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Grand unification remains unproven, hot, likely, and persuasive

Quanta Magazine's Natalie Wolchover wrote a status report about grand unification,

Grand Unification Dream Kept at Bay

Physicists have failed to find disintegrating protons, throwing into limbo the beloved theory that the forces of nature were unified at the beginning of time
whose subtitle – which includes words such as "failed" and "limbo" – is much more negative than the available evidence suggests. No smoking gun – such as the proton decay – that would "almost" prove grand unification has been found. But no good reason to stop loving grand unification has been found, either, and that's why lots of particle physicists keep on thinking about grand unified theories.




What is grand unification? The Standard Model has a gauge group, \(SU(3)\times SU(2)\times U(1)\), and the gauge bosons (gluons, photons, W-bosons, Z-bosons) that are almost unavoidable given a gauge group. Also, it works with fermionic fields that transform as a reducible representation – a collection of separate, irreducible representations – under the gauge group.

Some fermionic fields (quarks) are colored i.e. triplets \({\bf 3}\) under the \(SU(3)\) group, others (leptons) are color-neutral i.e. singlets \({\bf 1}\) under it. Some fermionic fields (the left-handed fermions) are doublets \({\bf 2}\) under the electroweak \(SU(2)\) factor, others (the right-handed ones) are singlets \({\bf 1}\). And all of these bits and pieces carry various values of the hypercharge \(Y\) under the \(U(1)\) factor of the Standard Model group.




You may feel it's a contrived setup. The gauge group has three factors. Why not one? And the quarks and leptons fit into several separate representations. Why not one or at least fewer? Well, the questions "why" may be legitimized because theories where the spectrum is simpler – more unified – exist.

The oldest example is the Georgi-Glashow or minimal \(SU(5)\) model. In 1974, when I was finding diapers helpful, they figured out that the Standard Model gauge group may be embedded into a simple group. The fermions may be organized into representations of this group and the Lagrangian may be written in such a way that we may obtain the Standard Model after a symmetry breaking that is very analogous to the electroweak symmetry breaking by the usual Higgs field in the Standard Model.

How does it work?

Well, \(SU(2)\) and \(SU(3)\) may be visualized as a group of \(2\times 2\) or \(3\times 3\) matrices, respectively. You may take a \(2\times 2\) matrix and a \(3\times 3\) matrix and turn them into blocks on the block diagonal of a larger, \(5\times 5\) matrix, cleverly exploiting the ingenious identity \(2+3=5\). The resulting \(5\times 5\) matrix will still be unitary. Moreover, if the blocks' determinants are one, the determinant of the \(5\times 5\) matrix will also be one. So \(SU(2)\times SU(3)\) is isomorphic to a subgroup of \(SU(5)\).

You may embed the hypercharge \(U(1)\) into the \(SU(5)\), too, by allowing matrices whose blocks' determinants aren't one but whose total determinant is one, namely matrices of the form\[

\begin{pmatrix}
f^{+3}&0&0&0&0\\
0&f^{+3}&0&0&0\\
0&0&f^{-2}&0&0\\
0&0&0&f^{-2}&0\\
0&0&0&0&f^{-2}
\end{pmatrix}, \quad f=\exp(i\alpha)

\] Both blocks are unitary matrices with determinants whose absolute value equals one. The blocks belong to \(U(2)\) and \(U(3)\), respectively, but not to \(SU(2)\) and \(SU(3)\). However, the \(5\times 5\) matrix does have the determinant \(f^{3+3-2-2-2}=f^0=1\).

Great. The dimension of \(SU(5)\) is \(5^2-1=24\). So it has 24 generators. The Standard Model gauge group only has \(8+3+1=12\) generators, so there are 12 new generators in \(SU(5)\). These generators – and their corresponding new gauge bosons – basically transform as the \(2\times 3\) complex rectangles under \(SU(2)\times SU(3)\) outside the diagonal. We haven't observed these W-like bosons, so they must be heavy if they exist. Just like the Standard Model's W-bosons are comparably heavy to the Higgs boson, the new bosons – X-bosons and Y-bosons, as they're sometimes called – should be heavy as the new generalized Higgs bosons that break \(SU(5)\).

