1.1 The Standard Model
The Standard Model (SM) of particle physics is a mathematically self-consistent gauge field theory that classifies all known types of elementary particles and describes their electromagnetic, weak and strong interactions. Within this fundamental theory, all known matter and energy phenomena can be explained in terms of the kinematics and interactions of elementary particles, which can in turn be understood as local excitations of different fields that permeate our universe.
Figure 1.1: Schematic overview of the particle content within the SM. Fundamental particles include fermions, further subdivided in quarks and leptons, and fundamental bosons, including the force mediators and the Higgs boson. Diagram adapted from MissMJ (CC BY 3.0 license).
From a historical perspective, this theory is the product of a succession of important theoretical developments and experimental discoveries over the last century [1], culminating with the discovery of the Higgs boson in 2012 [2], [3]. If a more principled viewpoint is taken, the SM can be thought of as the most general but mathematically consistent theory that respects a set of symmetries, namely a global Poincaré group symmetry (translational, rotational and relativistic boost invariance) and a local \[G_\textrm{SM} = SU(3)_{C}\otimes SU(2)_{L} \otimes U(1)_{Y}\qquad(1.1)\] gauge group symmetry. The \(G_\textrm{SM}\) symmetry group is essential to describe three of the four fundamental interactions observed in nature: strong interaction, weak interaction and electromagnetic interaction. In fact, the \(SU(3)_{C}\) is associated the strong force and the conservation of its charge, called colour, while the \(SU(2)_{L} \otimes U(1)_{Y}\) symmetry instead is related with electroweak interactions (i.e. unification of weak and electromagnetic) and the conservation of isospin and weak hypercharge. The SM is typically specified using the Lagrangian formalism and depends on a total of 19 parameters (not accounting for neutrino masses and mixing angles), which are not predicted by the theory from first principles, and thus can only be determined through experimental measurements.
In the context of the SM, excitations of the fundamental fields give rise to two types of elementary particles: fermions (characterised by having half-integer spin) and bosons (characterised by having integer spin). Fermions are the fundamental constituents of matter, and they are further subdivided into leptons and quarks depending on their interactions. A schematic overview of the fundamental particles of the SM and their properties is provided in Figure 1.1. Three particle generations are known for both quarks and leptons, each containing a pair of particles with different masses. For quarks, the heavier is referred to as up-type and the lighter as down-type. Instead, for leptons we distinguish the heavier charged particles (electron, muon and tau) from their corresponding light and uncharged neutrinos.
Regular matter is largely made of the first generation of quarks and electrons, given that higher generations rapidly decay quickly to lower generations characterised by smaller masses. All fermions interact via the weak force but only quarks carry colour charge and are subjected to the strong force. For each fermion in the SM, there is a another particle with identical properties but opposite quantum numbers, globally referred to as antimatter, and denoted for each particle with the anti prefix and a bar over the symbol (e.g. up antiquark \(\bar{u}\)) or by explicitly denoting the charge sign (e.g. positron \(e^+\)). Neutrinos are the only fermions that do not carry electrical charge and might be their own antiparticle.
The mediators of the strong, weak and electromagnetic fundamental interactions are referred to as gauge bosons, and are characterised by having spin 1. To model the strong interaction colour charge exchanges, a total of eight independent strong massless force mediators, or gluons, are needed. Gluons carry colour charge themselves and thus participate in colour interactions with other gluons, which leads to a phenomenon known as colour confinement, which will be discussed in Section 1.1.2 in more detail. The massless and neutral photon is the mediator of the electromagnetic force, while instead the massive \(Z\), \(W^+\) and \(W^-\) bosons mediate weak interactions. The last piece in the SM is the Higgs boson, the only fundamental known particle with spin 0. The Higgs boson is the quantum excitation of the Higgs field, which also couples with other fundamental particles such as the gauge bosons of the weak force, effectively generating their mass through their interaction. The Higgs boson and Higgs field play an essential role in the electroweak symmetry breaking (EWSB) mechanism, which will be discussed in more detail in Section 1.1.4.
