Grand Unified Theory

Grand Unified Theories, or GUTs (without which, is has been alleged, there is no glory) are theoretical attempts to unify the electromagnetic, strong and weak forces. Despite their name, they are actually less "grand" than Unified Field Theory (which hopes to add gravity into the mix as well).

Motivation
The Standard Model has been very successful experimentally, but it has various theoretical problems. Despite its complexity, it shows evidence of some underlying simplicity: The convergence happens around an energy of about 10^(16) gigaelectronvolts (GeV). That energy is so high that if one wanted to build a particle accelerator to reach it, it would have to be somewhere around the size of our Galaxy. Thus, it is unlikely that anyone will build such an accelerator anytime soon.
 * Interrelationships between particle multiplets and in their parameter values
 * Convergence of gauge-interaction strengths on a single value at high energies (gauge unification)

Such complexity with patterns has been discovered before, and in each case, it has been resolved by the discovery of some underlying simplicity. This gives hope that some underlying simplicity may be discovered for the Standard Model, a simplicity that is part of some GUT.
 * Chemical elements and atoms -- electrons and atomic nuclei
 * Nuclei -- protons and neutrons
 * Hadrons -- quarks

The smallest GUT: Georgi-Glashow SU(5)
Physicists Howard Georgi and Sheldon Glashow worked it out in 1974, as the Standard Model was being worked out. It is the GUT with the smallest gauge-symmetry group that is "simple" in the Lie-algebra sense (mathematician Sophus Lie, "Lee", not deliberate falsehoods).

All the elementary fermions fit inside a few multiplets: Notice the nice pattern -- the multiplet sizes are binomial(5,k) for k from 0 to 5.

All the Standard-Model gauge fields are parts of one gauge-field multiplet. It has size 24, and it has some extra gauge fields, the "leptoquarks". They have (anti)quarklike QCD color states, and they have electric charges +-1/3 and +-4/3. Likewise, the Higgs particle is part of a 5* multiplet, one that includes a down-quark-like "Higgs triplet". In the Minimal Supersymmetric Standard Model (MSSM), there are two unbroken-electroweak-symmetry Higgs particles, an "up Higgs", Hu, and a "down Higgs", Hd. The Hu is in a 5 multiplet and the Hd in a 5* multiplet, so the MSSM can also fit in as a subset of supersymmetric SU(5).

The leptoquarks and the Higgs triplets do something very interesting: they cause isolated protons to decay. They can also do that to isolated neutrons and bound protons and neutrons, but in many cases, this competes with other decay processes that are MUCH faster. In fact, most GUT's predict proton decay, with the rate being connected to the GUT energy scale.

This GUT also predicts partial mass unification, with down-type quarks and charged leptons having the same Higgs interaction strength at the GUT energy scale. This means that the effective masses of the bottom quark and the tau lepton will be the same at that energy scale. Effective meaning with quantum-mechanical effects when going from a few GeV to GUT energies.

Next up: SO(10)
Physicists Harald Fritzsch, Peter Minkowski, and Howard Georgi worked it out around 1974, around when Georgi and Glashow worked out the SU(5) model. This model is a superset of the SU(5) model, with its gauge-symmetry group breaking down as

SO(10) -> SU(5) * U(1)

The U(1) factor here is for a quantity related to B-L, (baryon number) - (lepton number). This breakdown goes as follows:

(Mplt = multiplet, Elem Ferm = elementary fermion) The gauge multiplet includes some additional leptoquarks and a sort of "B-L Z boson", but the particle content of SO(10) is otherwise the same as for SU(5).

Each generation of elementary fermions fits into one multiplet, the ultimate in unification. This makes possible complete mass unification, or more properly, complete Higgs-interaction unification. But it comes at a price: it has no cross-generation decays. These must be contributed by GUT symmetry breaking. Likewise, the elementary fermions include right-handed neutrinos, but they cannot have Majorana masses for the neutrino-mass seesaw effect. That also must be contributed by GUT symmetry breaking.

Likewise, all the Higgs bosons fit into one multiplet, a multiplet that accommodates the MSSM up Higgs and down Higgs.

Other GUT's
Here are various other Grand Unified Theories that have been proposed:
 * Pati-Salam: SU(4)*SU(2)*SU(2) ~ SO(6)*SO(4), a subset of SO(10). It treats leptons as a sort of fourth quark color.
 * E6, a superset of SO(10). Elementary fermions and Higgs bosons can fit into the same multiplet, if they are supersymmetric. That multiplet has size 27 and the gauge multiplet size 78.
 * Trinification: SU(3)^3, a subset of E6. It is unusual among GUT's for not predicting proton decay.
 * E8, a superset of E6. All of the Standard-Model particles can fit into one E8 gauge multiplet, though it must have supersymmetry and live in more than 4 space-time dimensions. This multiplet has size 248.

With E8, we get into string theory, in particular, the "HE heterotic superstring". It has supersymmetric gravity or "supergravity" along with two gauge fields, each one having symmetry group E8. It lives in a 10-dimensional space-time, so to get that to our Universe, 6 of those dimensions must be "compactified" into a tiny ball, about Planck-length-sized. This compactification produces symmetry breaking that breaks one of the gauge-field multiplets down through various GUT's to the Standard-Model fields.