MA PH 464 - Group Theory in Physics: Lecture Notes

Subsection 5.6.1 The Standard Model of particle physics

In this section we summarize some representation-theoretic aspects of the Standard Model in particle physics. For more on the mathematical formulation of the Standard Model, see for instance this wikipedia page

It has been emphasized a number of times in this course that states of a quantum mechanical systems can be labeled using irreducible representations of the group of symmetries. In quantum field theory, these states correspond to particles. So in the particle physics, we can label particles in terms of how they transform under the group of symmetries, i.e., in terms of the corresponding representations.

There are two sources of symmetries in the Standard Model of particle physics. One is the Lorentz group (or Poincare group) corresponding to symmetries of spacetime. So we can label particles according to how they transform under the Lorentz group. We have not seen yet the representations of the Lorentz group, but, roughly speaking, they are similar to the representations of \(SO(3)\text{.}\) In particular, this source of symmetry gives rise to the spin of a particle, which tells us in which representation of the Lorentz group it transforms.

Another source of symmetries is the gauge group of the Standard Model, which is a group of transformations of the theory that leaves the physics invariant (such as the \(U(1)\) transformations of the electromagnetic potential in electromagnetism). It turns out that the gauge group of the Standard Model is \(SU(3) \times SU(2) \times U(1)\text{.}\) The \(SU(2) \times U(1)\) factor comes from the electroweak force, while the \(SU(3)\) factor corresponds to the strong force. Therefore, particles are labeled by how they transform under this gauge group: in other words, they are labeled by representations of \(SU(3)\text{,}\) \(SU(2)\) and \(U(1)\text{.}\)

Recall the irreducible representations of \(U(1)\) are all one-dimensional (since \(U(1)\) is abelian), and take the form \(e^{i k \theta}\) with \(\theta \in [0,2 \pi)\) and \(k\) a non-negative integer. So they are indexed by an integer, which we call the charge. It is customary in the Standard Model to specify this charge as being the so-called “weak hypercharge”, which is not quite an integer, but rather a rational number.

So we usually specify a representation of \(SU(3) \times SU(2) \times U(1)\) as \((\mathbf{p}, \mathbf{q})_n\text{,}\) where \(\mathbf{p}\) is the \(p\)-dimensional representation of \(SU(3)\text{,}\) \(\mathbf{q}\) is the \(q\)-dimensional representation of \(SU(2)\text{,}\) and \(n\) is the weak hypercharge. With this notation, we can list the particles of the Standard Model, with their spin (representation of the Lorentz group) and their representation under \(SU(3) \times SU(2) \times U(1)\text{.}\) The following standard table is pretty much taken from wikipedia.

What is pretty cool is that you should now understand what the notation stands for, and be able to read this table!

A few things to note:

• The spin \(1\) particles, which correspond to the gauge bosons (that carry forces), all come in the adjoint representations of \(SU(3)\text{,}\) \(SU(2)\) and \(U(1)\text{.}\) This is a general feature, gauge bosons generally transform according to the adjoint representation.
• The matter particles, i.e. those that compose matter, such as electrons, muons, quarks, etc. all correspond to spin \(1/2\) particles (i.e. they are spinors).
• The Higgs boson is the only spin \(0\) particle.
• However, with the recent discovery of neutrino masses and neutrino oscillation, it is generally believed that one more particle should be added to the Standard Model, which would responsible for giving masses to the neutrinos. This particle is the “right-handed neutrino”, which would transform according to the trivial representation \((\mathbf{1}, \mathbf{1})_0\text{.}\) The existence of the right-handed neutrino has not been confirmed experimentally however.

This is all very cool. But, from a theoretical physics viewpoint, the representations that appear in the Standard Model appear rather random. Why are there particles transforming in the \((\mathbf{3},\mathbf{2})_{1/3}\) representation, but not in, say, the \((\overline{\mathbf{3}}, \mathbf{2})_{1/3}\) representation? Who chose this seemingly random list of representations?

One answer to this question could be that there is no answer, and that this is just what Nature says. That is a fine answer. But often, when there is something that seems rather random or mysterious, this means that we are missing something. And going a little deeper into representation theory, one sees that indeed, these representations are not random. This is the fundamental idea behind Grand Unified Theories (GUTs).

Subsection 5.6.2 \(SU(5)\) GUT

The idea of GUTs is that perhaps the gauge group of the Standard Model is in fact larger than \(SU(3) \times SU(2) \times U(1)\text{,}\) but is somehow broken to \(SU(3) \times SU(2) \times U(1)\) at some higher energy scale. Why would that be a good idea? You will see!

Perhaps the simplest embedding of \(SU(3) \times SU(2) \times U(1)\) as a subgroup of a larger group is

This embedding can be seen in terms of the defining representations. Roughly speaking, one constructs a subgroup of \(5 \times 5\) special unitary matrices by looking at those that are block diagonal, with a \(3 \times 3\) block with determinant one (\(SU(3)\)) and a \(2 \times 2\) block with determinant one (\(SU(2)\)). You also include diagonal matrices of the form \(\text{diag}(a,a,b,b,b)\) with \(a^2 b^3 = 1\text{,}\) which generates a copy of \(U(1)\text{.}\)

Now given irreducible representations of \(SU(5)\text{,}\) one can restrict to the subgroup \(SU(3) \times SU(2) \times U(1) \subset SU(5)\text{.}\) Those will generally now be reducible representations of the subgroup. One can work out how those decompose as direct sums of irreducible representations of the subgroups. Such restriction and decomposition are called branching rules in physics, and some textbooks include tons of table of such branching rules (see for instance Slansky's book Group Theory for Unified Model Building.)