The rest of this section will be devoted a more mathematically exhaustive review of the different components of the Standard Model, starting by reviewing the basic formalism of quantum field theories and incrementally building on it do describe the characteristics of both the strong and electroweak interactions that give rise to the diverse interactions dynamics of relevance in particle physics experiments. The mentioned review is heavily inspired by standard bibliographical references on the topic [4], [5], and which are recommended directly for a more detailed survey on the subject.
1.1.1 Essentials of Quantum Field Theory
As hinted in the previous section, in quantum field theory (QFT), observed particles are understood as excitations of fields that extend through the whole universe. Quantum field theory unifies the physical foundations of quantum mechanics and special relativity, and can be used to accurately describe phenomena in systems where relativistic and quantum effects are relevant, such as interactions between highly relativistic particles. In QFT, all the known physical processes in the universe are explained in terms of the state and dynamics of a set of fundamental tensor fields. A tensor field can be defined as a continuous and differentiable set of values, such a scalar or a vector, that exist for any given location and time. For simplicity, the fields in QFT are usually defined in a relativistic coordinate system \(x = (t, \boldsymbol{x})\) in order treat space \(\boldsymbol{x}\) and time \(t\) jointly.
To exemplify the fundamentals of the QFT framework, let us consider the simplest case, e.g. a single field that does not interact with any other field, which will be denoted as \(\phi(x)\). The dynamics of a field (or several fields) in QFT are specified by using the Lagrangian formalism, similarly to what can be done for systems in classical mechanics. However, instead of considering the Lagrangian \(L\) which depends the generalised coordinate vector \(\boldsymbol{q}(t)\) and its time derivatives \(\dot{\boldsymbol{q}}(t)\), in QFT the Lagrangian density \(\mathcal{L}\) is commonly used, which depends only on the field \(\phi (x)\) and its first derivative \(\partial_{\mu} \phi (x)\). In an analogous manner to what is done in classical mechanics to define the action functional \(S_{\textrm{classical}}\), we can define the action of the quantum field \(S_{\textrm{QFT}}\) as a function of the Lagrangian density \(\mathcal{L}\) as follows: \[ S_{\textrm{classical}} = \int L (\boldsymbol{q}(t) ,\dot{\boldsymbol{q}} (t)) dt \quad \Rightarrow \quad S_{\textrm{QFT}} = \int \mathcal{L}(\phi, \partial_{\mu} \phi)\ d^4 x \qquad(1.2)\] noting that the previous definition would also be valid when the Lagrangian depends on multiple fields and their derivatives instead of a single free field. Identically to what is done in classical systems, we can attempt to solve for the field that minimises the action, i.e. \(\delta S=0\). With the help of some functional calculus [6], it is possible to obtain the relativistic field theory version of the Euler-Langrange equation: \[\partial_\mu \left ( \frac{\partial \mathcal{L}} {\partial (\partial_\mu\phi)} \right ) - \frac{\partial\mathcal{L}}{\partial\phi} = 0 \qquad(1.3)\] where \(\partial_\mu=\partial/\partial x_\mu\) and the repetition of the coordinate index \(\mu \in \{0,1,2,3\}\) means summation over the product. The previous relation would still apply to each field in the case a Lagrangian including several fields was considered; therefore, given a Lagrangian, we can use Equation 1.3 to obtain their equations of motion. As an example, let us consider the following Lagrangian \(\mathcal{L}_\textrm{Dirac}\), which is a function of a bispinor field \(\psi\), a 4-dimensional complex vector field that can represent a field whose excitations behave like fermions of mass \(m\): \[\mathcal{L}_\textrm{Dirac} =\bar{\psi} (i\gamma^\mu \partial_{\mu} - m)\psi \qquad(1.4)\] where \(\gamma^\mu\) are the gamma matrices and \(\bar{\psi}=\psi^\dag \gamma^0\) is the spinor adjoint. As the chosen naming for the previous Lagrangian \(\mathcal{L}_\textrm{Dirac}\) gave away, the Euler-Lagrange relation obtained by minimising the action \(\delta S=0\) can be used to obtain field equations of motion that correspond to the Dirac equation [4] for the spinor field and its adjoint: \[ i\gamma^\mu \partial_{\mu} \psi - m \psi=0 \quad \textrm{and} \quad i\gamma^\mu \bar{\psi} \partial_{\mu} + m\bar{\psi}=0 \qquad(1.5)\] as well as the well-known Klein-Gordon equation component-wise \((\partial^\mu \partial_\mu + m^2)\psi=0\), where \(\partial^\mu=\partial/\partial x^\mu\). Both Dirac and Klein-Gordon equations were proposed in the context of a relativistic formulation of quantum mechanics.
