This is a post to describe how we construct the standard model lagrangian and generate the masses for gauge bosons and fermions. I will assume basic familiarity with QFT, lagrangians, gauge transformations and gauge groups (like U(1), SU(2)).

Recall that the mass term of a scalar field φ is simply proportional to φ². Vector fields have similar mass terms. For fermions, the mass term is written in terms of their left handed part and right handed part. All of these forms are given in the picture below. These facts should be rememebered.

Also, recall that in gauge theories, we can’t add explicit mass terms in the lagrangian because these terms will violate the gauge invariance of the action (actions should be gauge invariant).

We start by describing what Higgs mechanism is. Higgs mechanism is a mechanism where we generate mass terms for gauge bosons without explicitly adding mass terms for the gauge bosons in the lagrangian. We do this by introducing a scalar field (called the Higgs field) in the lagrangian and giving it a vacuum expectation value (vev). It means that if we calculate the average value of this field in the vacuum state, it won’t be zero. Instead, it is a number that we call v. We set it up such that the minimum is at some nonzero φ.

Please note that only scalar fields can have non-zero vacuum expectation values. If any other field (e.g. a vector field) has a non zero vacuum expectation value, it won’t be invariant under Lorentz transformations, which would be a problem.

We start with the case when the gauge group is Abelian, say U(1). In this case, we have some gauge transformations for φ and the gauge field that are shown in the picture below.

Since we have gauge freedom, we can fix a gauge before proceeding. Since φ is a complex field, we can spit it into two fields by writing it into the polar form of complex numbers. One field measures φ’s devitation from its vev and the other field is a phase. We fix the gauge by getting rid of the phase by a gauge transformation. This gauge fixing is called the unitary gauge.

Now, the term in the lagrangian that represents the interaction between φ and A_μ is the modulus squared of the covariant derivative of φ. We calculate this term and it gives us the mass terms of the A_μ boson.

So, we saw that the A_μ boson has aquired a mass and the G field has left the picture. We can think of this scenario as the massless A_μ eating the G field and becoming massive. This picture is motivated by the fact that the massless gauge bosons have two degrees of freedom (e.g. a photon has two possible polarizations which are perpendicular to the direction of propagation) but a massive boson has three degrees of freedom (DOF). In addition, a real scalar field like G has a single DOF. So, a massless A_μ ate G to aquire an extra DOF and become massive.

Now, we consider a case where the gauge group is non-abelian, say SU(2). The example of SU(2) is important because we will need this example in building the standard model lagrangian.

Before proceeding, let’s recall that fields can live in different representations of a gauge group. A field living in a representation of dimension N just means that under the gauge transformations, this field transforms as an object with N components. If N equals the dimension of the group, we say that this field lives in the adjoint representation. For SU(n) groups, the dimension is n²-1. For SU(n) groups, if N=n, we say that it is a fundamental representation. If a field doesn’t transform under a gauge group, we say that this field lives in a singlet representation of this group.

We also recall that for a gauge group, gauge bosons live in the adjoint representation and fermions (and scalars) live in either the fundamental or singlet representation of the group.

Now, let’s come back to the example of SU(2) gauge group. The adjoint representation of SU(2) is 3-dimensional and since gauge bosons live in the adjoint representation, there are 3 gauge bosons. The Higgs field φ lives in the fundamental representation of SU(2) (which is also called a doublet) which is 2-dimensional. We again do the process that we did in the Abelian case. We first choose a vev as follows.

Now, we need to choose the unitary gauge. Before proceeding, we ask the question that can we even choose a unitary gauge? The answer is yes the process is as follows.

Then we calculate the modulus squared of the covariant derivative of the φ field and we again get the mass terms for all the three gauge bosons. Please note that the mass terms could be obtained even if we calculated the modulus squared of the covariant derivative of the vev of φ instead of the full φ. This fact will be helpful in the future.

There is another very important observation that we have to make here. All the gauge bosons got a mass in the SU(2) case because none of the generators of SU(2) annihilated the vev, (i.e. when the generators are multiplied with the vev, they don’t give you zero). If some generator had annihilated the vev, the boson corresponding to that generator won’t have aquired a mass term. This fact will be important later on.

Now, we add fermions into the game and for simplicity, lets start with a single fermion. We recall that fermions have a left handed part ψ_L and a right handed part ψ_R . We again start with the Abelian gauge group U(1). Under this gauge group, the fermions transform by multiplication of a phase (because this is what a U(1) group will do). We choose these phases such that a Dirac mass term (which is proportional to ψ_Lψ_R) is forbidden, so that the mass of the fermion comes only from the Higgs field (recall that any term in the lagrangian should be gauge invariant). The interaction of the fermions and the Higgs field comes from Yukawa terms. These terms include a factor of the conjugate of ψ_L , a factor of ψ_R (or vice versa), and a factor of φ and thus, we want to allow these terms. These requirements dictate the phases by which the fermions and the Higgs field transforms.

Then, we can again choose the unitary gauge and write down the lagrangian with the Yukawa terms included. This gives us the Dirac mass terms for the fermion.

Now, we consider the case of a non-Abelian gauge group i.e. SU(2). Recall that the Higgs field φ lives in a doublet of SU(2). Now, the requirement that Dirac terms are forbidden but Yukawa terms are allowed can’t be satisfied if both ψ_L and ψ_R are doublets. One of them has to be a singlet. We choose ψ_R to be a singlet and ψ_L to be a doublet. Moreover, the transformations of Higgs field, ψ_L, and ψ_R are shown in the picture below. The Dirac mass terms for the fermions are expected to come from the vev of the Higgs field in the Yukawa terms. These terms are also shown in the picture below.

