Volume 125, number 1 PHYSICS LETTERS 19 May 1983

AN S U ( 3 ) M O D E L F O R E L E C T R O W E A K UNIFICATION

Mark SINGER, R. PARTHASARATHY 1 and K.S. VISWANATHAN Department of Physics and Theoretical Science Institute, Simon Fraser University, Burnaby, B.C., Canada VSA 1S6

Received 22 December 1982

We present a model of electroweak unification based on the group SU(3). We reproduce the standard SU(2) x U(1) ef- fective interactions with sin20w ~- 1/4. None of the gauge bosons need acquire a large mass in order to suppress unwanted interactions.

In two letters [1,2] Riazuddin has shown that one can identify the charged and neutral currents of the standard weak interaction [3] with the three compo- nents of the SU(2) group provided that

(i) the fundamental doublets under SU(2) contain a mixture of left-handed and right-handed fields, and

(ii) the basic interaction is axial vector rather than vector. In this letter we wish to show how a simple extension of this model to the group SU(3) can lead to electroweak unification [4,5] including the stan- dard weak neutral current interactions and the stan- dard W +--z mass ratio. We also predict that sin20 w

1

The original model [ 1] consisted of an SU(2) dou- blet

( " 1 = (cosa)!~ L + (sino0~R , (1)

and an SU(2) singlet ( - s i n a)~ L + (cos a)~R, where represents a charged lepton. The angle a is, for the moment , arbitrary. The interaction was axial vector rather than vector:

• /2IN T = i 2 - 1 / 2 g ~ a T l ~ T 5 w b ~ b (a,b = 1 , 2 ) , (2)

where Wgu; £eWCu = 0 are the gauge bosons. It was shown that this reproduced the standard neutral cur- rent interactions when one identified 2-1/2(W~u -

1 W ~) = Z u and sin2c~ = 2 sin20 w = g, but with the mass relationship rnZ(W -+) = ½mZ(Z) instead of the

1 Permanent address: Matscience, Madras-600 020, India.

standard relationship m2(W -+) --~m2(Z). Since this model does not contain electromagnetism, the abso- lute magnitude o fm2(W -+) is not known, but it has been shown in ref. [1] that if one t akesg = x /8e , then both this model and the standard SU(2)L × U(1) model give the same prediction for m2(W-+).

In extending the model to SU(3), we shall assume a lepton triplet of the form

= cosc0~ L + ( s i n a ) ~ R .

L(_sin ~)~L + (cos ~)~Lj (3)

This simply combines the original SU(2) doublet in (1) and singlet into an SU(3) triplet. The right-handed neutrino (if one exists) is a singlet. The interaction between the gauge bosons and fermions is still given by (2) but with a, b = 1, 2, 3.

Next we assume three sets of Higgs scalars whose vacuum expectation values (VEV) will generate fer- mion masses. First we have a Higgs sextet qSab = ~ba where the only non-zero VEV is (qSll) = a. This can generate a Majorana mass for the neutrino. Second, we have a Higgs triplet f a where ( f l ) = b, and third, a Higgs octet ~ , ZcXc c = 0 with (X32 f= (X 2) = k. The lepton mass lagrangian is then

-/~mass = A 1 t~Tac-ldPab t~b + h.c. + A2faj'aU R + h.c. (4)

+ A 3 ~ka x~ qJb ,

where A i are the Yukawa coupling constants, and C is

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Volume 125, number 1 PHYSICS LETTERS 19 May 1983

the charge conjugation matrix. From (3) and (4) we see that

m(~) = A 3 k cos 2c~ , (5)

so that if a = rr/4, the lepton £ cannot acquire mass. These three sets of Higgs scalars also give masses to

the gauge bosons wabu. Groupling the gauge bosons by charge and CP properties, we have

(i) W~,W3" wl W1 - - I ' - - 2 ' - - 3 '

(ii) Z = 6 - 1 / 2 ( - 2 W I + W ~ + W ~ ) ,

H = 2-1/2(W2 2 - W ~ ) , A = 2 - 1 / 2 ( w 3+w 2/ " 2 3 " ( 6 )