To shed some light on how a field like \(\psi\) can represent actual fermions in the universe, such as electrons or positrons, the field can be quantised by considering a plane wave expansion and defining annihilation operators \(a_{\boldsymbol{p}}^s\) and \(b_{\boldsymbol{p}}^s\), as well as creation \(a_{\boldsymbol{p}}^{s\dagger}\) and \(b_{\boldsymbol{p}}^{s\dagger}\) operators. The field and its adjoint, which can be thought of directly as operators instead of fields in this context, may then be expressed as:
\[ \psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\boldsymbol{p}}}} \sum_s \left ( a_{\boldsymbol{p}}^s u^s(p) e^{-ipx} + b_{\boldsymbol{p}}^{s\dagger} u^s(p) e^{ipx} \right ) \qquad(1.6)\]
\[ \bar{\psi}(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\boldsymbol{p}}}} \sum_s \left ( b_{\boldsymbol{p}}^s \bar{v}^s(p) e^{-ipx} + a_{\boldsymbol{p}}^{s\dagger} \bar{u}^s(p) e^{ipx} \right ) \qquad(1.7)\]
where \(u^s (p)\) and \(v^s(p)\) and its adjoints are the free particle solutions of the Dirac equation, \(s\) is their spin and \(E_{\boldsymbol{p}}\) their energy. The operators in the previous quantisations can be used to define arbitrary many-particle states. The vacuum state \(\left|{0}\right\rangle\) can be defined as the state for which \(a_{\boldsymbol{p}}^s\left|{0}\right\rangle=b_{\boldsymbol{p}}^s\left|{0}\right\rangle=0\). A single free fermion state of momenta \(\boldsymbol{p}\) and spin \(s\) can be obtained by applying the creation operators on the vacuum state \(\left|{\boldsymbol{p}, s}\right\rangle = \sqrt{2E_{\boldsymbol{p}}}a_{\boldsymbol{p}}^{s\dagger}\left|{0}\right\rangle\) - or alternatively an anti-fermion if the \(b_{\boldsymbol{p}}^{s\dagger}\) is used instead. Multi-particle free states in momenta representation can analogously be defined by the successive application of creation operators over momenta space.
In particle colliders, we are instead interested in interacting theories rather than free theories, given the we aim to compute total and differential cross sections. Interacting theories can also be characterised by their Hamiltonian density \(\mathcal{H} = \mathcal{H}_\textrm{free}+ \mathcal{H}_\textrm{int}\), which can be expressed as a function the Lagrangian density \(\mathcal{H}= \pi^a \dot{\psi}_a - \mathcal{L}\), where \(\dot{\psi}_a\) is the time derivative of the field and \(\pi^a\) is the conjugate momentum. The Hamiltonian density can divided in \(\mathcal{H}_\textrm{free}\), that is the part corresponding to the free theory, and \(\mathcal{H}_\textrm{int}\) that are the additional terms due to interactions. In interacting theories, time-dependence becomes more important and depends only on the \(\mathcal{H}_\textrm{int}\) component. Additionally, the ground state \(\left|{\Omega}\right\rangle\) can be different in interacting theories from the free theory vacuum state \(\left|{0}\right\rangle\).
