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The Standard Model and Electroweak Symmetry Breaking

\label{sec:theory}
The aim of this chapter is to briefly present the theory of the Standard Model (SM) by stating the Lagrangian we are using and commenting on the different pieces that it is made of. Thereby we want to especially place emphasis on the importance of the Higgs boson in this context. However, the discussions will be kept to a minimum since its theory can be found in almost every standard textbook of Quantum Field Theory (QFT) and would go beyond the aim and scope of this work.
With the SM and its notation fixed, we will then summarize the Feynman rules that are used throughout the following computations in this thesis.

Mathematical formulation of ‘the’ Standard Model

Before we state ‘the’ SM, we want to highlight that there actually is no unique consensus on what different authors consider as part of it or not — some also call the electroweak sector of the theory as the SM, a few others assume only massless, left-handed neutrinos, while again others include lepton flavor mixing described by the PMNS matrix, cf. [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
There exists a variety of different formulations, notations and conventions for its definition. The presentation given in this thesis is mainly based on [1].

The SM is defined by its action
\begin{equation} \label{eq:S_SM}
\Action[SM] [\vect{\psi}] = \int \dd[4]{x} \Lagrangian[SM] (\vect{\psi}(x), \partial\vect{\psi}(x))
\end{equation}
where $\Lagrangian[SM]$ is the Lagrangian of the SM and the vector $\vect{\psi} = (\psi_1, \ldots, \psi_n)$
collectively denotes the fields $\psi_i(x)$, $i = 1, \ldots, n$, for the different particles of the theory.
The SM is a Yang-Mills theory based on the gauge symmetry group $\SUc \times \SUL \times \UY$ where $\SUc$ denotes the gauge group of Quantum Chromodynamics (QCD) that describes the strong interaction among quarks and gluons and where $\SUL \times \UY$ represents the symmetry group of the electroweak interaction — the unification of the weak and electromagnetic forces — first described by Glashow, Weinberg and Salam [1]. Shortly we will outline that this symmetry group is spontaneously broken to the low-energy $\UEM$ symmetry associated with electromagnetism by the Higgs field acquiring a non-zero vacuum expectation value (VEV} [1].
Similar to the exposition in the aforementioned sources, we describe the SM Lagrangian in terms of covariant $R_\xi$ gauges with the Faddeev-Popov gauge fixing procedure and we include the terms for right-handed neutrinos and their hypothetical masses in the Yukawa sector, although these will not be of interest for our discussion, see [1] for further reference.

The complete Lagrangian for the SM from \cref{eq:S_SM} in this setup can be split as follows:
\begin{equation} \label{eq:L_SM}
\Lagrangian[SM] =
\Lagrangian[Boson] +
\Lagrangian[Fermion] +
\Lagrangian[Higgs] +
\Lagrangian[Yukawa] +
\Lagrangian[GF] +
\Lagrangian[Ghost] .
\end{equation}