As we see, we are able to generate the mass term for one of the fermions but not for the other fermion. To generate the other mass term, we need to include another kind of Yukawa term which is still invariant under SU(2) gauge transformations.

Now, we will start to construct the standard model lagrangian. The gauge group of the standard model consists of three gauge groups called SU(3), SU(2) and U(1). The SU(3) part describes strong interactions and the SU(2) and U(1) groups collectively describe weak interactions and electromagentism (we will see how this happens). The U(1) group is also called the hypercharge group.

What particles do we have in the standard mode? We have right handed leptons i.e. electrons, muons and tau (we will denote them as l), right handed +2/3 quarks, right handed -1/3 quarks. The +2/3 quarks are quarks with electrical charge 2/3 -in units of the charge of the proton- which include up, charm and top quarks. We will denote all these quarks as u. Whereas, the -1/3 quarks are down, strange and bottom quarks and we will denote all these quarks as d.

We also have left handed lepton doublets and quark doublets. In the lepton doublet, we have a left handed neutrino and the corresponding left handed lepton (we will denote them as e). In the quark doublets, we have a left handed u quark and the corresponding left handed d quark (i.e. up with down, charm with strange and top with bottom).

Lastly, we have the gauge bosons for SU(3) group (they are called gluons, denoted as G^a_μ and they are 8 in number and thus, a=1,…,8), gauge bosons for SU(2) group (that we denote as Wⁱ_μ and they are 3 in number and thus, i=1,2,3) and one gauge boson for U(1) (that we denote as B_μ ). The Higgs field is again a doublet of SU(2) but a singlet of SU(3).

How do all these fields transform under the hypercharge U(1) group? The fields transform by a phase e^iYα(x)/2 where Y is called the hypercharge of the field. I will call these transformations U(1) transformations. These hypercharges are choosen very carefully to provide the right theory. Y is related to the electrical and another quantity called T₃ (also called isospin). If a field is an SU(2) singlet, then T₃=0 but if a field is in an SU(2) doublet, then T₃=1/2 if it is in the upper entry and T₃=-1/2 if it is in the lower entry. For example, left handed neutrinos have T₃=1/2, left handed leptons have T₃=-1/2 and right handed leptons are singlets and thus, they have T₃=0. Now, Y=2(Q-T₃) where Q is the electrical charge. This formula is inspired by a similar formula for strong interactions called the Gell-Mann Nishijima formula.

Now we have the way to calculate the hypercharges. We can now tabulate the hypercharges and the representations in which all the fields live. We haven’t explained the hypercharge of Higgs field and its conjugate. We will show how to calculate them below. The table is given as follows.

Now, we recall that to get the mass terms of the gauge bosons, we needed to calculate the modulus squared of the covariant derivative of the vev of Higgs field and to calculate the mass terms of the fermions, we need Yukawa terms. Therefore, we will first calculate the covariant derivatives of all the fields Remember that in the covariant derivative of a field, we have a term for every gauge group, unless that field is a singlet of that gauge group. The derivatives are gives in the picture below.

We will now determine which Yukawa terms are allowed. Here, the main tool is the fact that the lagrangian should be gauge invariant and thus, every term in the lagrangian should have an overall hypercharge of zero (otherwise, that term would get multiplied by a phase under U(1) transformations). We assume that Y(Φ)=1 (and justify it shorlty) and then, the allowed terms are as follows.

We see that here we have assumed that Y of φ is 1 and Y of its conjugate is -1. We can see that if we choose Y(φ) to be anything but 1 or -1, then no Yukawa terms would be allowed. But what about Y(φ)=-1? Recall that when we were demonstrating how to generate masses for SU(2) fermions in the Higgs mechanism, the Yukawa term with φ in it generated the mass term for ψ₂ and not for ψ₁. The mass term for ψ₁ was generated by the Yukawa term that had the conjugate of φ in it. In the lepton doublet, ψ₁ is neutrino and ψ₂ is the left handed lepton. We don’t want a mass term for neutrino but only for the lepton. Thus, we want the Yukawa term with φ in it. This forces us to take Y(φ)=1 and not -1 because taking Y(φ)=-1 will forbid this Yukawa term.

Now, we have all the ingredients necessary to make the standard model lagrangian. It is shown in the picture below.

We can again see that all of the generators in SU(2) and U(1) don’t annihilate the vacuum but we can see that a particular combination of them does. We can see that this combination equals the electrical charge and this suggests that this combination hs something to do with electromagnetism. As discussed before, this combination of generators should correspond to a massless gauge boson. We will calculate this boson later.

As always, we calculate the modulus squared of the covariant derivative of the vev. For this purpose, we calculate the covariant derivative and its conjugate first. They are written in terms of newly defined W+ and W- bosons.

The lower entry in the covariant derivative can be written in a more transparent form by defining an angle called the weak mixing angle (also called the Weinberg angle). This allows us to define a combination of W₃ and B bosons as a new boson called the Z boson and an “orthogonal” combination which we call the A boson.

Now, if we calculate the modulus squared of the covariant derivative of the vev, then we see mass terms for the W+, W- and Z bosons but not for the A boson. It means that the A boson is massless and it is nothing but the photon of electromagnetism.

So, we have acheived our goal of generating three massive gauge bosons (W+, W- and Z) for weak interactions and one massless gauge boson (the photon) for electromagnetism. This is why the SU(2) and U(1) sector collectively is known as the electroweak sector.