Let us denote by \(\left|{i}\right\rangle=\left|{\psi(t \rightarrow - \infty)}\right\rangle\) and \(\left|{f}\right\rangle=\left|{\psi(t \rightarrow + \infty)}\right\rangle\) some arbitrary initial and final multi-particle states, temporarily far before and after the actual interaction being studied happened (i.e. around \(t=0\)), respectively. The observables of interest, which are discussed in Section 1.3, are a function of the transition amplitude \(\left\langle{i}\right| \mathcal{S} \left|{f}\right\rangle\) over all possible initial and final states, where \(\mathcal{S}\) is an operator describing the transition. The transition probability, which is expressed as the modulus square of the amplitude \(|\left\langle{i}\right| \mathcal{S} \left|{f}\right\rangle|^2\), is therefore also a function of \(\mathcal{S}\), fully describing the time-evolution from the initial the final state. The \(\mathcal{S}\) operator may be expressed as a perturbative series using the Dyson expansion: \[ \begin{aligned} \mathcal{S} &= T \left [ \exp \left ( -i \int_{-\infty}^{\infty} d^4 x \mathcal{H}_\textrm{int} (x) \right ) \right ] \\ &= \sum_{n=0}^{\infty} \frac{(-i)^n}{n!} \int_{-\infty}^{\infty} d^4 x_1 ... \int_{-\infty}^{\infty} d^4 x_n T \left [ \mathcal{H}_\textrm{int} (x_1) ... \mathcal{H}_\textrm{int} (x_n) \right ] \end{aligned} \qquad(1.8)\] where \(T\) is an operator ensuring that the Hamiltonian density factors \(\mathcal{H}_\textrm{int} (x_i)\) are ordered in time. Each time-ordered term in the series can be written as a sum of normal (i.e. not time ordered) products of permutations using Wicks theorem [7], which can become rather tedious for high orders. The formalism of Feynman diagrams can be used to simplify the computation of observables at a given order in the perturbative expansion.
Based on the previous perturbative series expansion, the transition amplitude \(\left\langle{i}\right| \mathcal{S} \left|{f}\right\rangle\) can be easily linked with scattering observables when denoted as: \[ \left\langle{i}\right| \mathcal{S} \left|{f}\right\rangle = \left\langle{i}\right| \boldsymbol{1} \left|{f}\right\rangle + i \mathcal{M} (2\pi)^4 \delta^4 \left ( \sum p_i - \sum p_f \right ) \qquad(1.9)\] where the first term corresponds to no interaction occurring, and the second includes the matrix element \(\mathcal{M}\) including all orders in the perturbative orders, and multiplied by a factor making explicit the conservation of momentum between the initial and final state particles. The matrix element \(\mathcal{M}\), which can be computed perturbatively as a function of the momenta of the particles given final state considered, can be used to define the differential cross section: \[ \frac{d \sigma}{d \Phi} \sim | \mathcal{M} |^2 \ \textrm{where} \ d\Phi = (2\pi)^4 \delta^4 \left ( \sum p_i - \sum p_f \right ) \prod_f \frac{1}{2 E_f} \frac{d^3 \boldsymbol{p}_f}{(2\pi)^3} \qquad(1.10)\] where the proportionality factor is a function of the initial state particles momenta and \(d\Phi\) is the full phase space differential element for which can be generally expressed as a product of the final state particle momenta differential elements. Total scattering rates can be obtained by summing over possible initial and final states and integrating over final states. Both differential and total cross sections can be truncated at a given perturbative order. The lowest expansion order is referred as leading order (LO), yet considering additional expansion can greatly increase the prediction accuracy so one (NLO) or two (NNLO) orders are often considered, higher orders often being too computationally challenging. A truncation at an additional order \(n\), relative to the lowest interaction order, will provide corrections proportional to \(\alpha=g^2/(4\pi)\), where \(g\) is the coupling constant characteristic of the interaction.
