# Spontaneous symmetry breaking in SU(5) GUT?

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At the end of this video lecture about grand unified theories, Prof. Susskind explains that there should be some kind of an additional Higgs mechanism at work, to break the symmetry between the $SU(2)$ and $SU(3)$ contained in a $SU(5)$ grand unified theory.

How is this additional Higgs mechanism assumed to work exactly and where do/does the additional Higgs come from? To avoid a too speedy proton decay in this theory, the $X$ and $Y$ gauge bosons of processes mixing the leptons and quarks must be huge (3 orders of magnitude smaller than the Planck mass). How can they get such a huge mass, compared to the gluons and the electroweak bosons?

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The simplest $SU(5)$ GUT Higgs transforms as ${\bf 10}$ under the gauge group, an antisymmetric tensor $5\times 4/2\times 1$ with two indices of the same kind (without complex conjugation). The 2-dimensional representation of $SU(2)$ has an antisymmetric invariant $\epsilon_{ab}$ and if you extend this antisymmetric tensor to 5-valued indices of $SU(5)$ and only make the $ab=45$ component nonzero, it will break the $SU(5)$ down to $SU(2)$ rotating $45$ and $SU(3)$ rotating the remaining $123$.

One could a priori think about other representations, for example ${\bf 15}$, the symmetric tensor with two indices $5\times 6/2\times 1$. It passes the basic test: You may imagine it determines a bilinear form on the 5-dimensional fundamental representation that has a different coefficient for the group of 3 basis vectors and different for the remaining 2 basis vectors among the 5, so something that tells you $$ds^2 = A(da^2+db^2+dc^2)+B(dd^2 +de^2)$$ where $A,B$ are different complex coefficients and $(a,b,c,d,e)$ is a complex 5-dimensional "vector" in the fundamental representation. It's easy to see that distinct values of $A,B$ break the rotational symmetry $SU(5)$ between all five $(a,b,c,d,e)$ to $SU(3)\times SU(2)$ between $a,b,c$ and $d,e$ separately.

It's hard to write realistic potentials for this one – and moreover, the hypercharge $U(1)$ which should be composed of the $U(1)$ factors in the $U(2)$, $U(3)$ subgroups – won't arise properly (the bilinear form above isn't invariant under any such $U(1)$) – but there exist other, larger representations for the Higgs in $SU(5)$ that can potentially do the breaking job.

In $SO(10)$ gauge theories, one usually needs a 16-dimensional representation to do the Higgsing to $SU(5)$. The $SU(5)$ is the subgroup preserved by a single chiral spinor. There may also be a 126-dimensional Higgs multiplet to do similar things (antisymmetric, self-dual, with 5 indices) but I don't want to list all group theory used in grand unification here.

In string theory, the breaking of the GUT gauge group often proceeds by non-field-theoretical mechanisms such as fluxes and the Wilson lines around some cycles in the compactified dimensions. The Wilson line is a monodromy, an element of the original unbroken gauge group, and the gauge subgroup that commutes with the monodromy remains unbroken. It has some advantages because the required Higgs fields in GUT theories (and their potentials) may be rather messy and moreover, the stringy approach may justify more structured Yukawa couplings for various quarks and leptons which is probably needed.

GUT theories have their characteristic energy scale, the GUT scale, so all massive things such as the $X,Y$ new gauge bosons as well as the new GUT Higgses are naturally this heavy, near $10^{16}\,{\rm GeV}$. There are other ways aside from the proton decay constraints to derive this energy scale - it's the scale at which properly normalized three Standard Model gauge couplings approximately unify (almost exactly when supersymmetry is added).

So before one answers your question, it must be reverted. The right question is why the other fields (and dimensionful parameters) are so immensely light relatively to the GUT scale. Because most of them are derived from the electroweak Higgs mass to one way or another (gluon is formally massless although it's confined at the QCD scale, and one explains the QCD scale as the scale at which the slowly logarithmically running QCD coupling just grows to 1 if we run from a reasonable value near the GUT scale), this question really asks why the electroweak Higgs boson is so much lighter than the GUT scale. This question is known as the hierarchy problem and it's been the primary mystery that was driving much of the work in phenomenology and model building although the LHC, by its seeing nothing new, is increasingly suggesting that there may be no "nice answer" to this puzzle at all.

This post imported from StackExchange Physics at 2014-03-12 15:31 (UCT), posted by SE-user Luboš Motl
answered Dec 9, 2012 by (10,278 points)
Thanks for these nice detailed explanations and clarifications, very helpful!

This post imported from StackExchange Physics at 2014-03-12 15:31 (UCT), posted by SE-user Dilaton

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