9.6 Action functional for the standard model

We have now all the necessary techniques to write down the action functional for the standard model of particle physics. Throughout the whole section we work on the \(d=4\) dimensional Minkowski spacetime \((\bbR ^4,\eta )\) and fix \(G = \U (1)\times \SU (2)\times \SU (3)\) to be the Lie group of the standard model. Note that the elements of this product Lie group are triples \((U_1,U_2,U_3)\in \U (1)\times \SU (2)\times \SU (3)\) consisting of elements of the individual factors, i.e. \(U_1\in \U (1)\) is a unitary \(1\times 1\)-matrix, \(U_2\in \SU (2)\) is a special unitary \(2\times 2\)-matrix and \(U_3\in \SU (3)\) is a special unitary \(3\times 3\)-matrix.

The action functional for the standard model takes the following form

where each of the individual terms can be constructed using the techniques developed in this chapter. Since the final result for \(S_{\mathrm {SM}}\) is quite lengthy, due to the many different particles/fields of the standard model (see the graphical visualization in Section 9.1), it makes sense to present the individual terms step by step. This is precisely what we shall do in the paragraphs below.

The action \(S_{\mathrm {gauge}}\): Since the standard model Lie group \(G = \U (1)\times \SU (2)\times \SU (3)\) is a product group, its Lie algebra decomposes into a direct sum \(\g = \mathfrak {u}(1)\oplus \mathfrak {su}(2)\oplus \mathfrak {su}(3)\). This means that a Yang-Mills field \(A_\mu \) for the Lie group \(G\) can be decomposed into three different components, which we shall denote as follows:

• (1) The \(\U (1)\) Yang-Mills field will be denoted by \(B_\mu \in \mathfrak {u}(1)\) and its field strength by \(B_{\mu \nu } = \partial _\mu B_{\nu } - \partial _\nu B_\mu + [B_\mu ,B_\nu ]= \partial _\mu B_{\nu } - \partial _\nu B_\mu \), where the last identity holds because \(\mathfrak {u}(1)\) is an Abelian Lie algebra, i.e. its Lie bracket is trivial.

• (2) The \(\SU (2)\) Yang-Mills field will be denoted by \(W_\mu \in \mathfrak {su}(2)\) and its field strength by \(W_{\mu \nu } = \partial _\mu W_{\nu } - \partial _\nu W_\mu + [W_\mu ,W_\nu ]\).

• (3) The \(\SU (3)\) Yang-Mills field will be denoted by \(G_\mu \in \mathfrak {su}(3)\) and its field strength by \(G_{\mu \nu } = \partial _\mu G_{\nu } - \partial _\nu G_\mu + [G_\mu ,G_\nu ]\).

Under a gauge transformation \(U = (U_1,U_2,U_3) : \bbR ^d\to \U (1)\times \SU (2)\times \SU (3)\) these components transform according to (9.36) as

\begin{flalign} (T_UB)_\mu &= U_1\,B_\mu U_1^{-1} + U_1\,\partial _\mu U_1^{-1} = B_\mu + U_1\,\partial _\mu U_1^{-1}\quad ,\\ (T_UW)_\mu &= U_2\,W_\mu U_2^{-1} + U_2\,\partial _\mu U_2^{-1}\quad ,\\ (T_UG)_\mu &= U_3\,G_\mu U_3^{-1} + U_3\,\partial _\mu U_3^{-1}\quad , \end{flalign} i.e. the each component transforms according to its corresponding factor of the product group. The action \(S_{\mathrm {gauge}}\) for the gauge bosons of the standard model is then given by summing the Yang-Mills actions (9.43) for the three components, i.e.

where \(g_1,g_2,g_3\in \bbR \) are three independent coupling constants for the three factors of \(G= \U (1)\times \SU (2)\times \SU (3)\) whose values have to be fixed by experiment. This action is invariant under the gauge transformations (9.77).