(iii) A = (1/xQi)(W 3 - W2),

where Lorentz indices have been omitted for ease of notation. The interaction terms between the Higgs scalars and gauge bosons are

"~INT = ½g2(<WgOcdC/fla + Wba dPbcWffl Oda + fawbW~fc

so that plugging in the appropriate VEV's for the Higgs scalars gives the following gauge boson mass spectrum:

m2(Wl 2) = m2(W~) = ½g2(a2 + b 2 + k 2 ) ,

m2(Z) _ a _2 4(2a2 + b 2) - ~ g ~ (8)

rn2(H) = m2(~,) = ½g2(4k2),

m2(A) = 0 .

Using the fact that

~L"/T5 ~R = ~R')")'5 ~L = 0,

t~LY)'5 ~)L = ~L'¥t~L, ~RTT5 ~R = --~R')'~R ,

we can write the fermion currents that couple to the gauge bosons in (6) as follows:

j2 = _ cot oe J~ = cos Oe~L3'V L ,

JZ = 6-1/2(2J0 - JEM), JH = - 2 - 1 / 2 COS 2aJEM,

J~ = 0 , JA = 2-1/2 sin2°eJEM , (9)

where

JEM =--~VJ~, J0 = ~LTt)L --~LT~L . (10)

We note that the gauge field A does not couple to

any fermions. We also see that

.~OEM = i2 -1 /2g A(2 -1/2 sin 2aJEM ) ,

so that

1 ( 1 1 ) lel = i s i n 2 a g .

The normal Fermi interaction is mediated by W 2 and W~ exchange, and since they are degenerate, we find

2-1/2GF = g2/Sm2(W2),

independent of ~ and the same as the standard model. Proceeding, the effective neutral current interaction can be written as

~OEF F ={g2[Jzm-2(Z)Jz+JHm-2(H)JH]. (13)

Pulling out the neutrino neutral currents, we see that

./9 EFF(V) = ivrSGF p~LTPL(j 0 _ 1 ~JEM) , (14)

where

p = m2(W2)/~ m2(Z).

Comparing (14) to the standard neutral current in- teraction lagrangian

~ws(V) = iVr2GF~L'YVL(Jo -- 2 sin20w JEM) ,

we see that the SU(3) model will reproduce the stan- dard model when sin20w = ¼, in reasonable agree- ment with experiment, and, from (8) and (12),

19=(a 2 + b 2+k2) / (2a 2+b 2) = 1. (15)

In order for (15) to be true, a 2 = k 2. This is perhaps unnatural in a technical sense, since (15) is not gua- ranteed for every set of VEV's, but it is not unreason- able. All the VEV's for the Higgs scalars can have the same magnitude, unlike the case in the standard model where the VEV of a Higgs triplet, needed in order to give a Majorana mass to the neutrino, must be small compared to the standard Higgs doublet VEV.

Next we look at the effective electron neutral cur- rent interaction:

1 1 .QEFF(e) = V/'2GF [p(--eL')'eL + ~ eye) (J0 - ~JEM)

+ p ' cos22a(½ gye) (JEM)] , (16)

where

p' ~ m2(W2)/m2(H) = (a 2 + b 2 + k2)/4k 2 . (17)

The first term in (16) is just the standard electron

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neutral current, while the second term, proportional to p' , is parity conserving. Thus it will not appear in parity violating electron neutral current experiments, but it will appear as a weak correction in e+e- tt+/a - . Experimentally [6]

p ' cos22c~ = -0.01 + 0.03 ,

so that c~ must be close to n/4, since p ' in (17) has a lower bound of ½ (when a 2 = k2). Ifc~ = rr/4 then we have exactly the standard model effective neutrino and electron neutral currents, but the charged lepton masses in (5) will be zero.

This raises the possibility that fermions do not ac- quire masses at this level. We shall now assume that the Higgs fields are needed only to give masses to the appropriate gauge bosons, and the fermions ac- quire masses elsewhere. Let us now assume that only the Higgs triplet fa occurs. The gauge boson mass spec- trum (8) then becomes

m2(W 2) = m2(W~) = ¼mZ(Z) = ½g2(b2),

(18) m2(H) = m2(A) = m2(a) = 0 .