Gauge boson sector

The first term denotes the Lagrangian for the gauge boson fields
\begin{equation} \label{eq:L_Boson}
\Lagrangian[Boson] =
– \frac{1}{4} G_{\mu \nu}^{a} G^{\mu \nu}_{a}
– \frac{1}{4} W_{\mu \nu}^{a} W^{\mu \nu}_{a}
– \frac{1}{4} B_{\mu \nu} B^{\mu \nu}
\end{equation}
with the summation over the $8 + 3 + 1 = 12$ generators of the gauge symmetry group $\SUc \times \SUL \times \UY$ in the adjoint representation being implicit [1]. The field strength tensors for the corresponding gauge groups read:
\begin{align*}
G_{\mu \nu}^a &= \partial_\mu G^a_\nu – \partial_\nu G^a_\mu + \gs f^{abc} G_\mu^b G_\nu^c , \\
W_{\mu \nu}^a &= \partial_\mu W^a_\nu – \partial_\nu W^a_\mu + \g \epsilon^{abc} W_\mu^b W_\nu^c , \\
B_{\mu \nu} &= \partial_\mu B_\nu – \partial_\nu B_\mu ,
\end{align*}
where $f^{abc}$ denote the structure constants for $\SUc$ satisfying $\comm{T^a}{T^b} = \I f^{abc} T^c$ and $T^a$ being the generators of the group which for quarks in the fundamental representation satisfy $T^a = \frac{\lambda^a}{2}$ where $\lambda^a$ are the eight Gell-Mann matrices [1]. Correspondingly, $\epsilon^{abc}$ is the totally antisymmetric $\epsilon$-tensor and coincides with the structure constants for $\SUL$ [1]. These satisfy $\comm{\tau^a}{\tau^b} = \I \epsilon^{abc} \tau^c$ where $\tau^a$ are the generators of $\SUL$ and are related by $\tau^a = \frac{\sigma^a}{2}$ to the Pauli matrices for fermions in the fundamental representation [1]. The fields $G^a_\mu$ are the $\SUc$ gauge bosons, i.e. the gluons, with $\gs$ being the strong coupling parameter, $W^a_\mu$ denote the $\SUL$ gauge bosons with the corresponding coupling parameter $\g$ and $B_\mu$ is the hypercharge gauge boson for the Abelian $\UY$ group with the coupling parameter $\gp$ [1].

Fermion sector

In the SM we distinguish the left-handed fermion fields which group up to doublets under $\SUL$ and are given by $\lh{P} = \frac{1}{2} (1 – \gamma_5)$ times the Dirac field from the right-handed fermion fields which pair up to $\SUL$ singlets and are obtained via $\rh{P} = \frac{1}{2} (1 + \gamma_5)$ times the corresponding Dirac field [1].
The left-handed quarks and leptons are grouped into three generations of $\SUL$ doublet pairs which transform in the corresponding fundamental representation [1]:
\[
\lh{Q}^\alpha
\in \qty{
\mqty( \lh{u} \\ \lh{d} ),
\mqty( \lh{c} \\ \lh{s} ),
\mqty( \lh{t} \\ \lh{b} ) }
\qand
\lh{L}^\alpha
\in \qty{
\mqty( \lh{{\nu_{e}}} \\ \lh{e} ),
\mqty( \lh{{\nu_{\mu}}} \\ \lh{\mu} ),
\mqty( \lh{{\nu_{\tau}}} \\ \lh{\tau} )
} .
\]
The index $\alpha = 1, 2, 3$ labels the generation. The right-handed fermions are uncharged under $\SUL$ and are therefore represented as singlets which are grouped into three generations similar to the corresponding doublets [1]:
\begin{alignat*}{2}
\rh{u}^\alpha &\in \qty{ \rh{u}, \rh{c}, \rh{t} } \qc
&&\rh{\nu}^\alpha \in \qty{ \rh{{\nu_{e}}}, \rh{{\nu_{\mu}}}, \rh{{\nu_{\tau}}} }, \\
\rh{d}^\alpha &\in \qty{ \rh{d}, \rh{s}, \rh{b} } \qc
&&\rh{e}^\alpha \in \qty{ \rh{e}, \rh{\mu}, \rh{\tau} } .
\end{alignat*}
The quark fields in $\lh{Q}^\alpha$, $\rh{u}^\alpha$ and $\rh{d}^\alpha$ are additionally charged under the $\SUc$ color subgroup of the SM and are therefore represented as triplets transforming in the corresponding fundamental representation, whereas the other fields are uncharged under the strong interaction so they happen to be $\SUc$ singlets [1].

In the SM, all fermions — except for the right-handed neutrinos having $Y_\nu = 0$ — and the Higgs boson couple to the hypercharge gauge boson and the quantities $Y_Q = \frac{1}{6}, Y_u = \frac{2}{3}, Y_d = – \frac{1}{3}, Y_L = – \frac{1}{2}, Y_e = -1, Y_\nu = 0$ and $Y_H = \frac{1}{2}$ denote the hypercharges of the corresponding fields [1].