The action \(S_{\mathrm {matter}}\): This part of the action describes the matter fermions of the standard model. As we have seen in Section 9.4, the coupling of matter fermions to Yang-Mills fields can be described by choosing a unitary representation \(\rho \) of the Lie group \(G\) on some complex vector space \(V\). Since \(G= \U (1)\times \SU (2)\times \SU (3)\) is a product Lie group, we can build such representations from a unitary representation \(\rho _1\) of \(\U (1)\), a unitary representation \(\rho _2\) of \(\SU (2)\) and a unitary representation \(\rho _3\) of \(\SU (3)\). Explicitly, denoting the underlying vector spaces of these representations by \(V_1\), \(V_2\) and \(V_3\), one checks that

\begin{flalign} \nn \rho \,:\,\U (1)\times \SU (2)\times \SU (3)& \longrightarrow \mathsf {GL}\big (V_1\otimes V_2\otimes V_3\big )~~,\\ (U_1,U_2,U_3)&\longmapsto \rho (U_1,U_2,U_3) := \rho _1(U_1)\otimes \rho _2(U_2) \otimes \rho _3(U_3) \end{flalign} defines a unitary representation. A left or right-handed Weyl field \(\Psi _{L/R}:\bbR ^4\to \bbC ^2\otimes V_1\otimes V_2\otimes V_3\) that takes values in such representation then transforms under gauge transformations as

\begin{flalign} \label {eqn:SMgaugetrafo2} T_U\Psi _{L/R} \,=\, \rho _1(U_1)\, \rho _2(U_2) \,\rho _3(U_3)\,\Psi _{L/R}\quad , \end{flalign} where on the right-hand side we suppressed the tensor products in \(\rho _1(U_1)\otimes \rho _2(U_2) \otimes \rho _3(U_3)\) to improve readability. The corresponding gauge covariant derivative reads as

\begin{flalign} D_\mu \Psi _{L/R} \,= \, \partial _\mu \Psi _{L/R} + \underline {\rho _{1}}(B_\mu )\,\Psi _{L/R} + \underline {\rho _{2}}(W_\mu )\,\Psi _{L/R} + \underline {\rho _{3}}(G_\mu )\,\Psi _{L/R}\quad , \end{flalign} where \(B_\mu \), \(W_\mu \) and \(G_\mu \) denote the three components of the Yang-Mills field that correspond to the three factors of \(G=\U (1)\times \SU (2)\times \SU (3)\).

So what we have to do is specify the various representations for the different standard model matter fermions in the graphical table from Section 9.1. It is important to emphasize that these representations can not be predicted theoretically, but they had to be determined by experiment. Recalling the specific representations from Examples 9.7 and 9.8, we start by listing the relevant representations for the matter fermions from the first generation:

The last column is a useful and efficient way to label the relevant representations that you will often find in the literature. The first entry of the triple is called weak hypercharge and it is computed from the \(\U (1)\) representation \(\rho ^{(k)}\) by \(\tfrac {k}{3}\). The second entry describes the \(\SU (2)\) representation and it is \(\mathbf {1}\) for the \(1\)-dimensional trivial representation \(\rho _{\mathrm {triv}}\) and \(\mathbf {2}\) for the \(2\)-dimensional defining representation \(\rho _{\mathrm {def}}\). So the bold number indicates the dimension of the representation. The third entry describes the \(\SU (3)\) representation and it is \(\mathbf {1}\) for the \(1\)-dimensional trivial representation \(\rho _{\mathrm {triv}}\) and \(\mathbf {3}\) for the \(3\)-dimensional defining representation \(\rho _{\mathrm {def}}\).

It is important to note that the left-handed quark \(q_L\) lives in the \(2\)-dimensional \(\SU (2)\) defining representation \(\rho _{\mathrm {def}}\), i.e. it consists of two components that can be identified respectively with the left-handed up quark \(u_L\) and the left-handed down quark \(d_L\). Similarly, the left-handed lepton \(\ell _L\) lives in the \(2\)-dimensional \(\SU (2)\) defining representation and its two components define the left-handed electron \(e_L\) and the left-handed neutrino \(\nu _L\). In our version of the standard model of particle physics, which agrees with the one that is typically presented in textbooks, there is no right-handed neutrino \(\nu _R\). Evidence from more recent experiments (e.g. neutrino oscillations) suggests that right-handed neutrinos exist, hence one should think about adding \(\nu _R\) to the table above. If it exist, one knows that the relevant representation for \(\nu _R\) must be \((0,\mathbf {1},\mathbf {1})\), which means that the right-handed neutrino participates trivially in all of the three fundamental interactions. (That’s why right-handed neutrinos are sometimes called sterile neutrinos.) Right-handed neutrinos will be discussed further in some of the student projects/presentations.

