m

22= 1.6× 10 -5

/~2=0.05m

z3=1.4× I0-4

/t3=0.09m

z4= 6.2 X 10 -4 , /t4--0.14m z 5 = 6 . 4 × 10 -4

/Zs=0.10m

z 6 = 6 . 2 X 10 - 3

fl6 = 0.22m

(6.3) 81

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PHYSICS LETTERS B

/.tT=0.16m,

z8 = 1.6× 10 -2 , /~8=0.18m,

(6.3 c o n t ' d )

where we have assumed all the couphngs flq from (5.4) to be equal to each other: flq= 1. From (6.3) we see that a minimal hierarchy o f the VEVs/tq is needed to obtain the right spectrum o f the quark and lepton masses:/to/#6 ~ 0.23. The mass m which characterizes the mass scale o f "new physics" remains a free parameter of the model. We can only find how the different experimental measurements of F C N C effects restrict the region of its values. The oscillations of K, Bd-mesons are the most sensitive to "new physics" phenomena. The electroweak interaction o f the SM reduces for these oscillations to the effective lagrangian: L#as=2= GK[g~u( 1 + 7 5 ) d ] 2 , 5a~b=2= GB [dyu( 1 + ~,5)b] z .

(6.4)

xK=0.48,

XB=0.73,

(6.5)

on the oscillation parameters

Xp=Amp/Fp=2Gpj~pmp/]"p,

Quark interactions with the scalars q~k generate in L~rf the additional F C N C terms ~OL=2 = G k ( g7sd) 2 ,

~q~*Ab=z=G~(aY75b) 2 .

P=K, B,

(6.6)

imply for Gp: Re GK-~ 7.4 X 10-13 ( 165 MeV/fK)EXK = 3 . 6 × 10 -13 GeV -2 ,

GB ~ 9.4 X - 12( 130 MeV/fB)ExB = 6 . 6 × 10 -12 GeV -2 .

(6.7)

The imaginary part o f GK is measured with much more accuracy: Im GK~ (2)1/2eK Re GK - 1.2× 10 -15 GeV -2 .

(6.8)

Taking into account many uncertainties, the calculable SM contribution to these oscillations can be less than is needed:

(6.10)

AS the matrix elements o f the operators from (6.4), (6.10) between the states of pseudoscalar mesons are equal, the values o f Gp in (6.7), (6.8) are essentially the upper bounds on the corresponding values of Gb, Re G k < 3 . 6 × 10 -13

Im G k < l . 2 × 1 0

-15

G h < 6 . 6 × 1 0 -12 (GeV -1)

(6.11)

YIs of d-type quarks in the limit o f unbroken Z, symmetry satisfy the conditions ( a ) - ( d ) o f section 2. Therefore, the oscillation parameters G' are suppressed by the small Zn-SB factors Zq. In ref. [4] we have shown that (6.11 ) are valid at m>> 5 T e V ,

The experimental data [ 6 ],

31 January 1991

if2=0.5, fi-0.1 ,

(6.12)

It is necessary to mention that no accidental reduction was used to obtain the condition (6.12). What was needed is the hermlticity of the YC matrices and the equality o f the moduli of all the elements m these matrices. Substituting m ~- 5 TeV in (6.3) we find the typical VEVs ]Aq:/t ~ 250-1000 GeV. Many generalizations o f the model with c Zn symmetry are possible, say, n = 2 9 ~ n' = 2 p. It is interesting, that when p = 12, 13 .... , it is possible to choose all Z,-breaklng VEVs/~q, q = 0 , p - 1 to be the same. In this case for the doublets VEVs Vq defined as in (5.7) we have Vq=(2q+l--1)VO

.

Thus all the elements of the Q L M M are some power o f two minus one (times the general scale Vo--me). These M M are in excellent agreement with what we observe in Nature [ 6 ]: mb(#) ~2m~(#) ~4mc(#),

/z "~ 100 G e V .

GKIcM--~ G 2 /47t2" ( Vc*s Vcd) 2 m 2 -~ 1 . 5 × 1 0 -~3 GeV - 2 ,

7. Conclusion

GBIcM--~G2/4112" ( Vtb Vta) * 2 m t2I ( m t /2m w )2 _~ 10-12-10 -11 GeV -2 .

82

(6.9)

Thus in this model the simple and realistic QLMM appear in a natural way. The only drawback of this

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"simplicity" is the existence o f 2p (p = 9, 13, ?) heavy Higgs doublets with mass m >> 5 TeV. The hierarchy of two mass scales v - 100 GeV and rn is a single hierarchy of the parameters m the model. The fact that the scale m is sufficiently high has two consequences: - the suppression of F C N C processes, - the smallness of all quark and lepton masses with respect to the t-quark mass. Let us note an interesting feature o f the model. The hghtest scalar H-"almost standard Higgs" has YC only with the t quark. But scalar mixing after Z,-SB generates the effective vertices with all other fermions,

LP~ff~- m--z(f,f)H, v

precisely as in SM. Note that the VEV v is not now fixed by the W-boson mass. It is related with that o f

31 January 1991

SM by the expression: v= ( 1 + b 2) - -

I/2Z)SM

.

I am grateful to K.A. Ter-Martirosyan for his interest in this work and many helpful discussions. References

[ 1 ] F Abe et al., Phys. Rev Lett 64 (1990) 142. [2] S. Welnberg, Phys. Rev. Lett. 29 (1972) 388, H. Georgl and S L Glashow, Phys Rev D 6 (1972) 2977, D 7 (1973) 2457, R.N. Mohapatra, Phys Rev. D 9 (1974) 3461, SM. Barr and A Zee, Phys. Rev D 15 (1977) 2652; D 17 (1978) 1854. [3] N.I. Polyakov, ITEP preprmt ITEP-65-90 (1990). [4] N.I Polyakov, ITEP preprlnt ITEP-90-90 (1990). [ 5 ] J Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77 [6] Particle Data Group, G.P Yost et al, Review of particle properties, Phys. Lett B 204 (1988) 1.

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