With all this one can explicitly write the Lagrangian for the fermion fields, the second term in \cref{eq:L_SM}, as [1]
\begin{equation} \label{eq:L_Fermion}
\begin{aligned}
\Lagrangian[Fermion]
&= \I \lh{\bar{Q}}^\alpha \left(
\slashed{\partial}
– \I \gs \slashed{G}^a T^a – \I \g \slashed{W}^a \tau^a – \I \gp Y_Q \slashed{B} \right) \lh{Q}^\alpha \\
&+ \I \rh{\bar{u}}^\alpha \left(
\slashed{\partial}
– \I \gs \slashed{G}^a T^a
– \I \gp Y_u \slashed{B}
\right) \rh{u}^\alpha
+ \I \rh{\bar{d}}^\alpha \left(
\slashed{\partial}
– \I \gs \slashed{G}^a T^a
– \I \gp Y_d \slashed{B}
\right) \rh{d}^\alpha \\
&+ \I \lh{\bar{L}}^\alpha \left(
\slashed{\partial}
– \I \g \slashed{W}^a \tau^a
– \I \gp Y_L \slashed{B}
\right) \lh{L}^\alpha \\
&+ \I \rh{\bar{e}}^\alpha \left(
\slashed{\partial}
– \I \gp Y_e \slashed{B}
\right) \rh{e}^\alpha
+ \I \rh{\bar{\nu}}^\alpha \left(
\slashed{\partial}
– \I \gp Y_\nu \slashed{B}
\right) \rh{\nu}^\alpha
\end{aligned}
\end{equation}
which contains the kinetic terms for the fermion fields and the interaction terms with the minimally coupled gauge fields. The terms in brackets correspond to the slashed covariant derivatives $\slashed{D} = \gamma^\mu D_\mu$ of the respective fermion fields which live in different representations of the SM gauge group [1]. During the course of this thesis, we will furthermore use the Feynman slash notation which is defined by $\slashed{p} = \gamma^\mu p_\mu$ with $\gamma^\mu$ being the Dirac $\gamma$-matrices, see [1] and \cref{subsec:DR_schemes}.

The hypercharge (operator) corresponding to the $\UY$ gauge symmetry is related to the electric charge (operator) of the $\UEM$ symmetry by $Q = \tau^3 + Y \id$, where $\tau^3$ is the matrix associated with $W_\mu^3$, i.e. the third component of the weak isospin [1]. The hypercharge values for the corresponding fields above have been obviously chosen in such a way that they coincide with the values that are observed in nature [1].

Higgs sector

So far no mass terms have appeared in the Lagrangians describing the gauge bosons $\Lagrangian[Boson]$ nor in the term describing the fermion sector in \cref{eq:L_Fermion}. However, we know from experiments that the $W^\pm$ and $Z^0$ bosons and most fermions are observed to be massive, cf. [1].
Adding these mass terms for the corresponding particles to the Lagrangians in \cref{eq:L_Boson,eq:L_Fermion} would immediately break $\SUL \times \UY$ gauge invariance because fermion masses would mix left-handed with right-handed fields in an $\SUL$ non-invariant way and because mass-terms for the vector bosons $W_\mu^a$ would not be invariant under the corresponding gauge transformations, see [1].

We do not want to give up any of the above principles because local gauge invariance of the SM is critical in proving renormalizability of the theory and the $W^{\pm}$ and $Z^0$ vector bosons and most fermions are known to have mass from experiments [1]. The solution to this problem on how to generate masses to the gauge bosons and fermions without breaking local $\SUL \times \UY$ gauge invariance is by the spontaneous symmetry breaking mechanism due to Anderson, Brout, Englert, Ginzburg, Guralnik, Hagen, Higgs, Kibble and Landau — also commonly known as the Higgs mechanism [1].