The representations for the second and the third generation of matter fermions are identical to the ones for the first generation because, besides different masses, the particles from different generations have identical properties. This means that we have to introduce three versions of the fields listed in the table above, which we shall do by introducing a subscript \(_i\) that runs over the three generations \(i=1,2,3\). The action \(S_{\mathrm {matter}}\) for the matter fermions is then given by a sum over the chiral actions (9.64) that takes into account all particles and all generations. Explicitly, this reads as

This action is invariant under the combined gauge transformations given by (9.77) and (9.80). Physically, it describes the dynamics of the matter fermions and their interaction with the gauge bosons.

The action \(S_{\mathrm {Higgs}}\): To introduce a Higgs field, one has to choose another unitary representation of the standard model Lie group \(G=\U (1)\times \SU (2)\times \times \SU (3)\), which also has to be determined from experiment. Given any representation of the form \(\rho =\rho _1\otimes \rho _2\otimes \rho _3\), the gauge transformation law for the Higgs field reads as

\begin{flalign} \label {eqn:SMgaugetrafo3} T_U\Phi \,=\, \rho _1(U_1)\, \rho _2(U_2) \,\rho _3(U_3)\,\Phi ~~,\quad T_U\Phi ^\dagger \,=\, \Phi ^\dagger \,\rho _1(U_1^{-1})\, \rho _2(U_2^{-1}) \,\rho _3(U_3^{-1}) \end{flalign} and the corresponding gauge covariant derivatives are given by

\begin{flalign} D_\mu \Phi \,&= \, \partial _\mu \Phi + \underline {\rho _{1}}(B_\mu )\,\Phi + \underline {\rho _{2}}(W_\mu )\,\Phi + \underline {\rho _{3}}(G_\mu )\,\Phi \quad ,\\ D_\mu \Phi ^{\dagger } \,&= \, \partial _\mu \Phi ^\dagger -\Phi ^\dagger \, \underline {\rho _{1}}(B_\mu ) -\Phi ^\dagger \, \underline {\rho _{2}}(W_\mu ) - \Phi ^\dagger \, \underline {\rho _{3}}(G_\mu ) \quad . \end{flalign} The standard model Higgs field comes in the following representation:

The action \(S_{\mathrm {Higgs}}\) for the Higgs field is then given by (9.73), which for completeness will be repeated here

This action is invariant under the combined gauge transformations given by (9.77) and (9.83). Physically, it describes the dynamics of the Higgs boson and its interaction with the gauge bosons. After spontaneous symmetry breaking, it will generate mass terms for the \(W^+\), \(W^-\) and \(Z\) bosons.

The action \(S_{\mathrm {Yukawa}}\): The last term in the standard model action (9.76) is an interaction term (called Yukawa interaction) between the Higgs field and the matter fermions. From a physical perspective this is needed to introduce fermion masses through spontaneous gauge symmetry breaking. The existence of the Yukawa interaction terms relies very sensitively on the specific details of the representations given to the matter fermions and the Higgs field (see the tables above). The best way I know to write down the Yukawa terms is to use the following partial index notation for only the \(\SU (2)\) representations: Since the left-handed fermions, i.e. \(q_L\) and \(\ell _L\), and the Higgs \(\Phi \) live in the \(2\)-dimensional \(\SU (2)\) defining representation \(\rho _{\mathrm {def}}\), we can pick a basis of \(\bbC ^2\) and write \(q_{L\,a}\), \(\ell _{L\,a}\) and \(\Phi _a\), with \(a=1,2\), for the two components. The other matter fermions \(u_R\), \(d_R\) and \(e_R\) live in the \(1\)-dimensional trivial \(\SU (2)\) representation, hence they do not get such an index related to \(\SU (2)\). I claim that the term

\begin{flalign} \label {eqn:Yukawa1} q_L^\dagger \,u_R\,\Phi \,:=\, q_{L\,a}^\dagger \,u_R\,\Phi _a \qquad \text {(summation over $a=1,2$ understood)} \end{flalign} is invariant under all \(G=\U (1)\times \SU (2)\times \SU (3)\) gauge transformations. This can be shown by considering the individual factors separately. Starting with \(U_3\in SU(3)\), we use that \(\Phi \) transforms in the trivial representation while \(q_L\) and \(u_R\) transform in the defining representation, hence

