P H Y S I C A L R E V I E W D V O L U M E 2 3 , N U M B E R 1 1 1 J U N E 1 9 8 1

SU(2) x U(l ) X Uf( l ) models which are slightly different f r o m the Weinberg-Salam model

Chong-Shou Gao* and Dan-di Wut Stanford Linear Accelerator Center, Stanford Uniuersity, Stanford, California 94305

(Received 2 September 1980)

We discuss SU(2)XU(I)xU' ( l ) models by a uniform formula which is convenient for their comparison with the standard Weinberg-Salam model. As examples, we give three interesting models which are based on different grand unification models. In one model, U'(1) does not contribute to the electromagnetic interaction; in the other two, both U(1) and U'(1) do contribute to the electromagnetic interaction. Also, the first two models can approach the standard Weinberg-Salam model as close as one wants; but the third model has limitations on it.

The s tandard Weinberg-Salam model1 of the electroweak interactions s e e m s consistent with the experiments . However, a s the neutral-cur- rent experiments a r e not s o accurate , the inter- mediate bosons of the SU(2) X U(1) gauge have not yet been found, and the Higgs sec tor may change the model in a very subtle way, other possibilities have s t i l l not been ruled out-especially those slightly different f rom the s tandard modeL2 In such a situation, when we study the low-energy electroweak phenomenology (EWP) of grand uni- fication models, e.g., the SO(10) model,3 the E, model, and other models unifying both vertical interactions and horizontal interactions, there i s no reason now to demand that it has to have the s a m e low-energy EWP a s dictated by the Wein- berg-Salam (wS) model [except the minimal grand unification ~ ~ ( 5 1 model4 which has a l a r g e d e s e r t between 10' GeV and 1014 GeV].

What is the possible low-energy EWP beyond the WS model? T h i s problem h a s been discussed c ~ n t i n u a l l y . ~ - ~ The next s imples t models beyond SU(2) X ~ ( 1 ) a r e S U ( ~ ) X ~ ( 1 ) x ~ ' ( 1 ) models. Many papers exis t on SU(2) x ~ ( 1 ) x ~ ' ( 1 ) while s o m e of them d iscuss the c a s e where the ~ ' ( 1 ) does not contribute to the electromagnetic interaction, o thers d i scuss where ~ ( 1 ) and ~ ' ( 1 ) do contribute to the electromagnetic interaction. Also, mos t papers d i scuss only "safe" models which can approach the s tandard Weinberg-Salam model a s close a s one wants s o i t will never be ruled out, except when the Weinberg-Salam model is ruled out. H e r e in th i s paper we will d i scuss al l SU(2) X ~ ( 1 ) X ~ ' ( 1 ) models in a uniform form- u la which is very convenient fo r comparing them with the Weinberg-Salam model. Also this form- u la can fit both sa fe and unsafe models.

The charac te r i s t i c of the s tandard SU(2), X ~ ( 1 ) model (with Higgs doublets only)I0 is that one p a r - ameter , the Weinberg angle, defines th ree things at the t r e e level: (i) It defines the ratio of the electromagnetic coupling constant and the weak-

charged gauge coupling constant

(ii) i t defines the m a s s ratio of the charged gauge boson and the neutral gauge boson

and (iii) i t defines the form of the low-energy neu- t ra l -cur ren t interaction

where

H e r e G,/42 = g,2/8m,2, a is the flavor index, T, = r3/2, and Q is the number of the e lec t r ic charge of the par t i c le a. Hereafter , f o r easy writing, we will omit a l l y matr ices and space-t ime indices.

