Spontaneous Symmetry Breaking
The symmetry breaking of SO(10) is usually done with a combination of (( a 45H OR a 54H) AND ((a 16H AND a ) OR (a 126H AND a )) ).
Let's say we choose a 54H. When this Higgs field acquires a GUT scale VEV, we have a symmetry breaking to Z2 ⋊ /Z2, i.e. the Pati-Salam model with a Z2 left-right symmetry.
If we have a 45H instead, this Higgs field can acquire any VEV in a two dimensional subspace without breaking the standard model. Depending on the direction of this linear combination, we can break the symmetry to SU(5)×U(1), the Georgi-Glashow model with a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), flipped SU(5) (diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)×SU(2)×U(1) (diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal left-right model (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)×SU(2)×U(1)×U(1) for any other nonzero VEV.
The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is called the Dimopoulos-Wilczek mechanism aka the missing VEV mechanism and it is proportional to B−L.
The choice of a 16H and a breaks the gauge group down to the Georgi-Glashow SU(5). The same comment applies to the choice of a 126H and a .
It is the combination of BOTH a 45/54 and a 16/ or 126/ which breaks SO(10) down to the Standard Model.
Read more about this topic: SO(10) (physics)
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