\begin{flalign} T_{U_3}\,:\,q_L^\dagger \,u_R\,\Phi \,\longmapsto \,\big ( \rho _{\mathrm {def}}(U_3)q_L\big )^\dagger \rho _{\mathrm {def}}(U_3)u_R\,\Phi = q_L^\dagger \,u_R\,\Phi \quad , \end{flalign} where in the last step we used that the defining representation is unitary. Considering now \(U_1\in \U (1)\), we look up the corresponding \(\U (1)\) representations from the table above and compute

\begin{flalign} T_{U_1}\,:\,q_L^\dagger \,u_R\,\Phi \,\longmapsto \,\rho ^{(-1)}(U_1)\,\rho ^{(4)}(U_1)\,\rho ^{(-3)}(U_1)\,q_L^\dagger \,u_R\,\Phi \, =\,U^{-1+4-3}_1\,q_L^\dagger \,u_R\,\Phi \,=\, q_L^\dagger \,u_R\,\Phi \quad , \end{flalign} where we used that \(\big (\rho ^{(1)}(U_1)\,q_L\big )^\dagger = \big (U_1 \,q_L\big )^\dagger = U_1^{-1}\,q_L^\dagger \) because \(U_1^\dagger = U_1^{-1}\) is a unitary \(1\times 1\)-matrix. Finally, for \(U_2\in \SU (2)\) we look again at the corresponding representations from the table above and compute in index notation

\begin{flalign} T_{U_2}\,:\,q_{L\,a}^\dagger \,u_R\,\Phi _a \,\longmapsto \,\big ({U_2}_{ab}\,q_{L\,b}\big )^\dagger \,u_R\,{U_2}_{ac}\,\Phi _c = {U_{2}}_{ab}^\ast \, {U_{2}}_{ac}\, q_{L\,b}^\dagger \, u_R\,\Phi _c = q_{L\,b}^\dagger \, u_R\,\Phi _b \quad , \end{flalign} where in the last step we used that \(U_2\) is a unitary matrix, i.e. \(U_2^\dagger \,U_2 = \mathbf {1}\), which reads in index notation as \({U_{2}}_{ab}^\ast \, {U_{2}}_{ac} = \delta _{bc}\). All of this together shows that the term (9.86) is gauge invariant under \(G=\U (1)\times \SU (2)\times \SU (3)\).

Let us now consider a second term of a similar form

\begin{flalign} \label {eqn:Yukawa2} q^\dagger _L\,d_R\,\widetilde {\Phi } \, :=\,\epsilon _{ab}\,q_{L\,a}^\dagger \,d_R\,\Phi ^\ast _b \qquad \text {(summation over $a,b=1,2$ understood)}\quad , \end{flalign} where \(\Phi _b^\ast \) denotes the complex conjugate of the \(b\)-component of the Higgs field and \(\epsilon _{ab}\) are the entries of the following \(2\times 2\)-matrix (the \(2\)-dimensional epsilon-tensor)

\begin{flalign} \epsilon \,=\,\begin{pmatrix} 0&1\\ -1&0 \end {pmatrix}\quad . \end{flalign} Invariance of this term under \(\U (1)\) and \(\SU (3)\) gauge transformations is easy to check by looking again at the table above. Checking invariance under \(\SU (2)\) gauge transformations is slightly more complicated. Given \(U_2\in \SU (2)\), we compute in index notation

\begin{flalign} T_{U_2}\,:\,\epsilon _{ab}\,q_{L\,a}^\dagger \,d_R\,\Phi ^\ast _b \,\longmapsto \, \epsilon _{ab}\, {U_{2}}^\ast _{ac}\,{U_{2}}^\ast _{bd}\,q_{L\,c}^\dagger \,d_R\,\Phi ^\ast _d\quad . \end{flalign} This means that the term (9.90) is gauge invariant under \(\SU (2)\) if and only if the matrix identity

\begin{flalign} U_2^\dagger \,\epsilon \,U_2^\ast \,=\, \epsilon \end{flalign} holds true, where \(U_2^\ast \) denotes entry-wise complex conjugation (without transposition!). One can show that every special unitary \(2\times 2\)-matrix can be written as

