State-of-the-art non-SUSY \(SO(10)\) grand unified models

Several recent hep-ph papers studied grand unification without supersymmetry, an approach that seems intriguing to a whole group of researchers. Grand unification theories (GUT) work nicely with supersymmetry (SUSY) – it's the supersymmetric version of the \(SU(5)\) and larger models that achieves the accurate unification of gauge couplings of the \(U(1)_Y\), \(SU(2)_W\), and \(SU(3)_c\) factors of the gauge group in the simplest, most natural way.



Super-Kamiokande has looked for a proton decay, a flagship prediction of grand unified theories

However, SUSY hasn't been experimentally proven yet so it's a legitimate possibility – at least from a phenomenologist's viewpoint – that SUSY isn't relevant for any low-energy phenomena (or isn't relevant for Nature at all). The Standard Model is a bit awkward, with its diverse groups and fragmented representations, so one should better unify those structures a little bit.

The people behind the papers below tend to assume that Nature needs a grand unified theory, the symmetry breaking is achieved by the ordinary Higgs mechanism (pure field theory, no stringy Wilson lines etc.), and a precision gauge coupling unification is achieved in some way, too. They typically require a dark matter candidate, too: it's usually an axion. GUT theories may imply lots of unobserved decays of particles (especially the proton decay) so the null results of all these experiments kill many GUT models and constrain the parameters of others.




Yesterday, C.P. Martin of Madrid released the paper

\(SO(10)\) GUTs with large tensor representations on Noncommutative Space-time

which argued that a recently proposed grand unified model is even more natural when combined with spacetime noncommutativity and reinterpreted in terms of a Clifford algebra, namely \(\CC\ell_{10}(\CC)\). I was initially intrigued but I couldn't understand what becomes nicer if the noncommutativity is added to the game (except for some new allowed interactions [e.g. Higgs' interactions] which follow from the broken Lorentz symmetry and I would call this loosening a vice, not a virtue). And much like in the case of most physics papers that place the phrase "Clifford algebra" at a prominent place, I just don't see any beef behind the phrase.

It seems to me that these people love to write the phrase "Clifford algebra" or the Greek letter \(\Gamma\) and that's it. It doesn't seem to me that anything linked to this mathematical structure is "exploited" in any physical way at all. At most, Martin noticed that the representations used in the model he promotes may be found in the decomposition of a bispinor of \(SO(10)\) – I mean the tensor product of a Dirac spinor with itself. That's great, it's a justification making these representations natural – but why is there so much ado about the "Clifford algebra" there? I have no clue.




Martin's paper uses the May 2013 model by Guido Altarelli and Davide Meloni (AM),

A non-supersymmetric \(SO(10)\) grand unified model for all the physics below \(M_{GUT}\),

which I will discuss for a while. At the end, I will mention a grand unified paper released today, too.

AM are extending the Standard Model and a possible strategy to plan a presentation of this model is the bottom-up approach, i.e. one starting at low energies.

First, let us look at the gauge groups and their breaking patterns.

Below the electroweak scale \(v=246\GeV\), we have the gauge group\[

SU(3)_c\times U(1)_{\rm em},

\] the QCD color group and the electromagnetic Abelian group. As we know very well, this is just an unbroken subgroup of a larger group operating above \(v=246\GeV\), namely the Standard Model group\[

G_{SM} = SU(3)_c\times SU(2)_W\times U(1)_Y

\] which includes the electroweak \(SU(2)\) and the hypercharge \(Y\). The breaking from the Standard Model group to the QCD+electromagnetism group is achieved by a Higgs doublet. This statement is pretty much an experimental fact by now, a Nobel-prize-winning one. In this grand unified model building, we ultimately want an \(SO(10)\) symmetry at really high scales so all fields, including the Higgs fields, have to be embedded into full representations of \(SO(10)\).

The simplest representation of \(SO(10)\) that provides us with the Nobel-prize-winning Higgs doublets is a \({\bf 10}\), the fundamental vector of \(SO(10)\), decomposing as\[

{\bf 10}_H \to ({\bf 1},{\bf 2},{\bf 2}) \oplus ({\bf 6},{\bf 1},{\bf 1})

\] under the Standard Model group. The symbol for the singlets and doublets is obvious; the 6-dimensional representation is the symmetric tensor of \(SU(3)\) (note that due to the complexity of the group, the "trace" cannot be separated like in \(SO(3)\)).

Great. Now we're above \(v=246\GeV\) and continue to raise the energy. AM are telling us that there is an intermediate scale, something like \(10^{10}\GeV\), where a larger group gets restored. The intermediate energy scale may be adjusted so that the gauge coupling unification is restored with the same precision we know from the simple SUSY GUTs. This is a bit ugly – we have to adjust one real parameter (the energy scale) to guarantee one real condition (the second coupling and the third coupling unify with the first one at the same energy scale) – but there's no guarantee that all types of unification that Nature manages are "breathtakingly beautiful and rigid".

Fine. So above the intermediate scale, the gauge group becomes larger, namely\[

G_{PS} = SU(4)_c \times SU(2)_L \times SU(2)_R.

\] We have enhanced the color group to \(SU(4)\), i.e. added the fourth basic color (it's like going from RGB to CMYK, if you like silly analogies). The electroweak \(SU(2)_W\) group was kept and just renamed to \(SU(2)_L\) because it couples to the left-handed fermions. And the hypercharge was extended to an \(SU(2)_R\) group where "R" stands for "right".