1.1.2 Quantum Chromodynamics
In a hadron collider such as the LHC, strong interactions between quark and gluons are dominant, and they can be modelled using quantum chromodynamics (QCD). The theory of QCD can be linked to a \(SU(3)\) symmetry group and is described by the following gauge invariant Lagrangian density:
\[ \mathcal{L}_\mathrm{QCD} = \bar{\psi} \left( \gamma^\mu D_\mu - m_f \right) \psi - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a, \quad \psi = \begin{bmatrix} \psi_r \\ \psi_g \\ \psi_b \end{bmatrix} \qquad(1.11)\]
where \(\psi\) is a spinor quark field for a given flavour \(f \in \{ \textrm{u}, \textrm{d}, \textrm{s}, \textrm{c}, \textrm{b}, \textrm{t}\}\) and quark mass \(m_f\), and each vector component represents a colour degree of freedom. Assuming that the Gell-Mann matrices \(\lambda^a\) are used to define a basis for the gluon field \(A_\mu = 1/2 \lambda^a \sum A_\mu^a\), the covariant derivative can be defined as \(D_\mu= \partial_\mu - i g_s \, A_\mu\), where \(g_s\) is the strong interaction coupling. In turn, the gluon field strength tensor \(G^a_{\mu \nu}\) is also related with the gluon field components:
\[ G^a_{\mu \nu} = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g_s f^{abc} A_\mu^b A_\nu^c \qquad(1.12)\] where \(f^{abc}\) are the structure constants of the \(SU(3)\) gauge group. The last term accounts for the self-interaction of the gluon, which are the massless and electrically neutral mediators of the strong force. There are two properties of QCD that play an important role from a phenomenological standpoint: confinement and asymptotic freedom.
The property of confinement has been postulated to explain why isolated quarks and gluons are not found in nature. Quarks have only been found as part of hadrons, that are colour-neutral composite particles. Even though confinement has not been understood from first principles, because the observables of bound states in QCD at low-energies cannot be computed in a perturbative manner, there exist extensive evidence both from lattice QCD calculations and experiments. In a bound state between quarks, the effective potential includes a term that increases proportional to their distance, so when the quarks are separated by an external energetic interaction, the additional potential energy generates an additional quark-antiquark pair, leading to the formation of bound states. Similar phenomena occur for isolated gluons, which generally are referred as hadronization, and can be understood as a consequence of colour confinement. In particle colliders, successive hadronization and radiation processes led to parton showers (see Section 1.3.4).
Quark are then only found in bound states, referred to as hadrons, which can either be mesons or baryons. Mesons are formed by quark-antiquark pairs \(\textrm{q}\bar{\textrm{q}}\), while baryons are composed of three quarks \(\textrm{q}\textrm{q}\textrm{q}\). Charged and neutral pions \(\pi^{+}\) (\(\textrm{u}\bar{\textrm{d}}\)) and \(\pi^{0}\) (\((\textrm{u}\bar{\textrm{u}}-\textrm{d}\bar{\textrm{d}})/\sqrt{2}\)), kaons \(\textrm{K}^{+}\) (\(\textrm{u}\bar{\textrm{s}}\)) and \(\textrm{K}^{0}\) (\(\textrm{d}\bar{\textrm{s}}\)) and the \(\textrm{J}/\Psi\) (\(\textrm{c}\bar{\textrm{c}}\)) are among the most common mesons produced at particle colliders. Baryons instead include the well-known proton (\(\textrm{u}\textrm{u}\textrm{d}\)) and neutron (\(\textrm{u}\textrm{d}\textrm{d}\)) that together with electrons are the constituents of most of the known matter in the universe. Many more short-lived baryons exist [8], in addition to the recently discovered exotic bound states referred as tetraquarks [9] and pentaquarks [10]. A detailed description of the compositeness of proton is an essential element for computing LHC observables, as reviewed in Section 1.3.2.
Asymptotic freedom is instead linked with the strength reduction of the strong coupling constant when higher energy scales are considered. Let us consider a renormalisation energy scale \(\mu_R^2\), which has to be often defined in order to compute physical observables which otherwise would be divergent due higher order perturbative corrections which cannot be easily calculated. This effect can be also understood as a coupling that varies with the energy scale, which is referred to as a “running” coupling constant. The strong force coupling \(\alpha_s=g_s^2/(4\pi)\) can thus be approximated as a function of the renormalisation energy scale \(\mu_R^2\) as follows: \[ \alpha_s(\mu_R^2) = \frac{\alpha_s(\mu_0^2)}{1+\alpha_s(\mu_0^2) \frac{33-2n_f}{12\pi} \ln \left ( \frac{\mu_R^2}{\mu_0^2} \right)} \qquad(1.13)\] where \(\alpha_s(\mu_0^2)\) is the measured coupling at a given energy and \(n_f\) is total number of quark flavours which are assumed to be massless in this approximation. The strong interaction thus becomes weaker at higher energies (or short distances) allowing the perturbative computation of observables related with high-energy interactions, as discussed in Section 1.3. The approximation from Equation 1.13 also provides a lower bound for the energy scale at which QCD can be treated perturbatively, i.e. the denominator becomes zero for an energy scale around \(200\ \textrm{MeV}\), leading to a diverging coupling constant.
