### The Algebra of Grand Unified Theories III

#### Posted by John Baez

If you’re a phenomenologist, here’s a question: is the nonsupersymmetric $SO(10)$ GUT still experimentally viable? I get the impression that it *is*.

If you’re a follower of Husserl or Heidegger, here’s a word of reassurance: I don’t mean *that kind* of ‘phenomenologist’. I mean an particle physicist who is not an experimentalist, but is nonetheless deeply concerned with experimental data.

And if you’re a physicist of a more theoretical or mathematical sort — the sort perhaps more likely to frequent the $n$-Café — please ask your phenomenologist pals to stop by and pass on their wisdom!

Let me start with a few words for nonexperts before asking my question to the experts.

In trying to go beyond the Standard Model of particle physics, grand unified theories try to make sense of the messy pack of particles we see and find elegant patterns lurking in the noise. The $SO(10)$ theory does an astoundingly good job of this. There are 32 particles and antiparticles in each generation of the Standard Model, if we include the elusive right-handed neutrino and its antiparticle — and these fit neatly in the spinor representation of $SO(10)$’s double cover $Spin(10)$ on the exterior algebra $\Lambda \C^5$, which has dimension $2^5 = 32$.

This is sort of amazing. In particular, the crazy-looking hypercharges of all 32 particles must be *exactly what we see* for this theory to make sense.

Let me explain this, in terms even a Ph.D. mathematician can understand.

Irreducible representations of the circle group, U$(1)$, are classified by integers. Physicists often call this integer the ‘charge’, because this is the modern explanation of why electric charge comes in discrete units. Any irreducible representation of the Standard Model group U$(1) \times SU(2) \times SU(3)$ thus has an associated ‘charge’. However, the U$(1)$ here is not associated to ordinary electric charge, so physicists use a different term in this case: ‘hypercharge’. They also divide this integer by 3, just to drive mathematicians crazy, but let’s not do that today.

Here are the 16 left-handed particles in the first generation of the Standard Model, and their hypercharges:

- left-handed neutrino: -3
- left-handed electron: -3
- left-handed up quarks (red, green and blue): 1
- left-handed down quarks (red, green and blue): 1
- left-handed antineutrino: 0
- left-handed positron: 6
- left-handed up antiquarks (antired, antigreen and antiblue): -4
- left-handed down antiquarks (antired, antigreen and antiblue): 2

It might seem hopeless to explain this list of numbers in an elegant way. But in fact $Spin(10)$ has a U$(1)$ subgroup such that when we restrict the spinor representation to this subgroup, it splits up as a direct sum of irreducible representations with precisely these charges!

Of course, there is a vast amount of particle physics data that a successful grand unified theory must fit. I’ve just skimmed off a tiny bit of the cream, the numerological stuff that even a Ph.D. mathematician can understand. But in fact, the $SO(10)$ theory does pretty well on quite a wide front.

Could it be true?

I want to know the current conventional wisdom on this question! John Huerta and I are writing an expository paper for mathematicians on the algebra of grand unified theories. Our paper doesn’t touch serious issues like proton decay or running coupling constants… but still, we want to know: *is $SO(10)$ without supersymmetry still a contender?*

I get the impression that this theory *is* still viable with a number of symmetry-breaking schemes, including ones where at intermediate energies the unbroken symmetry group is the Pati–Salam group $SU(2) \times SU(2) \times SU(4)$. This makes me happy.

The following review article compares 12 symmetry-breaking schemes where the symmetry group breaks down from $SO(10)$ (or really its double cover) to U$(1) \times SU(2) \times SU(3)$ through two intermediate steps:

- Stefano Bertolini, Luca Di Luzio and Michal Malinsky, Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal.

They use a notation for such schemes introduced here:

- N.G. Deshpande, E. Keith and Palash B. Pal, Implications of LEP results for SO(10) grand unification with two intermediate stages.

Is it true that all schemes with $SU(5)$ as an intermediate step are ruled out, while many models that pass through $SU(2) \times SU(2) \times SU(4)$ are viable?

Also: what might the Large Hadron Collider tell us about these issues? Suppose we find a Standard Model Higgs… then what else might we see, that would help confirm or disconfirm the $SO(10)$ GUT?

## right link

The right link seems to be Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal.