LHC: discovering grand unification

...aside from supersymmetry...

Apologies if the title was too dramatic. Supersymmetry and its possible looming discovery at the LHC has been discussed many times. But let's look at a different portion of modern model building in particle physics, Grand Unification. Grand Unified Theories (GUTs) are those that embed the Standard Model group\[

SU(3)_c\times SU(2)_W\times U(1)_Y

\] into a larger group, typically "simpler" (having fewer factors), ideally "simple" group in the technical sense (one factor). If it is possible, and it possible, the advantage is that the quarks and leptons may arise in a smaller number of multiplets (representations of the gauge group) and the lower number of factors in the gauge group implies a smaller number of adjustable coupling constants. So the GUTs are more constrained.

If they're verifiably right, they're more beautiful. But because they're more predictive, one may also be worried that they're less flexible and therefore less resilient towards falsification – a characteristic you may consider good or bad. However, the flexibility may be restored by adding various stuff and potentials at the GUT scale so the "qualitative difference" is somewhat debatable. At any rate, from a purely theoretical or aesthetic top-down perspective, it's rather natural to expect that Nature may want to unify the forces near the fundamental scale if She can.

And yes, She can. Many grand unified theories are naturally compatible with everything we may observe at low energies of doable experiments.




Grand unification has been discussed many times on this blog. In this text, I want to be somewhat more specific about the possible choices of the gauge groups and representations and possible particles that could show up at the accelerators (which is generally unlikely as the GUTs are associated with an insanely high energy scale, the GUT scale, which is not far from the Planck scale linked to quantum gravity).

\(SU(5)\): Georgi-Glashow model

The simplest gauge group we may pick is \(SU(5)\). Note that the rank – the maximum number of mutually commuting independent \(U(1)\) subgroups – is equal to four, much like for the Standard Model group. The Georgi-Glashow gauge group has dimension equal to \(d=5^2-1=24\). The adjoint representation, which is therefore 24-dimensional, may be decomposed into representations of the Standard Model subgroup:\[

24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}

\] In the parentheses, the first number represents the representation in which the piece transforms under the \(SU(3)_c\) colorful group of QCD. It may be a singlet (doesn't transform at all), a triplet (like quarks), antitriplet (like antiquarks), or octet (like gluons, the adjoint representation). More complicated representations may appear, too.

The second number in the parentheses is the representation under the \(SU(2)_W\) electroweak group. Note that the doublet \({\bf 2}\) is equivalent to its complex conjugate representation in this case – because the representation is pseudoreal (there's only one inequivalent spinor representation in 3 dimensions). The adjoint representation is 3-dimensional. The subscripts indicate the hypercharge i.e. charge under the \(U(1)_Y\) factor of the Standard Model gauge group.

You see that on the right hand side of the equation above, you find the adjoint representations of \(SU(3)_c\), \(SU(2)_W\), and \(U(1)_Y\). There are two new pieces that are complex conjugate to each other – they come with new gauge bosons that should however be insanely heavy (not accessible at the LHC) if the proton is supposed to preserve its longevity we know and love. The new bosons transform as color triplets as well as electroweak doublets. The latter fact implies that they have charges like an exotic quark doublet. We sometimes talk about new X-bosons and Y-bosons.

It's equally interesting to find out whether the leptons and quarks may be rearranged into representations of \(SU(5)\). The answer is YES and this fact is nontrivial. Let's look at all left-handed two-component complex spinor fields that are associated with a single generation of quarks and leptons (the whole field content or particle content is tripled at the end because there are three generations; and the Hermitian conjugate fields are added as well).

We find 15 such spinor fields in total: 2 (up, down: organized as an electroweak doublet) left-handed quarks, each in three colors (6 in total), 2 left-handed antiquarks (separate electroweak singlets), each in 3 colors (their Hermitian conjugates are right-handed quarks; again 6), left-handed electron and left-handed neutrino (they form a doublet), and a left-handed positron (an electroweak singlet). In total, \(6+6+2+1=15\). We may also add the left-handed antineutrino (so far unobserved and insanely feebly interacting if it exists) to raise fifteen to sixteen.

These 15 spinor fields are reorganized as the following representation of \(SU(5)\):\[

\mathbf{\bar{5}}\oplus\mathbf{10}\oplus\mathbf{1}

\] The decomposition of these pieces under the Standard Model group is\[

\eq{
\bar{5}&\rightarrow (\bar{3},1)_{\frac{1}{3}}\oplus (1,2)_{-\frac{1}{2}}\\
10&\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{-\frac{2}{3}}\oplus (1,1)_1\\
1&\rightarrow (1,1)_0
}

\] The last line, the left-handed antineutrino (whose Hermitian conjugate is the right-handed neutrino), is optional or uncertain, as we have mentioned. The first line contains the lepton doublet and the \(d\)-antiquark singlet (\(d^c\) and \(\ell\), to use some symbols) on the right hand side. The second line has \(q\), \(u^c\), and \(e^c\). The \(c\) superscript refers to the antiparticles (it stands for "charge conjugation"). Let me emphasize in advance that the clumping of the particles into these larger representations is inequivalent in flipped \(SU(5)\) models.