\begin{flalign} U_2=\begin{pmatrix} z & w\\ -w^\ast & z^\ast \end {pmatrix} \end{flalign} with some complex numbers \(z,w\in \bbC \) satisfying \(\vert z\vert ^2 + \vert w\vert ^2 =1\). From this we check

\begin{flalign} U_2^\dagger \,\epsilon \,U_2^\ast \,=\, \begin{pmatrix} z^\ast & -w\\ w^\ast & z \end {pmatrix}\, \begin{pmatrix} 0&1\\ -1&0 \end {pmatrix}\, \begin{pmatrix} z^\ast & w^\ast \\ -w & z \end {pmatrix} \,=\, \begin{pmatrix} 0 & 1\\ -1 & 0 \end {pmatrix} \,=\,\epsilon \quad . \end{flalign} There is a third and last gauge invariant term that one can form using the leptons

\begin{flalign} \label {eqn:Yukawa3} \ell _L^\dagger \,e_R\,\widetilde {\Phi } \,:=\,\epsilon _{ab}\, \ell _{L\,a}^\dagger \,e_R\,\Phi ^\ast _b \qquad \text {(summation over $a,b=1,2$ understood)}\quad . \end{flalign} The relevant checks are similar to the ones for the analogous quark term (9.90).

The Yukawa interaction term \(S_{\mathrm {Yukawa}}\) for the standard model is then given by considering the most general linear combination of the three terms (9.86), (9.90) and (9.96) over the three generations of matter fermions. Explicitly, this reads as

where \(+\,\mathrm {c.c.}\) means to add the complex conjugate to produce a real-valued action. This action is by construction invariant under gauge transformations. The three complex \(3\times 3\)-matrices \(Y\), \(Y^\prime \) and \(Y^{\prime \prime }\) (called Yukawa couplings) are free parameters of the standard model that have to be fixed from experiment. These parameters are in particular related to the masses of the quarks and leptons.

Summing up: This completes our description of the standard model action functional

\begin{flalign} S_{\mathrm {SM}} \,=\,S_{\mathrm {gauge}}+ S_{\mathrm {matter}} + S_{\mathrm {Higgs}} + S_{\mathrm {Yukawa}}\quad . \end{flalign} The individual terms in this action are displayed in (9.78), (9.82), (9.85) and (9.97). As you may expect from its complexity, the standard model is physically extremely rich. Some of the student projects/presentations will explore these physical aspects further.

The form of the standard model action functional discussed in this chapter is not quite yet ready for perturbative quantization. There are two further aspects that have to be dealt with:

• (1) Since the Higgs potential in (9.85) has non-trivial minima, one has to choose a classical vacuum \(\Phi _0\neq 0\) in order to set up a perturbative expansion. In practice, this amounts to writing the Higgs field \(\Phi (x) = \Phi _0 + \phi (x)\) in terms of perturbations \(\phi \) of the classical vacuum \(\Phi _0\), and then quantize these perturbations with our perturbative QFT techniques. Inserting this decomposition into the standard model action functional, one finds that the terms involving \(\Phi _0\) determine mass terms for the matter fermions and the \(W^+\), \(W^-\) and \(Z\) bosons. The details of this so-called Higgs mechanism for spontaneous gauge symmetry breaking will be the topic of some of the student projects/presentations.

• (2) To quantize the Yang-Mills fields in (9.78), one requires as in the case of the electromagnetic potential in Chapter 6 a gauge fixing and an analog of the Gupta-Bleuler formalism. For non-Abelian Lie groups, as those present in the standard model, these aspects are more subtle and involved than for the case of electromagnetism. There is a powerful tool, called the BRST formalism, that allows one to systematically deal with these issues. A new feature of non-Abelian gauge symmetries is that a consistent gauge fixing procedure requires introducing new fields, the so-called ghost fields, whose role is to cancel the unphysical degrees of freedom in quantum correlation functions and scattering amplitudes. Some of these aspects will be discussed in the student projects/presentations.

Once these two issues have been addressed, one can quantize the standard model of particle physics with our perturbative QFT techniques, which leads to a set of Feynman rules that can be used for making physical predictions for scattering amplitudes. This is how QFT connects to high-energy particle physics.

Further reading

For more details about the standard model of particle physics, see e.g. Srednicki (Chapters 69 and 87–89), Peskin/Schroeder (Chapters 15 and 17) and Schwartz (Chapters 25 and 29) from our reading list in Section 1.3.