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The $SO(10)$ group is spontaneously broken down into the Standard Model gauge group at around $10^{16}$ GeV and the electroweak scale is ~246 GeV. I think there should be 45 gauge fields predicted by an $SO(10)$ GUT (from $\frac{1}{2}N(N-1)$ generators, with $N=10$). How many of them would experience SSB and therefore acquire mass at scales above the electroweak scale?

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    $\begingroup$ Did you count? What is your guess? You know the number of them for the SM, right? $\endgroup$ Jan 8, 2022 at 23:22
  • $\begingroup$ @CosmasZachos Well I was hoping it wouldn't require a guess. To put my question more formally, how many of 45 generators are broken and thus result in massive gauge bosons? $\endgroup$
    – user319271
    Jan 8, 2022 at 23:24
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    $\begingroup$ How is this not a simple subtraction (45 - # of generators of the SM gauge group)? $\endgroup$
    – ACuriousMind
    Jan 8, 2022 at 23:26
  • $\begingroup$ @ACuriousMind Would that not just tell you how many gauge bosons in total are predicted above the electroweak scale but below the GUT scale - not the number of massive gauge bosons in that range? $\endgroup$
    – user319271
    Jan 8, 2022 at 23:33
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    $\begingroup$ But what does "GUT scale" mean if not that at that point all gauge symmetries except the usual SM gauge symmetries are broken, and hence massive? $\endgroup$
    – ACuriousMind
    Jan 8, 2022 at 23:38

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Above the SM SSB scale of 1/4 TeV, there are 8+3+1 = 12 massless gauge bosons; so, in your model, all other massive elementary vector bosons correspond to SSBroken generators, which makes them 45-12= 33 in all, all the way to "the" GUT scale, past which there are no more massive vectors. The GUT scale may involve several steps of partial braking to SU(5)$\times$ U(1), etc., but the takeaway point is that a few orders of magnitude below all of these scales, you must have 33 massive vectors.

Is this what you are after?

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  • $\begingroup$ Thank you! I had misunderstood the significance of the energy scales and didn't realise that any above the SM SSB scale must correspond to massive gauge bosons. $\endgroup$
    – user319271
    Jan 9, 2022 at 19:38

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