The \(SU(5)\) grand unified group may be broken down to the Standard Model if you realize that the Standard Model group is exactly the subgroup that commutes with the generator \(U(1)_Y\), the hypercharge. Up to an overall normalization (that sometimes omits the factor of \(1/2\)), it is given by the following traceless \(5\times 5\) matrix:\[

\frac{Y}{2} = \left(\begin{array}{rrr}
-\frac 13&0&0&0&0\\
0&-\frac 13&0&0&0\\
0&0&-\frac 13&0&0\\
0&0&0&+\frac 12&0\\
0&0&0&0&+\frac 12
\end{array}\right)

\] Equivalently, it is the subgroup that keeps a \(Y\)-like vev of a new GUT Higgs field transforming in \({\bf 24}\), the adjoint of \(SU(5)\), invariant. One may also achieve the breaking by Wilson lines (monodromies around non-contractible loops, to be more accurate) in stringy models with extra dimensions and by other means.

The minimum grand unified model predicted a rather slow but not too slow proton decay. Within a year or so, the prediction was ruled out. The simplest grand unified theory has been falsified. The broader concept hasn't been falsified and it has way too many positive features so that we shouldn't think we will kill it too quickly.

Flipped \(SU(5)\) i.e. \(SU(5)\times U(1)\)

The flipped \(SU(5)\) models are the "most comparable ones" to the Georgi-Glashow model. They're associated with the names Dimitri Nanopoulos, Stephen Barr, Ignatios Antoniadis, John Ellis, and John Hagelin and their research in the early 1980s. The realization of this previously overlooked possibility was linked to some string theory research in the 1980s and the 1990s: string theory arguably makes the flipped \(SU(5)\) models more natural than the Georgi-Glashow model.

Note that Sheldon Glashow had already omitted a \(U(1)\) factor once when he tried to construct a purely \(SU(2)\) theory of the electroweak force so it wouldn't be shocking if he has displayed the same excessive fanaticism for minimalism together with Howard Georgi again. ;-)

You could think that that \(SU(5)\times U(1)\) must be just a complicated extension of the original \(SU(5)\) in which we add possibilities and allow things to be deformed. You could think that the Georgi-Glashow model is a special case of the flipped \(SU(5)\) models. But you would be wrong. The flipped \(SU(5)\) models are qualitatively different from their unflipped cousins – and they're "equally robust", in some sense. Their arrangement of quarks and leptons into the \(SU(5)\) multiplets is permuted – therefore "flipped" – relatively to the Georgi-Glashow model.

To mention a visible example of the differences, note that the right-handed neutrino \(SU(5)\) singlet was optional in the Georgi-Glashow model. In the flipped \(SU(5)\) models you also have to use \(\bar{5}\oplus 10 \oplus 1\) but the last, singlet piece isn't optional at all. For a simple reason: it is not a neutrino. It is actually the left-handed positron! We surely need it. So the actual fermionic content we have to add in the flipped \(SU(5)\) models comes in the representation\[

\bar{5}_{-3}+10_{1}+1_{5}

\] where the subscripts indicate the charges under the \(U(1)\) factor of the flipped \(SU(5)\) gauge group. These subscripts are extremely natural from the viewpoint of an \(SO(10)\) or \(Spin(10)\) group in which \(SU(5)\times U(1)\) may be embedded. In that group, the three pieces combine to a 16-dimensional complex spinor representation; the \(U(1)\) charge is proportional to the sum of all the five weights. It is fair to say that the flipped \(SU(5)\) models become more natural than the ordinary Georgi-Glashow \(SU(5)\) if you look "beyond \(SU(5)\)" e.g. at the \(SO(10)\) models.

The decomposition of the relevant flipped \(SU(5)\) representations under the Standard Model group is\[

\eq{
\bar{5}_{-3}&\rightarrow (\bar{3},1)_{-\frac{2}{3}}\oplus (1,2)_{-\frac{1}{2}}\\
10_{1}&\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{\frac{1}{3}}\oplus (1,1)_0\\
1_{5}&\rightarrow (1,1)_1\\
24_0&\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{\frac{1}{6}}\oplus (\bar{3},2)_{-\frac{1}{6}}
}

\] The subscripts on the right hand side are the values of the hypercharge. The last line is the adjoint representation again; there is an extra generator of the \(U(1)\) aside from this line, too. The first line describes the up-antiquark left-handed singlet and the lepton left-handed doublet. The second line describes the quark left-handed doublet, the down-antiquark left-handed singlet; and the left-handed antineutrino (unobserved). The third line is the left-handed positron singlet and can't be omitted.

