Gauge boson mixing in an SU(3) × U(1) theory

Gauge boson mixing in an SU(3) × U(1) theory

Volume 70B, number 2 PHYSICS LETTERS 26 September 1977 G A U G E B O S O N M I X I N G I N A N SU(3) × U ( 1 ) T H E O R Y J. KANDASWAMY and J. SCH...

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Volume 70B, number 2


26 September 1977

G A U G E B O S O N M I X I N G I N A N SU(3) × U ( 1 ) T H E O R Y J. KANDASWAMY and J. SCHECHTER 1 Physics Department, Syracuse Umverstty, Syracuse, New York 13210, USA

and M. SINGER 2 Physics Department, University of Wisconsin, Madtson, Wisconsin 53706, USA

We describe the main features of an SU(3) X U(1) gauge theory m which the charged intermediate bosons mix with each other.

Unified weak-electromagnetic gauge theories based on the group SU(3) × U(1) have been considered for various reasons by a number of authors [ 1 - 3 ] . Recently, interest in this group has revived [4] since it may lend itself to a race explanation of the neutrino induced production [5] of/~-/a-/a +. One possible drawback of this explanation is that it requires, along with the/a-~t-~ +, associated production of a new heavy quark which may be kinematicaUy unlikely for some events [6]. Also, an inelegant feature of these models is that they can only naturally accommodate the Perl lepton r - at the expense of introducing a completely new lepton triplet. Here we show that both these objections can be overcome if the charged intermediate bosons mix with each other. Our notation for the intermediate vector bosons is as follows: singlet: Du octet: Wabu with WCu =0. (1) It is convenient to divide these into three groups with definite charge and CP transformation properties: W~, W~,


( W 3 + W2ul/Vr~,


I Work supported in part by the U.S. Energy Research and Development Administration, Contract No.EY-76-S-02-3533. 2 Work supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the U.S. Energy Research and Development Administration, Contract No. E(11-1)-881, C00-610. 204

In the absence of CP violation the bosons m group (b) should not mix with the one in (c). Evidently the mixing in general may be rather complicated. One may even contemplate the most general mixing pattern, allowing the mixing parameters to be determined by experiment. However, to simplify the analysis we shall here assume that the only appreciable mixings are between W2 and W] and between D and W]. The physical ~articles are defined through

-x/~wlu =


! Du


~ - sin ¢ oso


w~. / \-sin~


Au xs taken to be the photon field, ~bis the analog of f the Weinberg-Salam angle while Wu and Wu are the charged intermediate bosons, a is a new mixing angle. (Actually a mixing like (4) has been previously proposed [2] with a different hadron multiplet structure in attempts to explain the Cabibbo suppression.) The leptons are here taken to be distributed m four triplets and two singlets:











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We may picturesquely consider r - (~1.9 GeV) to be a heavy electron and M- (~8 GeV) [5] to be aheavy muon. The masses are ordered in such away that corresponding members of the muon family are least heavier than those of the electron family. Similarly M0 is a heavy muon neutrino (mass ~ 4 GeV) [5] while E 0 (mass < 4 GeV?) is a heavy electron neutrino. Note that no lepton mixing angles are required. Further note that we cannot get rid of the singlets without having E 0 and M0 be Majorana particles and consequently having excessive [7] neutrinoless double beta decay and similar reactions. The hadron multiplets are now almost Identical to the lepton ones, namely

~b d(0) /L ,


~(,0) L'

th d(0') R'

t' 1 s(o')J R




t L , t L , u R , cR •


26 September 1977

g 2 [ c°s2a + sin2a


\ m2(W)

G ' _- g2 cos a sin x/~ . G" ~=g2

m2(W')] '


m2(W ')

1 m2(W)




( cos2o~+ sm2a . \m2(W ') m2(W)]

Here G is the usual Fermi constant. If the mixing were to vanish (a ~ 0) G' would be zero, G would describe interactions mediated by W21,and G" would describe interactions mediated by W3. In this theory the charged current weak interactions of the well-known particles are the same as usual even though two intermediaries are involved. Note that the decays of the Perl lepton T- ---~/2- PitPe

e- ~eVe


0 is the Cabibbo angle. Of course, in addition to O' one could introduce Cabibbo mixings between t, t' and other corresponding pairs. It is tempting to speculate that the hadrons but not the leptons are Cabibbo mixed. This model is vector-like so no anomalies are present. The coupling constants of the theory may be conveniently specified by giving the covariant derivatives acting on the various fermion multiplets. For a lepton triplet Ca we thus have

are all proportional to G' so that this parameter is in principle measurable. The decay pattern of eq. (12) leads to P ( r - ~/2-- ~uVe) ,~, F ( r - ~ e-PeVe) with the usual "left × left" structure which is consistent with present experimental data [8]. From the above it is also clear why r - could not be economically accommodated in a theory without mixing. We find from (11) the inequality

(~u + 2ig'D~) Ca - 2igWab,~~ b '

G"/G >1(G'/G) 2


while the covariant derivative acting on a quark triplet qa is: auq a -- 2igWbtzqb.