At the t r e e approximation, these th ree defini- tions of the Weinberg angle, Eqs. (1)-(31, a r e equivalent and p = 1 in the WS model. However, a l l the p resen t experimental da ta a r e from the low-energy phenomena, especially the neutral- cur ren t interactions which a r e relevant to Eq. (4) only. So the consistency of the t h r e e definitions has not real ly been tested yet. The measured average value of sin29, i s2

We notice th ree points f rom the observation of these measurements: (i) T h e average has been taken over al l different experiments. This pro- cedure i s meaningful only when al l the relevant neutral-current interactions a r e governed by Eq. (4) with the s a m e sin28,. (ii) The typical experi- mental uncertainty of the value of sinz@, i s about & (in s o m e experiments , the uncertaint ies a r e even worse than i). (iii) Most of the experimen- t a l resul ts a r e model dependent; f o r example, Eq. (4) is used to fix p a r t of the experimental par -

23 - 2686 O 1981 The American Physical Society

23 - S U ( 2 ) X U ( l ) X U ' ( 1 ) M O D E L S W H I C H A R E S L I G H T L Y ... 2687

ameter, e.g., g,, and the parton model is used to deal with the hadrons.

In an S U ( ~ ) , X U(1) X U'(1) model, only the f i r s t definition, Eq. (I), remains unchanged; the other two definitions, Eqs. (2) and (3), change even at the t ree level, especially the low-energy behavior of the neutral-current interaction, viz., Eq. (3). Obviously, if an S U ( ~ ) , X U(1) X U ' (1) model i s really the underlying physics of low-energy EWP, some corrections up to $ to the standard WS model, which may have escaped present experimental ob- servation, a r e not impossible. What these correc- tions a r e like i s interesting for the phenomeno- logical analysis of more accurate experiments in the near future. The connection of these correc- tions to the underlying unification models i s an interesting problem from the theoretical stand- point.

The method we use here to deal with any SU(2) X U(1) X U'(1) model i s f i r s t making a suitable

rotation in the U(1) and U'(1) group space to make two new Abelian gauge groups U(l), x U ( l ) , , where ~ ( l ) , does not contribute to the electromag- netic interaction, and then discussing the mixing between the U ( l ) , gauge boson and the SU(2) x U(l), massive gauge boson to get the new formula for the neutral-current interactions +

Let us denote the operator of the second U(1) as Y', which has nothing to do with the electric- charge operator, and the f i r s t ~ ( l ) , a s Y a s in the WS model; then the electr ic charge i s

Now we may have a mixing between the SU(2) X ~ ( l ) , neutral massive gauge boson and the ~ ( l ) , , gauge boson [but, of course, no mixing between the photon field corresponding to the operator Eq. (6) and the ~ ( l ) , , field]. Thus the low-energy neutral-current interaction Hamiltonian becomes exactly

where P = Eq. (4), Jf is the current connected with the extra ~ ( l ) , symmetry, 0, is the mixing angle be- tween the two neutral gauge bosons, X, and X, a r e the diagonalized mass squares of the two neutral gauge bosons with XI < X, and normed by mw2/cos28w,

V = g"/(gL2 + g12)1/2, (8)

andg' and gN a r e the coupling constants of ~ ( l ) , and ~ ( l ) , , respectively. We call Eq. (7) the small de- viation form. Obviously, when the last five te rms a r e negligible, Eq. (7) becomes Eq. (3), the WS model.

The concrete form of Ji and the values of 0,, XI, and X, a r e model dependent; in other words, they de- pend on the meaning of t i e ext ra U ( l ) , and the structure of the Higgs sector, which in turn may connect with a grand unification model and i ts hierarchy pattern. Let u s discuss some interesting examples.

Model A. Suppose that above a very high energy scale M1 (Mr>>>Mw) the electroweak interactions a r e described by S U ( ~ ) , x U(l), x SU(~), , where the SU(2), i s the horizontal gauge group. The SU(2), i s bro- ken down to U(l), at the energy scale Mr. Thus we get the SU(2), X U(l), X ~ ( 1 ) ~ model below the energy scale M', which then i s broken down to ~ ( l ) , , . Now we study only the last symmetry breaking. We choose the effective Higgs sector as follows:

Then we get (when (u2 = vZ2/vl2 >> 1)

and the low-energy neutral-current interaction Hamiltonian to the f irst order of 1/a2 is

2688 C H O N G S H O I ; G A O 4 N D D A N - D I W U 23 -

with

J ' = 2$H$,

where the opera tor H/2 is the third component of the SU(2), group in the fermion representation. H has to be the s a m e number f o r a l l low-mass fermions to suppress the strangeness-changing neutral current . According to the p resen t experi- mental s ta tus , a ,>5 is already sa fe enough.