1.1.3 Electroweak Interactions
The remaining two fundamental interactions between elementary particles are the electromagnetic and the weak force. The description of the electromagnetic interaction in terms of quantum fields and gauge symmetries, leading to the development of quantum electrodynamics (QED) in the late 1940s, prompted a quest for an analogous theory for the weak force. The weak force, known to be responsible for the beta decay at the time, could effectively be modelled using Fermi theory using four-fermion interactions [11] but was not renormalisable and lacked the predictive capabilities and elegance of QED. A large theoretical effort lead to an alternative description based on a \(SU(2) \otimes U(1)\) symmetry, which unified electromagnetic and weak interactions [12], [13], and where the weak interaction was mediated by means of charged \(W^{\pm}\) and neutral \(Z\) massive vector bosons. Nevertheless, the theory did not provide an explanation for the mass of the weak mediators, until the so-called Brout-Englert-Higgs [14]–[16] mechanism for spontaneous symmetry breaking (SSB) was conceived. Higgs also noted explicitly that the mechanism would effectively create an additional scalar field, associated with a new scalar boson, whose existence could experimentally testable. The SSB mechanism was then combined with \(SU(2) \otimes U(1)\) unified theory [17] to give rise to what is now known as electroweak theory, which was then proved to be renormalisable [18].
The different testable properties of electroweak phenomena were verified by experiments including the existence of weakly-interacting neutral and charged currents [19] and the discovery of the massive \(W^{\pm}\) [20], [21] and \(Z\) [22], [23] bosons. Experimental evidence also showed that weak interactions were parity violating [24], thus in the electroweak theory the fermion fields are separated in their left-handed \(\psi_\textrm{L}\) and right-handed \(\psi_\textrm{R}\) chiral components as follows: \[ \psi_\textrm{L} = \textrm{P}_\textrm{L} \psi= \frac{1}{2} (1 - \gamma_5) \psi \quad \psi_\textrm{R} = \textrm{P}_\textrm{R} \psi = \frac{1}{2} (1 + \gamma_5) \psi \qquad(1.14)\] where \(\textrm{P}_\textrm{L}\) and \(\textrm{P}_\textrm{R}\) are the chiral projection operators and \(\gamma_5=i \gamma_0\gamma_1\gamma_2\gamma_3\) is the product of the gamma or Dirac matrices. For massless particles, chirality is equal to the helicity \(H=(\boldsymbol{p} \cdot \boldsymbol{s}) / | \boldsymbol{p} |\) which is the sign of the scalar product of momenta and spin. For massive particles, chirality is still defined but is not identical to helicity which cannot be invariantly defined.
Within the electroweak theory, fermion fields are broken in into their left-handed components, which can be expressed as doublets that would transform under \(SU(2)\), and can be denoted as: \[ L_q = \left \{ \begin{pmatrix} u \\ d \end{pmatrix}_L , \begin{pmatrix} c \\ s \end{pmatrix}_L , \begin{pmatrix} t \\ b \end{pmatrix}_L \right\} \quad L_l = \left \{ \begin{pmatrix} \nu_e \\ e\end{pmatrix}_L , \begin{pmatrix} \nu_\mu \\ \mu \end{pmatrix}_L , \begin{pmatrix} \mu_\tau \\ \tau \end{pmatrix}_L \right\} \qquad(1.15)\] and their right handed components, that instead can be expressed as singlets only transforming under \(U(1)\): \[ R_u = \left \{ u_R, c_R, t_R \right \} \quad R_d = \left \{ d_R, s_R, b_R \right \} \quad R_l = \left \{ e_R, \mu_R, \tau_R \right \} \qquad(1.16)\] where the right-handed neutrino components are omitted in the electroweak theory (and the SM), given they are electrically neutral and would not interact weakly when right-handed.
