Loss of perturbativity!

Electroweak Precision data still keeps room available to include one more chiral family into the standard model provided the quark and lepton masses are greater than some lower bound. The lower bound for the fourth generation quarks are around 200 GeV whereas for charged lepton it is around 100 GeV. The fourth generation neutrino should be more massive than

so as not to contribute to Z boson decay width which is experimentally measured very accurately and is in good agreement with three family Standard Model. Now in the standard model we have top quark yukawa coupling almost equal to 1 so as to account for its mass . Thus if we want to account for fourth generation quark masses, we have to take the corresponding yukawas large -->> Loss of perturbativity? In MSSM, the problem gets even more complicated. We have two Higgs doublet in this case, with vacuum expectation values and and their rations are denoted by . It turns out that with low value of ( close to unity) we can keep the yukawas perturbative at the electroweak scale and at the same time give rise to fourth generation quark and charged lepton masses above the experimental lower bound. However such a low value of will make the lightest Higgs boson mass at tree level very small and we have to check if loop corrections (including fourth generation) can make its mass greater than the LEP lower bound 114.5 GeV. Now suppose after taking loop corrections, we are getting Higgs mass greater than this limit as well as fourth generation masses are also above the lower bounds while keeping the yukawa perturbative. The problem is not yet solved, because when we evolve those yukawas under renormalization group, at every stage upto the grand unification scale (GUT) (assuming there is no new physics between MSSM and the GUT scale). But it turns out that (although I haven't checked it yet but there are works related to this in the literature)yukawas become non-perturbative near the TeV scale if there is not new physics between MSSM and GUT scales. People then incorporate new physics at the TeV scale which keeps the yukawas perturbative till the GUT scale. So far I have seen only one paper arXiv:0806.2064 where they have talked about adding some new vector like matter particles at the TeV scale. I personally find these vector like matter particles quite ad-hoc although they are serving this particular purpose here, I don't know how to incorporate them within the framework of some higher theories like Grand Unified Theories. But vector fields contribute differently to the beta function compared to chiral fields. Their contribution comes with opposite signs and may be that could be the reason why vector particles help the yukawas to slow down their running to keep them perturbative till the GUT scale. Anyway these are theoretical issues, but experimentally also fourth generation might have interesting signatures from colliders to dark matter search experiments as well. Will update about those issues next time :)

The Dead Man's running :P

For last couple of days I was trying to see how the Yukawa (The dead man :P) couplings of a particular model run under renormalization group evolution. Although at one loop level the equations are pretty simple, the whole sets of equations look messy since they are coupled to each other. So if there are say 23 parameters, there will be 23 coupled linear differential equations. For a complicated model where there are many parameters, lots of approximations need to be taken since we don't have as many input parameters at the electroweak scale. That's why I believe top down approach is much simpler although I never tried that. I always follow bottom up approach where I give low energy parameters as input and predict the high energy parameters. In top down approach since there are very less number of parameters, we can choose them so as to arrive at the low energy expected parameters after the running. Since I don't yet know the high energy theory properly, I can't say whether the high energy parameters I am getting after the running will fit inside a specific GUT model. May be it's time for me to switch to hard core GUT models and use the top down approach :)