Not too grand unified groups

Let me mention that there are other GUT-like gauge groups that are far from simple and that have been proposed with various justifications. The electroweak gauge group only treats the left-handed particles as doublets. One may add another \(SU(2)\) under which the right-handed particles are doublets. These left-right models have the gauge group \(SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}\).

The Pati-Salam model has the gauge group \(SU(4)\times SU(2)\times SU(2)\). The two \(SU(2)\) factors are analogous to those in the previous paragraph. However, \(SU(3)_c\) is embedded into an \(SU(4)\) where the "quark of the fourth color" is identified with a lepton.

Also, \(SO(10)\) models may be flipped to \(SO(10)\times U(1)\). The electroweak \(SU(2)_W\) may be extended to another \(SU(3)\) in the 331 models (Paul Frampton was a pioneer). \(SU(6)\) models have been constructed, too.

Trinification has a gauge group composed of three equal pieces, \(SU(3)\times SU(3)\times SU(3)\).

Truly large and simple GUT groups

However, the truly motivated "master gauge groups" are \(SO(10)\) and \(E_6\), an exceptional group. Note that the \(E_6\) group is the only exceptional group that has complex representations at all – something we need to treat left-handed and right-handed fermions separately.

This \(E_6\) is naturally embedded into the \(E_8\) that arises in the heterotic string and heterotic M-theory and it contains pretty much all the GUT groups above as a subgroup, including the group of trinification. The decompositions of all the representations under various subgroups are interesting and you should try to find all of them if you haven't done so. Also, \(SO(10)\times U(1)\) – where the \(U(1)\) may participate or not, depending on whether we have flipped models – may be embedded into \(E_6\).

The fermionic content of \(SO(10)\) models combines into the 16-dimensional spinor representation of \(Spin(10)\) I have already mentioned. But it's even more interesting to see what happens for \(E_6\) because the minimum nontrivial representation is the 27-dimensional fundamental representation, \({\bf 27}\). This inevitably contains some new spinor fields aside from the quark and leptons we know (and different from the right-handed neutrino, too)!

The decomposition of this representation under \(SO(10)\times U(1)\) is\[

27\rightarrow 16_{1}+10_{-2}+1_4.

\] Note that the trace of this \(U(1)\) in this 27-dimensional representation is \(16\times 1-10\times 2+ 1\times 4 = 0\). It has to vanish because it's the trace of a generic generator of a non-Abelian group, \(E_6\).

Now, the \(U(1)\) labels are funny. You may notice that only the 16-dimensional spinor of quarks and leptons has an odd label. If you correlated this \(U(1)\) charge with the R-parity in supersymmetric models (instead of using \(B-L\), and this is my idea), you could say that we will easily observe fermions from the 16-dimensional representation (we do) but bosons from the remaining, 10- and 1-dimensional representations!

So it's funny to ask what are the Standard Model charges of the additional representations. If you use the flipped \(SU(5)\) models, you will find out that \({\bf 10}\) decomposes into \({\bf 5}\oplus \bar{\bf 5}\) and these pieces contain both Higgs doublets of the MSSM (but this field content is tripled because there's a pair of Higgs doublets in each of the three generations!) plus a new electroweak singlet squark with the charges of the down-squark.

Symmetry has been a powerful tool in our successful efforts to understand Nature at least for a century. Supersymmetry tells us that we may expect superpartners of all known particles – although some of them may be much less accessible than others. But we could actually find traces of another, ordinary "bosonic" symmetry, by finding the (broken) symmetry partners of the known quarks and leptons.

For example, if the LHC ultimately started to find the new Higgs doublets predicted above as well as an exotic quark or three, it would be a huge hint of an underlying greater symmetry operating in Nature. Many of these models are heavily constrained and virtually all models with new light charged particles destroy the simplest "intriguing accident" of gauge coupling unification. But I view the simplest match for the MSSM gauge coupling unification to be just a 2-sigma intriguing bump. It may still be a misleading guide and there may be many more fascinating things we may discover if we sacrifice this one. Of course, it's up to us what we believe but on a sunny day in the future, Nature may very well force us to sacrifice the belief in the simplest scenario of gauge coupling unification.

So far, no official discovery of physics beyond the Standard Model has been announced by the LHC; just to be sure, the Higgs boson is almost as old physics as Peter Higgs himself. But this situation isn't guaranteed to last forever. It isn't even guaranteed to survive the Moriond 2013 conference in early March.

Stay tuned.