The main idea is to not break $\SUL \times \UY$ explicitly, but rather spontaneously. This means that the action and the Lagrangian of the SM are still invariant under the gauge symmetry group $\SUc \times \SUL \times \UY$ — probably realized in a different, hidden way —, but the ground state of the theory is not [1].
In the following lines we want to briefly describe how the Higgs mechanism spontaneously breaks the high-energy electroweak force associated with $\SUL \times \UY$ down to the weak and electromagnetic forces at low energies, i.e. describe the symmetry-breaking of $\SUL \times \UY \rightarrow \UEM$ where $\UEM$ is the symmetry group associated with electromagnetism [1].
For this we have to introduce the Higgs part of the SM Lagrangian from \cref{eq:L_SM} which reads [1]
\begin{equation} \label{eq:L_Higgs}
\Lagrangian[Higgs] =
( D^\mu H )^\dagger (D_\mu H)
+ \muH^2 H^\dagger H
– \lambdaH (H^\dagger H)^2
\end{equation}
where the Higgs multiplet $H$ is a complex $\SUL$ doublet with hypercharge $Y_H = \frac{1}{2}$ and where the covariant derivative is given by
\begin{equation*}
D_\mu H = \qty( \partial_\mu – \I \g W^a_\mu \tau^a – \I \gp Y_H B_\mu ) H .
\end{equation*}
This Lagrangian is obviously invariant under the gauge symmetry group of the SM.
One observes that the potential energy density of the Higgs multiplet $V(H) = -\muH^2 H^\dagger H + \lambdaH (H^\dagger H)^2$ contains a negative mass-term which for $\muH, \lambdaH > 0$ is minimized when \[
\pdv{V}{(H^\dagger H)} = – \muH^2 + 2 \lambdaH H^\dagger H = 0
\quad \Leftrightarrow \quad
H_0^\dagger H_0 = \frac{\muH^2}{2 \lambdaH} = \frac{v^2}{2} .
\]
Therefore, we have an infinite number of equivalent ground states $\ket{\Omega_U}$ with a non-vanishing VEV for $H$ which are related to each other by a global $\SUL \times \UY$ transformation $U$, see [1].
However, although all ground states are equivalent, one $\ket{\Omega}$ has to be chosen which leads to spontaneous symmetry breaking [1]. Without loss of generality one can take the VEV for $H$ so that it is real and in the lower component:
\[
\expval{H}{\Omega} = \mqty(0 \\ \frac{v}{\sqrt{2}})
\qc
\qq{with} v = \sqrt{\frac{\muH^2}{\lambdaH}}
\]
which is not invariant under general $\SUL \times \UY$ gauge transformations. However, it is (constructed to be) invariant under $\UEM$ gauge transformations — which are a special subgroup of the general $\SUL \times \UY$ transformations — so that the electromagnetic gauge symmetry associated with $\UEM$ is unbroken, the photon remains massless and the electric charge is conserved [1].
This is exactly the symmetry-breaking pattern $\SUL \times \UY \rightarrow \UEM$ that has been indicated before.

In order to have a perturbative QFT, one has to expand the Higgs doublet around its VEV. We will parametrize it in terms of a complex scalar field $w^{+}(x)$ and two real scalar fields $z(x)$ and $h(x)$:
\begin{equation*}
H(x)
= \mqty(w^{+}(x) \\ \frac{v + h(x) + \I z(x)}{\sqrt{2}})
\end{equation*}
By applying the charge operator on the individual components of the Higgs doublet one can derive that the complex scalar field $w^{+}(x)$ is positively charged whereas the fields in the lower component are neutral. The field $h(x)$ corresponds to the Higgs field and gives rise to massive, electrically neutral spin-$0$ particles — these are the Higgs bosons [1]. The remaining fields correspond to the Goldstone bosons and in the unitary gauge ($\xi_W, \xi_Z \rightarrow \infty$, see later) these are gauged away and are absorbed by the $W^{\pm}$ and $Z^0$ gauge boson where they give rise to their longitudinal modes [1]. For our computations, we will work in the unitary gauge where the Goldstone bosons do not appear in the spectrum [1].