The additional ~ ( l ) , , coming f r o m an SU(N) (N > 5) o r GL(n, c ) has been discussed in Ref. 7.

Model B. The SO(10) group can be decomposed down to

SU(21, x SU(2), x SU(3), x U( l ) , c SO(10). (1 3)

The ~ ( 1 1 , means B - L U(1) gauge symmetry, where B is the baryon number and L is the lepton number. We notice that a t the grand unification level, the rat io of the coupling constants is

2 = 1:1:;. g ~ ' : BR :&TE (14)

We suppose that a t the energy sca le 111 - 1014 GeV, the SO(10) symmetry is broken down to the sub- group Eq. (13). Because of the different behaviors of the running coupling constants of the non-Abe- l ian gauge group S U ( ~ ) ~ x S U ( ~ ) , - SO(4) and the ~ ( 1 ) ~ gauge group, the rat io changes a s the en- ergy goes down. At a s t i l l ve ry high energy sca le M' we reach

gL2:gRZ:gB2= 1.8: 1.8: 1 , (15)

where the SU(2), is broken down to ~ ( 1 ) ~ and at the energy sca le about 1 TeV we get a model8

SU(21, x U(l), x U(l), , with, say,

In this model we have Y in Eq. (6) as

We choose the effective Higgs s e c t o r a s follows:

( x ) = v2 with T L = 0, T, = $, B = - 1.

Then we get the s i m i l a r resul t as Eq. (11) fo r the low-energy neutral-current interaction Hamilton- ian when a,' = ~ ~ ~ / v ~ ~ cos4q >> 1 with

J? = 4 J,, - t an2va + )) , ( (19) where tan2q =gB2/gR2. Equation (1) becomes

~ i n ~ 8 ~ = g ~ ~ g ~ ~ / ( g ~ ~ g ~ ~ f g L ~ g B ~ + g L k R ~ ) , (20)

and Eq. (8) becomes

q2 = gR4/(&TE2gR2 + g L 2 g E 2 + gL2gRz) ' @ I )

Substituting Eq. (16) into Eq. (20) we get

The numbers in Eqs. (151, (16), and (22) a r e given for definiteness; they have no definitely physical meaning. The reason is that s ince many unknown things may happen at the energy a r e a of 1014 GeV down to 1 TeV, we do not want to go into the de- t a i l s of the hierarchy here. However, we would s t r e s s that these numbers a r e reasonable if the hierarchy of the SO(10) model is in the way men- tioned above, which is very different f rom the s tandard h i e r a r ~ h y . ~

The models A and B can approach the s tandard WS model c loser and c loser by adjusting the par- ameter (only one) a, bigger and bigger. We can indeed choose such a type of Higgs sec tor to build a model, which has m o r e than one adjustable par - ameter ; however, even if we adjust a l l the p a r - ameters , we s t i l l have to face an appreciable de- viation from the WS model. This i s the model C.

Model C. In model B, we choose different Higgs- field contents

(c$)=(:2) with T L = + , T S R = t , B = - 2 .

The neutral-current Hamiltonian Eq. (7) has the following parametrization:

with a = 1 + 2 tan2v, f f 2 = vI2/(ul2+ vZ2), and P 2 = (1 - a'). If we choose the parametr izat ion a s Eq. (161, then q= 0.64, 77% 0.41, and we get (when aZ =

Then in Eq. (7) every t e r m in the l a s t five t e r m s is s m a l l e r than of the f i r s t t e rm. However,this may not be the best parametr izat ion.

T h e r e a r e two problems in this model: (i) The total s t rength of the neutral-current interaction is almost two t imes bigger than what is in the s tan- dard model, though this can be improved a lit t le. (ii) One use of the Higgs c$ , is to give fermions m a s s e s through the Yukawa couplings. Because i t s vacuum expectation value (VEV) decreases

23 - S U ( 2 ) X U ( 1 ) X U ' ( 1 ) M O D E L S W H I C H A R E S L I G H T L Y ... 2689

five times more than that in the WS model, we have to increase the coupling constants five times to get the same fermion masses. The Yukawa in- teractions of Higgs part icles and fermions have not yet been detected. Whether this increase will bring bad news for such a model, we have to wait and see.

