Resources that helped me understand Grand Unified Theories

I recently finished my master’s thesis on dark matter in Grand Unified Theories. Here are some resources that I found particularly helpful.

Group Theoretical Preliminaries

Unification means that we embed the standard model gauge group $G_{SM} \equiv SU(3)_C \times SU(2)_L \times U(1)_Y$ in a larger gauge group $G_{GUT}$.

Thus the first important questions for me were:

• Which groups $G_{GUT}$ can be used?
• How can we embed $G_{SM}$ in $G_{GUT}$ and what does this actually mean?
• How can we describe and compute the breaking of $G_{GUT}$ to $G_{SM}$? (Although there are other methods to break a symmetry, I restricted myself to the usual Higgs mechanism.)

The answer to the first question is, that we use simple groups that have non-self-conjugate representations and that are large enough such that $G_{SM}$ can be embedded. I’ve written a long post about the classification of all simple groups. A representation of a Lie group is called non-self-conjugate if it is not equivalent to the corresponding conjugated representation. This means the group elements are represented by complex matrices $R^a$ and it is impossible to get the conjugated matrices $\overline{R}^a$ from the original matrices using a similarity transformation $U R^a U^\dagger \neq \overline{R}^a$. If there is a map $U R^a U^\dagger= \overline{R}^a  $ the representation is called self-conjugate. In physics, it is conventional to call a self-conjugate representation real and a non-self-conjugate representation complex, although these notions have in mathematics a different meaning. The short version is that GUT models that put the fermions in a self-conjugate representation “will not give the observed chiral structure of weak interactions”. I was quite confused about this for several weeks because I couldn’t find a good explanation. What made it finally click for me was

The answer to the second question is that embedding means that we identify the standard model Cartan generators among the Cartan generators of $G_{GUT}$. An embedding is best described by so-called charge axes. Unfortunately, I don’t know any good resource that explains how we can find an embedding of $G_{SM}$ in a given GUT group and I’ve planned to write about it as soon as I find some time. The standard resource on this kind of thing is

but I wasn’t able to understand his explanations. For the concept of charge axes I found

very helpful.

Unfortunately, I also don’t know any good resource that answers the third question. The breaking of $G_{GUT}$ can be described either using tensor methods or using Dynkin’s methods. In tensor language, a vev is given by a matrix $V$, which is a member of some representation $R$. Then one acts on this matrix with every generator $G$ of the group via the usual Lie algebra product $[G,V]$ (the commutator).

• If the result $M$ of the commutator: $[G,V]=M$ is an element of $R$, i.e. of the same representation as $V$ then the generator $G$ is broken.
• If the result $M$ of the commutator: $[G,V]=M$ is not an element of $R$, then the generator $G$ is unbroken because $G$ annihilates (by definition) the vev $V$. If acting with a generator on an element of some representation $R$ does not yield another element of $R$, the result is by definition zero. This is analogous to the ladder operators for the angular momentum in quantum mechanics. We can apply these ladder operators only until we reach the highest state. If we act with a raising operator on the highest state, we would get a state which is not part of the representation and therefore we say that the state gets annihilated.

The main task is always to write down the correct explicit matrices for the various representations. The rest of the computation, i.e. the commutator and then check if the result is in the same representation as the VEV, can be easily implemented in Mathematica. While tensor methods are easier to explain, I find Dynkin’s methods most of the time easier, especially for a non-trivial group like $SO(10)$ or $E_6$. (However, a colleague of mine recently published a Mathematica package for explicit matrix expressions of $E_6$ representations and this should simplify the computations in $E_6$ models significantly.) I don’t know any good paper or book that explains how symmetry breaking can be described in Dynkin’s framework and therefore I will write about it as soon as I find the time.

For the group-theoretical aspects of GUTs is found the following resources really helpful:

Grand Unified Theories in General

To get an overview I found the famous review

really helpful, although it is quite old and some things are outdated.

I know of three books about GUTs and all of them are quite old, too. Nevertheless, I think that

is still a great book and I found some chapters in

really helpful.

The third GUT book I know of is

but it didn’t help me.

Specific Topics

To understand many subtle effects in GUTs I found the review

incredibly helpful. It helped me finally understand, why the gauge couplings should unify at all and what the GUT scale actually is.

Two other excellent papers that were immensely helpful and elucidate many important GUT problems are

Renormalization Group Running

I’ve written three long posts about the renormalization group running and most papers that I found helpful are linked there. The most helpful paper was

and for threshold effects:

Scalar Masses

In this context, the extended survival hypothesis is extremely important. We actually don’t understand scalar masses, but we need to know them in order to compute the renormalization group equations. Therefore on usually invokes the extended survival hypothesis which states that “at every stage oft he symmetry breaking chain only those scalars are present that develop a vacuum expectation value (VEV) at the current or the subsequent levels of the spontaneous symmetry breaking.” The two standard papers on this topic are

Breaking Chains

The scalar sector is usually too complicated in GUTs and thus one must stick to rules of thumb like the extended survival hypothesis. Another very important such rule is Michel’s conjecture. This conjecture states that states that minima of a Higgs potential correspond to vacuum expectation values that break a given algebra to a maximal subalgebra. This conjecture is explained nicely in

Another important aspect explained there is the necessary condition that in order for a scalar representation to be able to break a group to the subgroups in a given breaking chain that it “must contain singlets with respect to the various subgroups G’, G”…”

Induced Vacuum Expectation Values

A third important observation in the scalar sector is that even if we assume that some scalar field that is allowed to get a VEV does not get a VEV, it is possible that is gets a small induced VEV. This is shown nicely in