Finally for two typical singlets the covariant derivatives are. a~EOL


(a~ - 2ig'D~)u R.


Using (3) one gets the relations tan ¢ = x / ~ g/g',

lel = - x / 6 g cos ¢,


where lel is the magnitude of the electron charge. Next we discuss the novel properties of the present theory. The weak interactions mediated by charged intermediate bosons are characterized by three effective coupling constants:



showing, for example, that there is no inconsistency m having both [G'/GI and G"/G smaller than unity. The quantity G" makes its first appearance in connection with the M-. We can express the mixing angle as G' tan 2~ = 2 G" - G"


The old lower bound [1] for W2 in the unmixed theory is now replaced by the condition m2(W)mE(W') 1> (43 GeV) 2. m2(W) sin2a + m2(W ') cos2ot


This means that the lighter of the two charged bosons (say W) can be somewhat less massive than 43 GeV. The trimuon events produced by vu on nucleon targets can be given a fairly straightforward explanation in this model by assuming [5] that they result from 205

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the cascading decay of an initially produced M - . Because of the mixing M- can be produced either with an associated ordinary quark or with an associated new quark (t or t' in eq. (6)). The ratio of these two rates is very roughly

o(vuN-+ M - u ) _[ G"12 O(VuN~ M - t ) ~ 3 ~ ] X (phase space ratio), (16) the factor 3 arising because of a right handed coupling for t production. In principle this ratio could be extracted from experiment and furnish information about G'/G". M- is next supposed [5] to decay into /2- (or a source of/2- like r - ) and either M0 or ~ 0 . The latter in turn can decay Into/2+/2- plus neutrino. The experimental ratio [5]

o(vuN -~ /2-/2-/2+~) o(v, N -->/2- g )

5 X 10 - 4 ,


can then be estimated as

[*I(G'/G) 2 +} %(G"/G) 2] B1B2,


where *1 is the phase space ratio of M- production with associated ordinary quark to/2- production, *2 is the same ratio for an associated heavy quark, and

P(M- ~ # B1 =

+ [M0 or M0])

P(M- ~-all) .......


B 2 - P(M° -+/2+/2-) - P(l~° --*/2+/2-) F(M 0 ~ all) C(M 0 -* all)


A rough idea of the branching ratios B 1 andB 2 can be gotten by counting available channels which are kinematically important. We also weight the phase space for, say, M- ~ M°/2-~ u one fifth as heavily as the phase space for say, M- -+/2- vu~u (this factor is the suppression one finds for ordinary muon decay in an unphysical world where rn(e) = {m(/2)). The results are G 2 + G"2 B 1 ~ 5G 2 + 40G '2 + 6G"2 ,

5G'2 + G"2/5 B 2 ~ 28G, 2 + G, 2 ,

(20a, b) where we have assumed that E 0 does not contribute appreciably. The factor *1 is estimated by Barger et al. [9], as 0.05 to 0.18 depending on energy, while they give [6] " 1 / ' 2 ~ 3.5 as an average over the relevant energies. Taking *1 ~ 0.1 and equating (18) to (17) gives an equation for G' in terms of G". Some solutions are


26 September 1977



0.37 0.50 0.78 1.47

0.60 0.40 0.20 0

(G"/G) 2 can be no higher than 1.47 (no mixing solution) without driving (G'/G) 2 negative. Also (G'/G) 2 can not be larger than 0.60 without violating the inequality (13). Of course, the above analysis should be refined, as more data become available to take into account the energy variation o f * l and *2- The precise values of the M- and M0 masses will also improve the estimate o f B 1 and B 2. Note that the total decay width of the Perl lepton r - would be reduced by the factor (G'/G) 2 compared to a an SU(2) X U(1) theory in which the r - and a new neutrino couple universally to vu/2- etc. If(G'/G) 2 1s too small the decays o f t will no longer be prompt. We mention that the deca~,s M- ~ r%-+/2- which are mediated by the W3 -+W~bosons do not contribute to trimuon events. One of these evidently leads to /2-/2-e ÷ (wrong-sign dlmuons) as does the charged current mediated sequence M - ~ / 2 - M ° ~ u, M0 ~ / 2 + hadrons. The similar sequence M- ~ M0 + hadrons, M0 ~/2-/2+u u leads to "symmetric" dimuon events. In this model the E 0 is unstable, decaying with characteristic strength G' into e-/2+v u etc. and (assuming m(E 0) > m(r-)) into r-/2+vu etc. with characterxstic strength G. Recently experiments involving neutral current reactions have Lrnposed severe new constramst on possible models. While the mixing of eq. (4) has no direct effect here we think it worthwhile to briefly show the consistency of this model with the present data. First note that the lack of panty violation in atomic physics experLrnents [10] agrees with the prediction of our effective Lagrangian: £eff(atom. physics)=