Our conclusion i s that when energy of the ac- celerators goes up, we may find one more Z bo- son, whereas in the currently reachable energy region, we may find some signature of the other Z boson. If that i s true, the low-mass Z boson will be lighter than that expected in the standard WS model at the t ree level. We notice that the mass of the Z boson will become heavier if we consider radiative corrections in the standard WS

*Permanent address: Department of Physics, Beijing University, Beijing, China.

?On leave from Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138. Per- manent address: Institute of High Energy Physics, P. 0. Box 918, Beijing, China.

IS. Weinberg, Phys. Rev. Lett. 2, 1264 (1967); A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Almqvist and Wiksell, Stock- holm, 1968), p. 367; S. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

'see, for instance, C. Baltay, in Proceedings of the 19th International Conference on High Energy Physics, Tokyo, Japan, 1979, edited by S. Homma, M. Kawa- guchi, and H. Miyazawa (Phys. Soc. of Japan, Tokyo, 1979), p. 882.

3See, for instance, H. Fritzsch and P. Minkowski, Ann. Phys. (N. Y.) 93, 193 (1975); M. S. Chanowitz et a l . , Nucl. Phys. w, 506 (1979); H. Georgi and D. V. Nanopoulos, ibid. E, 52 (1979).

4 ~ . Georgi and S. Glashow, Phys. Rev. Lett. 32, 438 (1974).

5See, for instance, R. N. Mohapatra and D. P. Sidhu, Phys. Rev. Lett. 3, 667 (1977); E . Ma, Nucl. Phys.

model.g If the exotic Higgs m ~ l t i p l e t s ' ~ acquire (VEV'S), it may cause the ratio rn,/m, to be smal ler than that in the standard model.g How- ever, the form of the neutral-current interaction Eq. (3) will not change except when there i s more than one neutral gauge boson.

One of the autbors (D.-d. W.) wishes to thank the people at SLAC for their warm hospitality, S. Weinberg and S. D. Drell for making his stay possible at SLAC, and Bambi Hu, N. Snyderman, and J. Bjorken for helpful discussions. This work was supported in part by the Department of Energy under Contract No. DE-AC03- 76SF00515 and by the National Science Foundation under Grant No. PHY77-22864.

B121, 421 (1977); V. S. Mathur and T. Rizzo, Phys. - Rev. D 17, 2449 (1978); Kwang Chao Chou and Chong Shou ax Report NO. SLAGPUB-2445, 1980 (unpub- lished).

6 ~ . Kang and 3. E. Kim, Lett. Nuovo Cimento 16, 252 (1976); Phys. Rev. D 2, 1903 (1976); 2, 3367 (1978); V. Barger and R. J. N. Phillips, ibid. 18, 775 (1978); M. ~ a s u ; , h o g . Theor. Phys. 1, 269 (1978); Chanowitz et a l . (Ref. 3); G. G. Ross and T. Weiler, J. Phys. G 2, 733 (1979) [this paper dis- cussed the idea which may lead to Eq. (7)J; and others.

TA. Zee and J. E. Kim, Phys. Rev. D z , 1939 (1980): J. de Groot et a1 ., Phys. Lett. E, 399 (1979).

'N. G. Deshpande and D. Iskandar have discussed the SU(2),xUR(l)xU(1), model very closely; see N. G. Deshpande and D. Iskandar, Phys. Lett. E, 383 (1979); Phys. Rev. Lett. 42, 20 (1979); Nucl. Phys. B167, 223 (1980); N. G. Deshpande, University of - Oregon Report No. OITS-141, 1980 (unpublished).

9 ~ . Nn, private communication: M. Veltman. Nucl. Phvs. - .

B123, 89 (1977). 'OFormodels with nondoublet Hings fields, s ee , for ex- --

ample, T. P. Cheng and Ling- Fong Li, ~ a r n e g i e - Mellon University Report No. 3066-152, 1980 (unpub- lished).