Let us then look at the simplest irreducible representations of \(SU(5)\text{.}\) We know what those are: they are the tensor representations that we constructed earlier. Here are the first few, with how they decompose as direct sums of irreducible representations of the subgroup \(SU(3) \times SU(2) \times U(1) \subset SU(5)\text{:}\)

The last representation is the adjoint representation, whose decomposition always includes the adjoint representation of the subgroup, and extra stuff.

Now the really cool thing here is that all the seemingly random representations corresponding to the matter content of the Standard Model (see Table 5.6.1) are in fact the irreducible representations that appear in the decomposition of the representation \(\overline{\mathbf{5}} \oplus \mathbf{10}\) of \(SU(5)\text{!}\) (Note that the weak hypercharge in the decomposition above has been rescaled by \(2\) in comparison with Table 5.6.1, which is just a choice of convention.) Thus the representations do not appear so random anymore. This is so nice that it cannot simply be a coincidence.

As for the other particles of the Standard Model, the gauge bosons arise from the decomposition of the adjoint \(\mathbf{24}\) of \(SU(5)\text{,}\) and the Higgs arises from the \(\mathbf{5}\) and \(\overline{\mathbf{5}}\text{.}\) The right-handed neutrino, if observed, would need to be added as coming from the trivial representation \(\mathbf{1}\) of \(SU(5).\)

This is what motivated Georgi and Glashow to try to come up with a gauge theory based on \(SU(5)\) that would underly the Standard Model. However, the simplest such models run into trouble; for instance, they predict proton decay with a lifetime that is too short to be consistent with experiments (proton decay has never been observed). But this just means that the simple Georgi-Glashow model cannot be correct. The beauty of the representation-theoretic unification may appear in a different context. It is just too nice to be coincidental.

Subsection 5.6.3 \(SO(10)\) GUT

In fact, one can keep going and look at further embeddings into larger gauge groups. The next level of embedding consists in looking at the subgroup

As you will see, things become even nicer!

As before, we look at irreducible representations of \(SO(10)\text{,}\) and see how their restriction to the subgroup decomposes as a direct sum of irreducible representations of the subgroup. The first few non-trivial representations (we include the spin representation \(\mathbf{16}\)) decompose as follows.

Amazingly, all the matter content, including the right-handed neutrino, arises from the decomposition of a single representation, the spin representation \(\mathbf{16}\) of \(SO(10)\text{!}\) Isn't that amazing? This cannot be a coincidence.

As for the rest of the particles, the gauge bosons as always come from the decomposition of the adjoint \(\mathbf{45}\) of \(SO(10)\text{,}\) while the Higgs comes from the \(\mathbf{10}\) of \(SO(10)\text{.}\)

The \(SO(10)\) GUTs do not suffer from the proton decay problem of \(SU(5)\text{.}\) However, they have other issues, such as the so-called “doublet-triplet splitting problem”.

Nevertheless, the point here is that the representation-theoretic “unification” of the particles of the Standard Model is really quite elegant. What it points to is that the larger groups \(SU(5)\) and/or \(SO(10)\) may perhaps play a role in whatever theory of physics goes beyond the Standard Model. If this is the case, it would be a tremendous achievement of representation theory in particle physics.

Subsection 5.6.4 Beyond \(SO(10)\)

Why stop at \(SO(10)\text{?}\) In fact, there is a natural chain of embeddings:

where \(E_6, E_7, E_8\) are exceptional Lie groups. The fundamental representation of \(E_6\) is \(27\)-dimensional, and it turns out that for the subgroup \(SO(10) \times U(1) \subset E_6\) it decomposes as

In particular, it now gives rise to not only the matter content of the Standard Model (\(\mathbf{16}_1\)), but also the Higgs (\(\mathbf{10}_{-2}\)). All those unify into a single representation of \(E_6\text{!}\) Let me state that one more time: all the matter content of the Standard Model, including the Higgs and the right-handed neutrino, all combine into the fundamental representation of \(E_6\text{.}\) Wow!

As for the gauge bosons, as always, they come from the decomposition of the adjoint of \(E_6\text{.}\)

Why then stop at \(E_6\text{?}\) Well, now there is a good reason to stop. \(E_6\) is the only exceptional group that has complex representations, and those are needed to give rise to chiral fermions as in the Standard Model through standard symmetry breaking by a mechanism of Higgs-type. So standard GUT theories must somehow stop at \(E_6\text{.}\) It is no good to try to construct standard GUT theories with gauge group larger than \(E_6\text{.}\) In any case, all the matter content has already unified into a single representation, so there is no reason really to look at larger gauge groups.

However, in string theory we can go further, and in fact we often must. But in string theory we can break symmetries using other mechanisms, such as string mechanisms, so we are allowed to go further. In fact, some flavours of string theory (such as heterotic string theory) naturally come with an \(E_8\) gauge group. So it seems like this representation-theoretic unification does take place in string theory, at least in some string models. Of course, we do not know yet whether string theory is a valid description of Nature. But it is quite nice that it naturally gives rise to a representation-theoretic unification of the particles of the Standard Model.

In any case, I hope that I have convinced you in this section that representation theory is essential to understand the Standard Model of particle physics. And that it points towards some sort of unification of the matter content into representations of a larger gauge group, which, if physically correct, would be a striking prediction of representation theory.