This mass scale is likely to be huge, \(10^{13}-10^{16}\GeV\) or so, because with this rather extreme choice, one explains the values of the \(SU(3),SU(2),U(1)\) couplings that need to unify into a single \(SU(5)\) gauge coupling at the symmetry breaking scale. Just like the Standard Model uses the Higgs doubles, the Georgi-Glashow theory may use some more complicated representations of \(SU(5)\) with a potential that wants to break the gauge group from \(SU(5)\) to the Standard Model group.

The fermions may be arranged to \(SU(5)\) representations, too. Each generation of leptons and quarks in the Standard Model contains 15 (or 16 if we include a right-handed neutrino) two-component fermionic fields. They get arranged to \({\bf 5}\oplus \bar{\bf 10}\), the sum of the fundamental 5-component representation of \(SU(5)\) and the antisymmetric 2-index tensor with \(5\times 4 / 2\times 1 = 10\) components (complex conjugated, therefore the bar).

There are actually two ways how to clump the numerous lepton-and-quark representations into these 5- and 10-dimensional representations of \(SU(5)\). Georgi and Glashow found the old way where the lepton doublet teams up with the right-handed up-quark singlets to create the 5-dimensional representation, and the remaining fermions belong to the 10-dimensional one.

Surprisingly, people had to wait up to 1982-1984 when the flipped \(SU(5)\) model was found by Stephen Barr, Dimitri Nanopoulos, and with some important additions by Antoniadis, Ellis, and Hagelin, the U.S. presidential candidate for the Maharishi Natural Party. In the flipped \(SU(5)\) models, some pairs of representations are, you know, flipped. So the 5-dimensional representation arises from the lepton doublet and the right-handed up-quark (not down-quark) singlet. The correct hypercharges are obtained as long as you normalize the \(5\times 5\) matrix displayed above differently and correctly.

These original models weren't supersymmetric. You can add supersymmetry and this gives the models some virtues that go beyond the virtues of grand unification and supersymmetry separately. One of them is the gauge coupling unification that seems to work rather well in SUSY GUT. Two straight lines (graphs of \(1/g^2\) as a function of \(E\) or \(\Lambda\)) almost always intersect somewhere in the plane. But a third line only goes through the two lines' intersection if one real parameter – e.g. the slope of the third line – has the right value. In the minimal non-supersymmetric \(SU(5)\) model, the three lines almost intersect at one point but not quite. In the supersymmetric version of this model (and many models that behave almost indistinguishably), the lines exactly intersect, within the known error margins. So at least one real parameter measured experimentally indicates that the three gauge groups want to get unified, and within a supersymmetric theory.

Also, people found other grand unified groups that may do the job. \(SU(5)\) may be represented as a \(10\times 10\) matrix: recall that the complex number \(a+bi\) basically behaves under multiplication just like the \(2\times 2\) matrix \(((a,b),(-b,a))\). So \(SU(5)\) may be embedded into \(SO(10)\), a somewhat larger group. Its additional advantage is that the 5- and 10-dimensional representations of \(SU(5)\) may be embedded into a single irreducible representation of \(SO(10)\), the 16-dimensional spinor – a right-handed neutrino becomes mandatory in \(SO(10)\).

Alternatively, you may pick an even larger simple group, the exceptional group \(E_6\), where the fermions come from the fundamental 27-dimensional representation. There exist various schemes to break the symmetries by Higgs fields in various large representations of the gauge group, or by stringy Wilson lines that don't behave as simple fields in point-like particle-based field theories, and there's a lot of fun to study here.