With this, let us come back to the original question regarding the origin of the masses for the $W^{\pm}$ and $Z$ bosons. By expanding the Higgs doublet around its VEV, re-expressing it in terms of the Higgs and Goldstone fields and plugging it back into the Lagrangian in \cref{eq:L_Higgs} one can observe that the terms in $( D^\mu H )^\dagger (D_\mu H)$ — after diagonalizing the vector bosons fields to the mass basis — give mass to the vector bosons $W^{\pm}$ and $Z^0$.
Diagonalization is performed by rotating the gauge bosons as follows: [1]
\begin{align*}
\mqty( B_\mu \\ W_\mu^3 )
= \mqty(
\cos \thetaW & – \sin \thetaW \\
\sin \thetaW & \cos \thetaW )
\mqty( A_\mu \\ Z_\mu )
\quad \Leftrightarrow \quad
\mqty( A_\mu \\ Z_\mu^3 )
= \mqty(
\cos \thetaW & \sin \thetaW \\
– \sin \thetaW & \cos \thetaW )
\mqty( B_\mu \\ W_\mu^3 )
\end{align*}
where the Weinberg angle $\thetaW$ is defined by $\tan \thetaW = \frac{\gp}{\g}$ [1].
After re-expressing the gauge fields $W^3_\mu$ and $B_\mu$ in terms of the mass eigenstates $Z_\mu$ and $A_\mu$ corresponding to the $Z^0$ boson and photon fields and by rewriting $W^1_\mu$ and $W^2_\mu$ using [1]
\[
W^{\pm}_\mu = \frac{1}{\sqrt{2}} \qty( W^1_\mu \mp \I W^2_\mu )
\]
in terms of the corresponding charged complex vector fields for the $W^{\pm}$ fields, one can easily read off the masses from the $( D^\mu H )^\dagger (D_\mu H)$ terms in \cref{eq:L_Higgs} as
\[
\mW = \frac{1}{2} \g v \qc
\mZ = \frac{1}{2 \cos \thetaW} \g v = \frac{\mW}{\cos \thetaW} .
\]
As expected, the photon field $A_\mu$ remains massless since the electromagnetic gauge symmetry $\UEM$ is unbroken.

Yukawa sector

We have seen how the $W^{\pm}$ and $Z^0$ bosons have acquired masses, but the formalism so far was not enough to also give rise to massive fermions.
For fermions to become massive we have to additionally add Yukawa coupling terms to the Lagrangian in \cref{eq:L_SM} [1]:^[1]
\begin{equation} \label{eq:L_Yukawa}
\begin{aligned}
\Lagrangian[Yukawa] =
&- Y^d_{\alpha \beta} \lh{\bar{Q}}^\alpha H \rh{d}^\beta
– Y^u_{\alpha \beta} \lh{\bar{Q}}^\alpha \tilde{H} \rh{u}^\beta
– Y^e_{\alpha \beta} \lh{\bar{L}}^\alpha H \rh{e}^\beta
– Y^\nu_{\alpha \beta} \lh{\bar{L}}^\alpha \tilde{H} \rh{\nu}^\beta
+ h.c. ,
\end{aligned}
\end{equation}
where $\tilde{H}$ is defined by $\tilde{H} \equiv \I \sigma^2 H^{\star} = – \I \qty[ H^\dagger \sigma^2 ]^{\mathrm{T}}$ [1].
Each of these terms is obviously invariant under the symmetry gauge group $\SUc \times \SUL \times \UY$ of the SM. These terms will generate mass terms for the fermions only after electroweak symmetry breaking when the Higgs doublet $H$ acquires a VEV [1]. We do not want to go into much detail on the specific terms since we assume the reader to be familiar with this formulation, otherwise we refer to [1].
The matrices $Y^d_{\alpha \beta}$, $Y^u_{\alpha \beta}$, $Y^e_{\alpha \beta}$ and $Y^\nu_{\alpha \beta}$ are the $3 \times 3$ Yukawa matrices for the different types of quarks and leptons which using a singular value decomposition can be diagonalized to bring the quarks and leptons to the mass basis. Applying this change of basis also to the fermion Lagrangian $\Lagrangian[Fermion]$ in \cref{eq:L_Fermion} introduces flavor mixing effects for the coupling of fermions with $W^{\pm}$ bosons which are described by the CKM matrix for quarks and by the PMNS matrix for leptons with massive neutrinos.
These flavor mixing effects have to be considered in our computations and enter the calculations via the corresponding Feynman rules for our processes.

