What are the problems that physicists face while searching for the grand unified theory (GUT)? I would like to know the missing links in the grand unification and what are the possible candidates for those missing links.
 A: There are numerous problems which physicists hope to address while solving for a grand unified theory. The premise of the theory is that there exists an energy scale beyond which the running of the strong and electroweak coupling constants is described by the same equation i.e. the strng and electroweak forces are "unified." YOu can find a list of these problems on the wikipedia site for grand unified theories here. SOme of the problems with the standard model are:


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*Why are there three fermion families? And why does it seem like most of the universe is made up of only the first family?

*Why are the values of coupling for the strong, electromagnetic and weak forces the way they are?

*Why are there so many free parameters in the SM?

*Why is the electric charge not quantized?


The idea is to go away from the $SU(3)\times SU(2) \times U(1)$ construction of the standard model which represents the various symmetries of the different sectors of the SM and enhance the size of the symmetry group. Enhancing the symmetry of the SM could explain why some of these "problems" occur and therefore scratch them off as effective phenomena when the larger symmetry is broken to the standard model. The problem with this is that so far, the approach needs the "one size fits all scheme" i.e. you should be able to explain everything the standard model sees from this one theory. In principle this is a very difficult thing to do. For this there are many candidates.


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*The $SU(5)$ aka the Georgi-Glashow model. In this model, the SM is a maximal subgroup of the $SU(5) $ group. It tells us why there is no right handed neutrino. It quantizes the electric charge. The proton is very stable. It makes prdictions about the mixing of Weinberg angles correctly. Can be useful in explaining the baryon asymmetry.  But it does not explain why there are 3 families of fermions, it still has many free parameters, and they allow of the existence of magnetic monopoles. 

*$SO(10)$ model (with Pati-Salam): There are a bit more interesting to work with. I would recommend reading Rabindra Mohapatra's book to find out more about this. 

*So you see, the pattern keeps going on and on. I.e. we need to investigate all models which can in principle yield the standard model (with or without supersymmetry) so we cannot settle on which higher gauge group is correct. Is it an exceptional group? If so, which one? 


But there are issues which the GUT's must address and if your model can address all those problems then, well, you're my hero. A good indicator for the study of grand unified theories could come from the bounds for proton decay. Protons are expected to be stable in the SM but several GUT's predict their eventual decay (takes longer than the age of the universe by many orders of magnitude though). So there are many experiments too which could aid the theory and help narrow down the problem to analysing just a few handful of models. 
I would urge you to read the book by Mohapatra if you are interested in such problems.