~-~ (~-T~Tse)(dT~Tsd) m (H)

(21) 2 g2 1 -- 3 cos2~b ~ae[fi3, (co s 2q~ 1 -- ~5)u 3 m2(Z) sin2¢ + sm2q) d'yad],

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The H mediated term is parity conserving while m the Z medmted term the electron current is purely vector which leads to a drastic suppression of the parity violatmg matrix element. In thas model only the Z mediates the neutrino reduced neutral current reactions. The relevant terms in the Lagranglan are X / ~ sTng~ Za ( - v u T a ( 1 +T5)vu - ~eTa(1 + T5)Ve + (I - 3 cos2~b)~q%e + fiya(cos 2¢ 1 - 75)u


+ sin2¢ dT~d). The inelastic neutrino induced cross sections can be estimated from (22) by using the quark parton model in the standard way [e.g., 11 ]. The relevant result is -

°(vuN-~ vuX-)

very sensitively on the exact choice of ¢ so we should not take them too seriously at present). Finally we gwe a brief discussion of the Higgs mesons needed for a model of this type. The Hlggs' sector of the Lagrangian is far from umque and in fact perhaps not even necessary if one allows the possibility of dynamical symmetry breakdown. Fortunately the main experimental predictions of the weak-EM gauge theories do not depend very directly on the detailed choice of Higgs' bosons. Consider, for example, a Higgs triplet fa with the same charges as the le~0,,ton triplet, three hermitean Higgs octets ~ab , ~ab' , ~ a even under CP and one hermitean Higgs' octet, X~ odd under CP. Then the most general allowed pattern of vacuum values consistent with CP conservation and charge conservation is as follows:

tan4~ + 3/2

°(ruN -~ vu-X) ~

2 tan4q~ + 1/2'


(f)0 =

where ~ is the angle defined in (3). From (23) we immediately get

o(~,N ~ ~,,X) 3 -e -~NS__ ~vuX) >~ 1/2.

(~I/)o= (24)

The experimental result [12] for the ratio in (23) is clearly different from unity and is about equal to 0.6 with a fairly large uncertainty. Taking the value 0.6 gives ~ ~ 56 °. This value is also roughly consistent with the elastic neutrino proton data [13]. In a similar approximation the analog of (23) is

o(vuP ~ ~uP) a(vup ~ vup)


tanaq~ + 2


7 tana¢ + 2

Another neutral current experiment to be checked is Vee -* ~e e. The effective Lagrangian is £eff(~ee ~ Fee )

= -(G/V~)[Ve~'~ (1 + "r5)%1 [e~'~(Cv + CA'rs)e] CA = 1

Cv = 1

(26) 2 x/r2 g2 1 - 3 cos2q~ 3 G m2(Z) sin2~b

The experimental data [14] for this reaction together with the data for Pue ~ ~ue lead for ¢ ~ 56 ° to 23 GeV m(Z) <~ 42 GeV. (However those bounds depend

26 September 1977

!1 ,


(qb)0 = 0



k' - a ' - b '

(×>o =

0 -iK


0 k

(~")0 =



( 00! b"



0 -a"-b"


where all constants are real. Note that SU(3) symmetry transformations are similarity transformations in the above space. We have thus used the freedom to make SU(3) transformations to diagonalize (qb")0. In general, of course, (qb>0 and (qb')0 are not sunultaneously diagonalizable. It turns out that to generate a pattern of intermediate boson masses similar to the one postulated here f, qb, and ~b" are all that is required. To get the correct pattern of fermion masses we need in addition qb' and X as well as the elimination of the bare fermlon mass terms by a discrete symmetry. It is easy to work out that the photon-Z mixing given in (3) follows from the Higgs field f. If a different boson triplet - with charge assignment (1,0, 0) - were used we would have a slightly more complicated structure in which the first three fields of (2b) could mix. To see explicitly how the mixing (4) arises it may be helpful to write out the contributions to the vector boson mass terms in the Lagrangian which result from just ~. (The contributions from other Higgs' scalars are simply 207

