Closing the $SU(3{)}_L\otimes U(1{)}_X$ Symmetry at Electroweak Scale

Abstract

We show that some models with $SU(3{)}_C\otimes SU(3{)}_L\otimes U(1{)}_X$ gauge symmetry can be realized at the electroweak scale and that this is a consequence of an approximate global $SU(2{)}_{L+R}$ symmetry. This symmetry implies a condition among the vacuum expectation value of one of the neutral Higgs scalars, the $U(1{)}_X$'s coupling constant, $g_X$, the sine of the weak mixing angle $\sin {\theta }_W$, and the mass of the $W$ boson, $M_W$. In the limit in which this symmetry is valid it avoids the tree level mixing of the $Z$ boson of the Standard Model with the extra $Z^{\prime }$ boson. We have verified that the oblique $T$ parameter is within the allowed range indicating that the radiative corrections that induce such a mixing at the 1-loop level are small. We also show that a $SU(3{)}_{L+R}$ custodial symmetry implies that in some of the models we have to include sterile (singlets of the 3-3-1 symmetry) right-handed neutrinos with Majorana masses, being the see-saw mechanism mandatory to obtain light active neutrinos. Moreover, the approximate $SU(2{)}_{L+R}\subset SU(3{)}_{L+R}$ symmetry implies that the extra non-standard particles of these 3-3-1 models can be considerably lighter than it had been thought before so that new physics can be really just around the corner.

32 pages, no figure, RevTeX. Some typos corrected

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