The most well-known experimental prediction of grand unified theories is the proton decay. The conservation of the lepton and baryon numbers \(L\) and \(B\) no longer holds so with an intermediate X-boson or a Y-boson, you allow some processes in which the proton turns into the (equally charged, equally spinning) positron – plus pure energy (some particles, like photons, whose total conserved charges are zero and the spin is integer).

In Japan, huge tanks with purified water have waited for flashes carrying the gospel of the proton decay for some 20 years. They have found nothing so far. They could have thrown a light-emitting fish into the water tank and receive the Nobel prize but their scientific pride tells them not to use Al Gore's methods to win the prize. ;-) If I trust Wolchover's number which is hopefully up-to-date, the lifetime of the proton is greater than\[

t_{\rm proton} \gt 1.6\times 10^{34}\,{\rm years}.

\] You see that the proton lifetime is a bit longer than the average human's life expectancy. Well, even since the Big Bang, only a very tiny fraction of the protons had enough time to decay (not more than 1 proton in a gram of matter has decayed, roughly speaking) – if they can decay at all. The rather stable proton, as indicated by the experiments, excludes some very specific models based on the grand unification. A picture from the Quanta Magazine:



You see some theories that are either killed (red) or remain viable (green). Supersymmetry tends to prolong the predicted lifetime of the proton which is a good thing. As you can read in Wolchover's article or Brian Greene's The Elegant Universe, Glashow immediately abandoned grand unification in all forms once their first model, minimal \(SU(5)\), was ruled out by the proton decay experiments.

That's a very unwise reaction, I think, because there's really no good reason why exactly the original Georgi-Glashow model should be the right grand unified model in Nature. (In the 1960s, Glashow wasn't lazy and tried a different model to represent the W-bosons before he ended up with the Standard Model representation. In the case of grand unification, he chose the strategy try-fail-and-relax.) Stephen Barr, the original father of the flipped \(SU(5)\) model, describes the so far null results of the search for the proton decay as follows. You're waiting for your wife and she's already some 10 minutes or 1 hour late. Will you already call the company to organize the funeral? Is Glashow organizing the funerals for his wife this quickly?

There are lots of interesting ideas in grand unification – various ways to deal with the proton decay, low neutrino masses, the doublet-triplet splitting problem etc. And grand unified theories may be naturally realized in several major classes of semi-realistic string compactifications which bring some even better (and perhaps more viable) possibilities to the research.

Grand unification is natural in string theory – but string theory may also lead to models (especially various braneworlds) which don't incorporate the idea of grand unification, at least not in any obvious way. Given the fact that I feel almost certain about the validity of string theory, do I believe in grand unification? I probably think it's more likely than not. But I am in no way certain that grand unification is realized in Nature.

However, I am sure that it's wrong to say that the theory was thrown into "limbo". Just look at the charts above taken from the Quanta Magazine. A significant portion (about one-half) of the theory bars remains green. The wife hasn't arrived in the first 1/2 of the possible times when she could have arrived. It's not a terribly strong piece of evidence that she is dead. Maybe it's sensible and ethical to only plan the funeral party once your wife is really falsified and you see her dead body. And of course, Nature may prefer variations of the grand unification that make the proton decay significantly more long-lived, perhaps by some extra discrete symmetries or other structures. Most of these sub-theories' parameter spaces could have been basically untouched by the proton decay experiments.

So be sure that phenomenologists keep on working on grand unified theories. None of them is terribly sure about the right value of the proton lifetime. But the proton's lifetime may be finite. I think that most phenomenologists believe that it is. If neither the baryon number nor the lepton number \(B,L\) is a gauge symmetry, and they're probably not, then the proton may be considered an ultratiny black hole which may evaporate – decay – and only the truly conserved, gauged charges are conserved. So it should be able to decay to the positron plus neutral stuff (such as a photon or two), at least by some (very slowly acting) Planck-scale operators.

It would be extremely sloppy to evaluate the current situation in the proton decay experiment by saying that the proton is certainly stable – or that there can't be any grand unification in Nature.

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