Light Higgs!

We all know that in minimal supersymmetric standard model (MSSM), we naturally arrive at a light Higgs whose mass is of the order of Z boson mass. Its only after taking loop corrections into account we can make it as heavy as the LEP limit which is around 114.5 GeV. The question of light Higgs bosons always puzzled me in Supersymmetric Left-Right(SUSYLR) models which forced me to work out the Higgs boson mass in MSSM at least. Only after the calculations I got the idea how things become so different when we add supersymmetry to the standard model. In standard model we have a Higgs potential involving the mass term and the quartic coupling term. However in MSSM we have only a bi-linear term in the superpotential which can give rise to the mass term in the scalar potential. But the quartic coupling term can come from only the D-terms. Hence the quartic coupling parameter in this case is a combination of the gauge coupling constants, not a free parameter like in the case of standard model. We have no choice in MSSM to fine tune these couplings to make the Higgs mass heavy. In richer theories like SUSYLR, things become more complicated.

Suppose we have triplet Higgs fields $ \triangle_L, \triangle_R $ in addition to the doublet Higgs fields which gives masses to the fermions. These triplets are added to break the SUSYLR gauge symmetry $ SU(2)_L \times SU(2)_R \times U(1)_{B-L} $ down to the MSSM gauge symmetry $ SU(2)_L \times U(1)_Y $. Here L, R denotes left handed and right handed respectively, B, L and Y denote baryon number, lepton number and weak hypercharges respectively. These triplet fields have B-L charge $\pm 2$ and hence we need to add two more such fields in SUSYLR model to cancel the anomalies. Now the common intuition will say that these triplet fields will acquire masses of the order of $ SU(2)_R $ symmetry breaking scales because these fields are introduced for this symmetry breaking. This intuition is sometimes called survival principle which I mentioned in one of my previous posts. Now if we actually calculate the mass spectra, we see that this principle is not always true. For example, consider these triplets. Gauge symmetry does not allow us to write the bi linear or trilinear terms of the same triplet field in the superpotential. We can however write bi linear mass terms involving one triplet and another of opposite B-L charge. Now when we write the scalar potential we only have the mass terms and no quartic terms. And unlike in MSSM where we could find some quartic terms coming from the D-terms, here we can't have it because that will break supersymmetry. We are breaking SUSYLR to MSSM which means that supersymmetry is not broken, but only the gauge symmetry is broken. Hence the D-term scalar potential will decouple at this stage. Thus we need to incorporate non-renormalizable quartic terms which will give rise to non-zero mass of these Higgs fields. The masses come out to be light naturally since these non-renormalizable terms are highly suppressed by the Planck scale or GUT sale. I think, if the $ SU(2)_R $ breaking scale is around the supersymmetry breaking scale, we can have quartic couplings. But in this case also existence of light Higgs will still be true. I was reading some papers by Mohapatra et al. where they describe the scenario in terms of some enhanced global symmetry in the superpotenal in the absence of some terms, then they break this approximate symmetry resulting in pseudo-goldstone bosons which are not exactly mass-less but have light masses. I am still not confident about those enhanced global symmetries and all, but the existence of light Higgs in such models is something unavoidable with minimal field content. Such light particles creates problems in gauge coupling unification since they keep contributing to the evolution till the electroweak scale which forces the couplings to reach Landau pole (non-perturbative) much before the GUT/Planck scale.