The electroweak interactions then can be made explicit by introducing additional boson fields \(W= \{ W^1, W^2, W^3 \}\) and \(B\) which will interact with the fermions. Similarly in structure to QED (and also QCD as described in Section 1.1.2), the electroweak Lagrangian before spontaneous symmetry breaking is composed by interaction terms for the previous doublet and singlet fields, characterised by a covariant derivative, and kinematic terms for both boson fields: \[ \begin{aligned} \mathcal{L_\textrm{EW}} = & \sum^{\psi \in \{ L_q, L_l \}} \bar{\psi} (i \gamma_\mu D_L^\mu)\psi + \sum^{\psi \in \{ L_q, L_l \}} \bar{\psi} (i \gamma_\mu D_R^\mu)\psi \\ & - \frac{1}{4} W_{\mu\nu}W^{\mu\nu} - \frac{1}{4} B_{\mu\nu}B^{\mu\nu} \end{aligned} \qquad(1.17)\] where the covariant derivatives for left-handed \(D_L^\mu\) and right-handed \(D_R^\mu\) fermion fields are respectively defined as: \[ \begin{aligned} & D_L^\mu = \partial^\mu - \frac{1}{2} g_B Y B_\mu - \frac{1}{2} g_W \sigma W_\mu \\ & D_R^\mu = \partial^\mu - \frac{1}{2} g_B Y B_\mu \end{aligned} \qquad(1.18)\] where \(\sigma = \{ \sigma_1 , \sigma_2, \sigma_3\}\) are the Pauli matrices and \(g_B\) and \(g_W\) are the coupling constants. The \(W_{\mu\nu}\) and \(B_{\mu\nu}\) field strength tensors from kinematic terms can in turn be obtained as: \[ \begin{aligned} & W^i_{\mu\nu} = \partial_\mu W^i_\nu - \partial_\mu W^i_\mu - g_W \epsilon^{ijk} W^i_\mu W^k_\nu \\ & B_{\mu\nu} = \partial_\mu B_\nu - \partial_\mu B_\mu \end{aligned} \qquad(1.19)\] where \(\epsilon^{ijk}\) is the Levi-Civita symbol for each permutation, which is the structure constant for \(SU(2)\).
1.1.4 Symmetry Breaking and the Higgs Boson
The problem with the electroweak theory as described by the Lagrangian from Equation 1.17, which is based on Yang-Mills gauge theory formulation, is that it is not possible to directly add mass term for the fermions nor the weak bosons to the Lagrangian density without breaking the \(SU(2)\) invariance. At the time the mentioned theory was developed, there was extensive evidence not only for lepton masses but also for the weak bosons being massive; the mass required to explain why the weak interaction was short-ranged. The issue of lacking a theoretical mechanism that could explain the mass of fermions and weak boson was solved by the spontaneous symmetry breaking mechanism [14]–[16], which is based on postulating the existence of an additional complex scalar field \(\phi\), which is a \(SU(2)\) doublet with the following structure: \[ \phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix} = \begin{pmatrix} \phi_3 + i\phi_4 \\ \phi_1 + i\phi_2 \end{pmatrix} \qquad(1.20)\] where we made the component notation explicit because it will be relevant later. This scalar field is expected to interact with the electroweak fields \(W\) and \(B\) by means of the following Lagrangian: \[ \mathcal{L}_\textrm{scalar} = (D^H_\mu \phi)^\dagger (D^\mu \phi) - V(\phi) \qquad(1.21)\] where the covariant derivate in this case is defined as: \[ D_H^\mu = \partial^\mu - \frac{1}{2} i g_B Y B_\mu - \frac{1}{2} i g_W \sigma W_\mu. \qquad(1.22)\] The minimal form for a scalar field potential \(V(\phi)\), constructed ad-hoc to provide a degenerate vacuum states and a local maximum - a required condition for spontaneous symmetry breaking, may be expressed as: \[ V(\phi) = - \mu^2 \phi^\dagger \phi + \frac{1}{2} \lambda (\phi^\dagger \phi )^2 \qquad(1.23)\] where both the quadratic \(\mu^2\) and