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additwe). We have the terms in £' -4g2{W12W 1 [(a - b) 2 + k21 + W~W 1 [(2a + b) 2 + k21 - 3 k a ( W ~ W 1 + w l w ~ ) + 2k2H 2

2 +-~(2b+a)2\


w3 w22


) +[} (2b+a)2


Note that the third term is the one which gives the mixing o f W 2 and W 3. It evidently requires (~)0 to have b o t h diagonal and off-diagonal parts. From our previous discussion it is clear that trying to get rid of this term by diagonalizing <~)0 would result in a similar term popping up again in, say, the transformed <~")0" We see from (28) that if (2b + a) :~ 0 there will also be some mixing o f the fields H and (W32 + W23)/x/2. Such a mixing would not affect any decays of the wellknown particles b u t it would lead (with strength dependent on the amount of mixing) to decays like M -->/l-/l+g - , / 1 - e + e - and r - --> e - e + e - , e-g+/1 - . The latter have not been seen [8] so we must have 2b + a = 0, etc.

References [1] J. Schechter and Y. Ueda, Phys. Rev. D8 (1973) 484; C.H. Albnght, C. Jarlskog and M. Tjm, Nucl. Phys. B86 (1975) 535; L.K. Pandit, Pramana 7 (1976) 291; F. Gursey, P. Ramond and P. Sikwie, Phys. Rev. D12 (1975) 2166; P. Ramond, Nucl. Phys. B l l 0 (1976) 214; M. Yoshimura, Prog. Theor. Phys. 57 (1977) 237, G. Segle and J. Weyers, Phys. Lett. 65B (1976) 243. [2] J. Schechter and M. Singer, Phys. Rev. D9 (1974) 1769;


26 September 1977

L. Clavelh and T. Yang, Phys. Rev. D10 (1974) 658; V. Gupta and H. Mare, Phys. Rev. D10 (1974) 1310. [3] Theories based on the simple group SU(3) have recently been gwen by. H. Fritzsch and P. Minkowski, Phys. Lett. 63B (1976) 99; J. Kandaswamy and J. Schechter, Phys. Rev. D15 (1977) 251; Y.M. Cho, N.Y.U. report TR2/77 (1977); J. Kim, Brown Univ. report HET-334 (1976). [4] B.W. Lee and S. Weinberg, Phys. Rev. Letters 38 (1977) 1237; P. Langacker and G. Segre, Pennsylvania report UPR0073T (1977); D. Horn and G.G. Ross, Caltech report 68-597 (1977); R.M. Barnett and L.N. Chang; SLAC report PUB-1932 (1977) [5] A. Benvenuti et al., Phys. Rev. Lett. 38 (1977) 1110, 1183; B.C. Barish et al., Phys. Rev. Lett. 38 (1977) 577; C.H. Albright, J. Smith and J. Vermaseran, Phys. Rev. Lett. 38 (1977) 1187; V. Barger et al., Phys. Rev. Lett. 38 (1977) 1190; F. Bletzacker, H. Nieh and A. Soni, Phys. Rev. Lett. 38 (1977) 1241; A. Zee, F. Wilczek and S. Treiman, Princeton preprint (1977). [6] V. Barger, T, Gottschalk, D. Nanopoulos and R. Phillips, University of Wisconsin report C00-600 (1977). [7] A. Halpin, P. Minkowski, H. Pnmakoff and S.P. Rosen, Phys. Rev. D13 (1976) 2567; T.P. Chen, University of Missouri report (1975), unpublished. [8] M.L. Perl, SLAC PUB-1923 (1977), to be published in: Proc. of the XII Recontre de Monond, ed. T.T. Van. [9] V. Barger et al., Wisconsin report C00-596 (1977). [10] P. Baird et al., Nature 264 (1976) 528. A recent review is given by C. Bouchlat, J. Phys. G3 (1977) 147. [11] M. Barnett, Phys. Rev. D15 (1977) 675. [12] A. Benvenuti et al., Phys. Rev. Lett. 37 (1976) 1039. Also see ref. [11]. [13] D. Cline et al., Phys. Rev. Lett. 37 (1976) 252. W. Lee et al., Phys. Rev. Lett. 37 (1976) 186. D. Chne et al., Phys. Rev. Lett 37 (1976) 648. [14] F. Remes, H. Gurr and H. Sobel, Phys. Rev. Lett. 37 (1976) 315.