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Unification and distance scales

This section uses units where (Planck's constant)/2p and the speed of light = 1. This choice of units is called natural units. With this choice, mass has units of inverse length, and vice versa. The conversion factor is 2x10-7 eV = 1/meter.

Unification and group theory

   The success of spontaneous symmetry breaking in explaining electroweak physics led physicis to wonder whether the three particle theories of the SU(3)xSU(2)xU(1) model could be the spontaneously broken version of a higher unified theory at some higher energy scale, a single theory with only one gauge group and one coupling constant. This type of theory is called a Grand Unified Theory, or GUT for short.
   The quantum behavior of the known particle coupling constants supports the idea of Grand Unification. Because of renormalization, the electromagnetic coupling constant grows larger at high energies, whereas the coupling constants for the weak and strong nuclear interactions grow smaller at higher energies. At the mass scale

Grand unification energy scale

the three coupling constants become equal. Therefore, this ought to be the mass scale where the single gauge symmetry of a Grand Unified Theory would become spontaneously broken into the three distinct gauge symmetries of the SU(3)xSU(2)xU(1) model.
   The single gauge group of a GUT has to be mathematically capable of containing the group product SU(3)xSU(2)xU(1) of the three gauge groups relevant to low energy particle physcs. The best candidate for such a theory is unitary group SU(5), which would give 24 gauge bosons mediating the single unified force, but there are also other GUT models based on other groups, such as the orthogonal group SO(10), which would give 45 gauge bosons and contain the SU(5) theory as a subgroup.
   The problem with Grand Unification is that the unified gauge bosons allow quarks to couple to leptons in such a way that two quarks can be converted into an antiquark and an antilepton. For example, two up quarks would be allowed to turn into a positron and and a down antiquark.
   A proton consists of two up quarks and and down quark. A neutral pion consists of a down quark and a down antiquark. Therefore the unified gauge boson in a GUT could mediate proton decay by the interaction

and other related decays.
   The proton lifetime predicted in a GUT is about

Proton lifetime in a GUT

whereas the current best measurement of the proton lifetime is

Observed proton lifetime

It's important to note here that proton decay can happen through radiative corrections even in the Standard Model, so we don't expect the proton lifetime to be infinite.
    However, it seems that the proton doesn't decay as quickly as predicted by a GUT. This situtation is improved when supersymmetry is added to the GUT, and this will be explained in next section.

What about gravity?

    Einstein's elegant and experimentally tested theory of gravity called General Relativity is not a normal gauge theory like electromagnetism. The symmetry is not a unitary group symmetry like U(1) or SU(3), but instead a symmetry under general coordinate transformations in four spacetime dimensions. This does not lead to a renormalizable quantum field theory, and so gravity cannot be unified with the other three known physical forces in the context of a Grand(er) Unified Theory.
   But string theory claims to be a unified theory encompassing all known forces including gravity. How can that be? The main symmetry apparent in string theory is conformal invariance, or superconformal invariance, on the string world sheet. This symmetry dicates the spectrum of allowed mass and spin states in the theory. The spin two graviton and the spin one gauge bosons exist within this framework naturally as part of the tensor structure of the quantized string spectrum.
   This is another reason why physicists have become so impressed by string theory. There exists a completely novel way of putting gravity and the other known forces together in the context of a single symmetry, that is much more powerful than the ordinary quantum gauge theory of particles. But the question is -- is this really the way that nature does it? The answer to that may take a long time to sort out.

Symmetry breaking in string theory

   The two string theories that have shown the most promise for yielding a pattern of symmetry breaking that is like Grand Unification plus gravity are the heterotic superstring theories based on the groups SO(32) and E8xE8. However, these are supersymmetric theories in ten spacetime dimensions, so the symmetry breaking scheme also has to be involved with breaking the supersymmetry (because fermions and bosons don't come in pairs in the real world) and dealing with the extra six space dimensions in some manner. So the possibilities, and the possible complications, are much wider in string theory than in ordinary quantum gauge field theories.
   Forgetting these complications for a moment, focus on the group theory of the E8xE8 model. The group E8 is an exceptional group with interesting properties too complex to explain here. The common suppostion is that one of the E8 groups remains unbroken, and decouples from physical observation as a kind of shadow matter. The other E8 has the right mathematical structure to break down to an SU(5) GUT via E8 -> E6 -> SO(10) -> SU(5).
   The symmetry breaking scale would presumably start somewhere near the Planck scale

Planck mass

and end up at the GUT scale of about 1014 GeV. The spontaneous symmetry breaking mechanism would presumably be scalar field potentials of the form shown above, where a subset of the scalar fields with normal modes like the radial mode become massive, and the remaining massless scalar fields become longitudinal modes of massive gauge bosons to break the gauge symmetry down to the next level.
   But -- in string theory, at the level of perturbation theory where the physics is most understood -- the scalar potentials seem to be flat in all directions and hence the scalar fields all remain massless. The solution to symmetry breaking in string theory has to be nonperturbative and is still regarded as an unsolved problem.

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