What we have done is to break SU(3) down to SU(2), so that we are left with three massless gauge bosons. At first this appears to be a model with too many "photons", but (6), (9) and (18) tell us that the only interaction is

.1 "~INT = 17g(--cos 2a H + sin 2a A)JEM ,

so tha tg = - 2 e and

photon = cos 2c~ H - sin 2c~ A .

The other two SU(2) massless gauge bosons, sin 2a H + cos 2c~ A and ~,, do n o t interact with ordinary mat- ter. In addition, we have exactly reproduced the Weinberg-Salam model with p = 1 and the prediction

n2 1 that si 0 w 4- Quarks can be incorporated in this model if we as-

sume that they are integrally charged Han-Nambu quarks [7]. The first color, say red, can be placed in a triplet

I " 1 ~rea = (cos~) d L + (sina)d R

L(-s in a)dL + (c°sa )dRJ red

while the next two colors are in antitriplets

t~g reen = (COS O0UL + (sin a) u R

L(--sin ~)UL + (c°s°0uRJ green

~blue = (COS 00UL + (sin °0UR

--sinc0uL + (c°sc0uR blue

The appropriate right-handed fields are assumed to be singlets. This then reproduces the standard model since J0 in (10) becomes

Jo = VL Tv --~L'Y~L + co~lor (ULTUL -- dLTdL)

and JEM is

JEM = --~7 Q -- (37 d)red + (u TU)green + (uTU)blue •

JEM is not a color singlet. All physical hadrons are color singlets, so that only the color singlet part of JEM will contribute in a physical process. The color singlet part of JEM can then be written as

~colorsinglet)_ ~ 2 - ld,),d) --~7~ EM - color (3UTU 3 '

and we have reproduced the standard SU(2)L X U(1) effective lagrangian. The addition of quarks in this manner also cancels the triangle anomalies [8,9].

We also must point out that this "axial" SU(3) electroweak unification is very similar to the SU(3) model in [5]. In that model, though, both particles and anti-particles were placed in the same representa- tion, i.e.

= (VL, ~L, J ~ ) ,

where ~c is the charge conjugate of ~. Because of this, unwanted processes, such as muonium-ant i-muonium oscillations, had to be suppressed by requiring that some of the gauge bosons get large masses. In addition, both quarks and antiquarks are present in the same multiplets, so that weak interactions also change color. Neither of these features occur in our model.

One of us (RP) wishes to thank Professor A. Ramakrishnan for encouragement and support, and the Department of Physics at SFU for their warm

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Volume 125, number 1 PHYSICS LETTERS 19 May 1983

hospitality. This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

[1] Riazuddin, Phys. Rev. Lett. 45 (1980) 976. [2] Riazuddin, Phys. Lett. 116B (1982) 407. [3] S. Glashow, Nucl. Phys. 22 (1961) 579;

S. Weinberg, Phys. Rev. Lett. 19 (1967 ) 1264 ; A. Salam, in: Elementary particle theory: relativistic groups and analyticity (Nobel Symposium No. 8) ed. N. Svartholm (Almqvist and WikseU, Stockholm, 1968) p. 367.

[4] H. Ftitzsch and P. Minkowski, Phys. Lett. 63B (1976) 99; J. Kandaswamy and J. Schechter, Phys. Rev. D 15 (1977) 251.

[5] R.E. Puhg, Phys. Rev. D 21 (1980) 815. [6] B. Naroska, Invited talk second Intern. Conf. on Physics

in collisions (Stockholm, June 1982), DESY preprint. [7] M.Y. Han and Y. Nambu, Phys. Rev. 139 (1965) B1006. [8] S.L. Adler, Phys. Rev. 117 (1967) 2426;

J.S. Bell and R. Jackiw, Nuovo Cimento 51 (1969) 47. [9] H. Georgi and S. Glashow, Phys. Rev. D 6 (1972) 929.

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