The group above has been known as the Pati-Salam group. It's a "partial unification" group because the number of factors is the same as it is in the Standard Model but at least the smaller two factors became isomorphic to one another and may be related by a discrete symmetry.

This Pati-Salam group is broken to the Standard Model by some Higgs fields analogous to those in the Standard Model. The quartic and quadratic interactions are analogous, just the group theory is a bit more complicated. The required Higgses transform as \[

\overline{\bf 126}.

\] What is this representation? Well, consider the antisymmetric tensor field with five indices, \({\bf 5}\wedge {\bf 5}\wedge {\bf 5}\wedge {\bf 5}\wedge {\bf 5}\), if you wish. Its dimension is \[

\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1} = 252

\] but one may also "Hodge-dualize" this tensor field with a 10-index "epsilon symbol" which allows us to split this 252-dimensional representation to two 126-dimensional ones, the self-dual and the anti-self-dual ones, which are complex conjugate to each other.

This tensor field \(T_{abcde}\) may remember which 5 of the 10 (complexified) directions in the \(SO(10)\) vector are the "holomorphic" directions of the fundamental 5-dimensional representation of \(SU(5)\) and which of them are the antifundamental ones. Consequently, this field might break \(SO(10)\) to \(SU(5)\).

However, when we start with the Pati-Salam group instead of \(SO(10)\), it's also able to break the group to the Standard Model group. The group theory is not too complicated and we actually need a Higgs field in another representation, \({\bf 45}\), as well. This is the antisymmetric tensor with two indices, \({\bf 10}\wedge {\bf 10}\), if you wish, with dimension \(10\times 9/2\times 1 = 45\). Such a field simply picks a preferred complex 2-plane inside the 4-dimensional space symmetric under \(SO(4)=SU(2)\times SU(2)\) and this complex 2-plane is the one that preserves the \(SU(2)_L\) symmetry.

Finally, there's another scale, the highest one: the GUT scale. Above the scale, \(SO(10)\) is restored. Beneath the scale, it is broken to the Pati-Salam group. The breaking is done by \({\bf 210}_H\), an antisymmetric tensor with four indices whose dimension is understandably\[

\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1} = 210.

\] Now, there is no Hodge duality. Note that all these antisymmetric tensor products of copies of \({\bf 10}\) may be found in the tensor product of two spinors, \({\bf 16}\otimes {\bf 16}\), of \(SO(10)\), except that sometimes we need the same chirality and sometimes we need the opposite chirality of the two 16-dimensional spinors. That's also the case of the 120-dimensional antisymmetric tensor with three indices that is not used in this construction.

It's easy to understand why \(SO(10)\) may be broken to the Pati-Salam group by the antisymmetric tensor with four indices. It defines a volume form for 4-dimensional submanifolds of the 10-dimensional space so it splits "ten" to "four plus six" and \(SO(6)=SU(4)\) while \(SO(4)=SU(2)\times SU(2)\) which produce all the factors of the Pati-Salam group.

So the representations used in the AM paper have dimensions \(10,16,45,126,210\) where \(16\) is the usual "single generation of fermions" including an active right-handed neutrino. The 120-dimensional representation isn't used in this particular grand unified model (it is used in other \(SO(10)\) models, however) while the 45-dimensional representation is used and produces an axion, a dark matter candidate. If dark matter is composed of axions only, the underground experiments will probably find nothing, at least for quite some time.

They discuss many features of the model and show it is viable. They also identify 15 high-scale parameters that are pretty much manifested as 15 parameters of the Standard Model. (My understanding is that the Higgs quartic couplings etc. are assumed to be irrelevant in all this fitting; they don't seem to know anything about their values.) Their best fit (see an appendix in that paper) leaves no freedom and you may see that the Standard Model with the required values of the parameters may be realized within their model. Some of the "best fit" values are extremely unnatural (tiny) numbers, highlighting the fact that grand unification without SUSY seems more awkward than one with SUSY.

However, it seems that aside from the gauge coupling unification and a dark matter candidate, they also manage to suppress all the dangerous decays of various particles that rule out many other generic GUT models.

Today, a Spanish-Portuguese-Czech collaboration (Carolina Arbeláez, Martin Hirsch, Michal Malinský, Jorge C. Romão: AHMR; I have never met Michal Malinský, I believe) released another non-supersymmetric \(SO(10)\) grand unified paper,

LHC-scale left-right symmetry and unification

AHMR study not just one grand unified model but a big class of models "similar" to the AM model above. They have some intermediate left-right (Pati-Salam) gauge group and the condition of gauge coupling unification is imposed.

AM adjusted the Pati-Salam scale to be an intermediate one; it was needed for gauge coupling unification. AHMR do something else. They show that in many models, the Pati-Salam scale may be brought down, close to the LHC scale so that there's no continuous adjustment needed to restore the gauge coupling unification.

However, what's needed as an extra price to pay are new multiplets of matter – which should or might be accessible by the LHC. They seem to conclude that to guarantee a long enough lifetime of the proton that is compatible with the current lower bound of order \(10^{34}\,{\rm years}\), or at least to respect this lower bound "safely", most of their models need to add new colored states at the LHC scale. That would be very exciting, of course (although arguably less exciting than an experimental discovery of SUSY), if such new GUT-predicted states were found.