Gauge fixing terms

The Equations of motion (EOMs) of the gauge fields are not invertible without adding gauge-fixing terms to the SM Lagrangian. Since we want to compute the gauge field propagators in covariant $R_\xi$ gauges, one furthermore has to introduce Faddeev-Popov ghosts as an artifact from keeping Lorentz invariance in our Lagrangian when deriving them properly using the path integral. These terms are part of the last term in \cref{eq:L_SM} and will be discussed in the next paragraph below.
The gauge-fixing Lagrangian for covariant $R_\xi$ gauges reads [1]:
\begin{equation} \label{eq:L_GF}
\Lagrangian[GF] =
– \frac{1}{2 \xi_G} F_G^a F_{G}^a
– \frac{1}{\xi_W} F_{-}F_{+}
– \frac{1}{2 \xi_Z} F_Z^2
– \frac{1}{2 \xi_A} F_A^2
\end{equation}
where
\begin{equation} \label{eq:gauge_fixing_conditions}
\begin{aligned}
F_G^a &= \partial^\mu G_\mu^a , \\
F_A &= \partial^\mu A_\mu , \\
F_Z &= \partial^\mu Z_\mu + \xi_Z \mZ z , \\
F_{\pm} &= \partial^\mu W_\mu^{\pm} \pm \I \xi_W \mW w^{\pm} .
\end{aligned}
\end{equation}
The parameters $\xi_G, \xi_W$, $\xi_Z$ and $\xi_A$ are arbitrary gauge parameters and we are free to choose them as we like in our computations since — due to gauge invariance — they have to drop out of physical quantities at all orders in perturbation theory [1]. Our computations will be performed in the unitary gauge for the massive vector bosons for which $\xi_W, \xi_Z \rightarrow \infty$. For this gauge choice, as already explained before, the Goldstone bosons are removed from the physical spectrum of the SM and do not have to be considered in our computations.

Ghost sector

As stated in the previous paragraph, the last part of the Lagrangian of the SM contains the terms that arise from applying the Faddeev-Popov procedure to formulate the SM in covariant $R_\xi$ gauges.
These include the ghosts $c^a$ associated with the $\SUc$ gauge transformations and the electroweak ghosts $c_{\pm}, c_Z, c_A$ associated with the $\SUL \times \UY$ gauge transformations. For the linear gauge fixing conditions from \cref{eq:gauge_fixing_conditions} they read (see [1]):
\begin{equation} \label{eq:L_Ghost}
\Lagrangian[Ghost] =
\sum_{a,b = 1}^{8} \bar{c}^a \pdv{(\var{F_G^a})}{(\var{\alpha^b})} c^b
+ \sum_{a = 1}^{4} \qty(
\bar{c}_{+} \pdv{(\var{F_{+}})}{(\var{\alpha^a})}
+ \bar{c}_{-} \pdv{(\var{F_{-}})}{(\var{\alpha^a})}
+ \bar{c}_{Z} \pdv{(\var{F_{Z}})}{(\var{\alpha^a})}
+ \bar{c}_{A} \pdv{(\var{F_{A}})}{(\var{\alpha^a})}
) c_a
\end{equation}
where $\alpha^a$ are the numbers that parametrize the group elements of $\SUc$ or $\SUL \times \UY$.
This Lagrangian can be determined from computing the corresponding Faddeev-Popov functional determinants. An excellent discussion and explicit derivation of these terms can be found in \cite[Ch.~12]{Pokorski:GaugeFieldTheories2000} \cite[App.~A.1.]{Romao:FeynmanRulesSM}.