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.

Towards a more and more complete theory..an endless journey it seems!

The hunt for a complete fundamental theory of Physics is not new, it was one of Einstein's unfulfilled dream or may be many Great minds also dreamed of it earlier. Of course it sounds so amazing that just one theory could possibly explain everything around you, but sometimes I feel if this expectation or dream is just a wish or is there any physical necessity. I mean why can't we describe two different observed phenomena using two different theories. Yes we can, but there is no guarantee that these two observed phenomena always happen separately. There may be some situations where both these phenomena are happening simultaneously (in a non-independent way because the two different systems may interact with each other) and we have no clue which theory we should use to describe them. Hence we always wish if we had a theory which can explain everything using one master equation, say. But the question is how far have we been in this really long journey of four or five decades. String theorists are doing a great job of course in developing such an elegant theory which can describe all the four fundamental forces(gravity, electromagnetic, strong and weak nuclear force) in the Universe together. There is no other theory where we can talk about all four forces. In particle physics we can talk about the later three using gauge theories, but gravity is a real bitch in such theories which (if we include) make everything meaningless. Particle physicists that's why live with these three forces only which serve their purposes also because within the region of their interests gravity is negligible. So in particle physics, a complete theory means a theory which can describe these three forces together, the so called Grand Unified Theories (GUT). There have been tremendous works in the last three and half decades on these theories. They look beautiful of course since they have strong predictivities. You write your GUT theory at very high energy and see what it predicts at low energy accessible to experimental detection. There are many models based on different gauge groups which can give rise to standard model of particle physics at low energy and also predict most of the experimental results so far. The best thing is that you start with a theory where there is only one force, only one type of particles at very high energy, and as a output you get a theory with three different forces and many different particles at low energy. But still lots of works need to be done to have a complete theory. Although the achievements in both string theory and such GUT models are tremendous so far, I am still confused whether the real goal of having such a fundamental theory is achieved (even partially) or not. Like GUT people do not talk about gravity, string theories also do not uniquely predict the observed phenomena at low energy. Recently there have been lots of enthusiasm in arriving at a GUT model starting with string theory. But to make the theory internally consistent the GUT models or any particle physics model people arrive at from string theory constructions contain many more particles than a minimal particle physics model. Thus predictivity is lost if we consider such non-minimal models. Hope people will continue such works with more and more enthusiasm and soon we will see some success in this direction.

Assuming Survival principle was not a very good idea

I was doing some calculations regarding renormalization group evolutions of masses, couplings etc within the framework of a model. For simplicity I took the mass of various Higgs fields same as their mass terms appearing in the superpotential. After doing all the analysis, I came to know that the physical masses of the Higgs fields need not always be same as the bare mass terms in the superpotential or the Lagrangian. In fact the assumption I took was quite ad hoc. Before calculating the physical masses of all the Higgs fields, I can not assume their masses, but that's also one way to study the RG running, but it might not give the correct result always. In some cases the physical masses comes out to be very different from the bare mass terms and hence will affect the RG substantially. What happens is that when we write the F-term scalar potential of a model, sometimes the quartic term of a scalar field remains absent because the corresponding trilinear term was not there in the superpotential so as to obey the internal gauge symmetries. Thus the mass terms remain arbitrary at the renormalizable level and higher dimensional terms have to be considered. This leads to lighter Higgs masses than the bare mass terms. And even if the quartic terms are there in the superpotential, it is not to straightforward to say that the mass will be same as the bare mass terms, because the various Higgs fields mix with each other, and it's not unusual that the diagonalization of their mass matrix will give a small eigenvalue. So better calculate the physical masses of all the particles (not just fermions ) and then proceed with the RG evolutions.

Conphused 2!

I could make some improvement in the unification plots than before. I am using the U(1) normalization to be $ \surd{3/8} $ in my calculations. For a specific choice of intermediate symmetry breaking scale I see the SU(3) and the SU(2) line meeting near $ 10^{12} $ GeV, which is not enough to prevent proton decay however. But I guess this tension can be eased if we add some new color degrees of freedom and make the two lines meet at a higher scale. But one serious problem is coming from the $ U(1)_{B-L} $ line which is hitting the Landau pole much before the expected unification scale. One way to get rid of this is may be to shift the intermediate symmetry breaking scales or may be to add some new $ U(1)_{B-L} $ charged particles. But adding new such particles might make the $ U(1)_{B-L} $ line meet the Landau pole even before the present case. May be I need to play around with these two possibilities and see how things get changed.

Conphused!

I am getting confused regarding $U(1)$ normalization while trying to check unification of gauge coupling constants in a class of models. For $ U(1)_Y $ the normalization constant square is $ \surd{3/5} $ if we take the GUT group SU(5) . For $ U(1)_{B-L} $ its according to some convention $ \surd{3/8} $ and according to some others $ \surd{3/2} $ if we take the GUT group $ SO(10) $. Initially I was following a paper by Manfred Lindner where he has taken $ \surd{3/8 } $. Today I have seen a paper by R. Mohapatra where he has taken $ \surd{3/2} $. I am not being able to derive any one of them from the $ SO(10) $ multiplets. Depending on the normalization the boundary condition for intermediate symmetry breaking also changes and that is going to affect the running. My calculations are giving some nonsense results now where the U(1) coupling is hitting the Landau Pole much before the preferred unification scale. May be I have to do all the calculations more carefully.

Coupling Constant Unification

Currently I am working on gauge coupling unification in certain specific models. I studied basic renormalization in abelian gauge theory as well as scalar field theory during my MSc coursework but did not study renormalization group evolution properly. My supervisor advised me to look at Weinberg's QFT volumes, and believe me I liked the way this renormalization topic is written in that book and that has probably given me enough enthu to spend 3500 bucks to buy all the three volumes. The renormalization group evolution is given by $ \mu (d g/d \mu) = \beta(g) $, where g is the coupling constant and $ \mu $ is the mass scale. Thus this equation basically predicts the evolution of the gauge coupling constant with energy scale. I have not yet looked into the derivations of the beta functions for the most general case. I am just using the standard formula for one-loop beta function from text books and using it in my own model. The new thing I have learnt is that choosing a specific gauge group does not decide the beta function completely. Its the particle content of that group: the fermions, gauge bosons as well as Higgs scalars which decide the form of beta function. Thus for the same gauge group if we modify the particle content, the beta function will also change and accordingly the evolution of the coupling constants.