the quartic \(\lambda\) self-interaction parameters are defined positive with this sign convention. The resulting shape for the potential is often referred as mexican hat, and is depicted in Figure 1.2. The presence of a potential minimum different from the origin gives rises to a non-zero vacuum expectation value for the scalar field: \[ \langle \phi \rangle_0 = \frac{\mu^2}{\lambda} = v^2 \qquad(1.24)\] whose values depend on the \(V(\phi)\) potential parameters \(\mu^2\) and \(\lambda\), and it is denoted as \(v^2\) for convenience. The non-zero vacuum expectation value is thus said to spontaneously break the the \(SU(2) \otimes U(1)\) symmetry, the consequences made more clear when the field is expanded around the minimum: \[ \phi = \frac{1}{\sqrt{2}} \exp(i \frac{\sigma \cdot G}{v}) \begin{pmatrix} 0\\ v + H \end{pmatrix} \qquad(1.25)\] as a product of a scalar field \(H\) and a complex exponential of the scalar product of a three-component field \(G=\{G_1, G_2, G_3\}\) with the Pauli matrices \(\sigma=\{\sigma_1, \sigma_2, \sigma_3\}\). The complex exponential phase can be then removed by a \(SU(2)\) group rotation, a transformation that is often referred as unitary gauge. The resulting scalar field can simply be expressed as: \[ \phi = \frac{1}{\sqrt{2}} \begin{pmatrix} 0\\ v + H \end{pmatrix} \qquad(1.26)\] where three of the four degrees of freedom in Equation 1.20, which correspond the field \(G\) which would otherwise give rise to the so-called Goldstone bosons, have been removed after the gauge transformation.
Substituting the rotated scalar field from Equation 1.26 in the Lagrangian described by Equation 1.21 leads to mass-like terms for linear combinations of the \(W\) and \(B\) fields. In order to obtain the physical bosons observed in nature, the mass terms have to be made independent by the following transformations: \[ W^\pm_\mu = \frac{1}{\sqrt{2}} \left ( W_\mu^1 \mp i W_\mu^2 \right) \quad \begin{pmatrix} Z_\mu \\ A_\mu \end{pmatrix} = \begin{pmatrix} \cos \theta_W & -\sin \theta_W \\ \sin \theta_W & \cos \theta_W \end{pmatrix} \begin{pmatrix} W^3_\mu \\ B_\mu \end{pmatrix} \qquad(1.27)\] where the fields \(W^+\) and \(W^-\) are associated with the charged weak bosons, the field \(Z\) with the neutral weak boson, the electromagnetic field \(A\) with the photon, and \(g_W\) is the Weinberg angle which is related with the electroweak couplings according the relation \(\tan \theta_W = g_B/g_W\). Omitting for now the terms related with the \(H\) field, the Lagrangian in Equation 1.21 leads to the following mass terms for the electroweak force mediators after the unitary gauge and the transformation described in Equation 1.27 have been applied: \[ \begin{aligned} \mathcal{L}_\textrm{EW bosons} =& \frac{1}{2} \underbrace{\left ( \frac{g_W^2 v^2}{4} \right )}_{m_{W^{+}}^2} W^{+}_{\mu} W^{+\mu} + \frac{1}{2} \underbrace{\left ( \frac{g_W^2 v^2}{4} \right )}_{m_{W^{-}}^2} W^{-}_{\mu} W^{-\mu} + \\ & \frac{1}{2} \underbrace{\left ( \frac{g_W^2 v^2}{4\cos \theta_W} \right )}_{m_Z^2} Z_{\mu} Z^{\mu} + \frac{1}{2} \underbrace{(\ 0\ )}_{m_{\gamma}^2} A_{\mu} A^{\mu} \end{aligned} \qquad(1.28)\] resulting in mass terms for the massive weak bosons which depend to the weak coupling, the Weinberg angle and the vacuum expectation value of the Higgs field. The last term for the electromagnetic field has only been included to make explicit that no mass term is associated with the electromagnetic force carrier \(\gamma\). The terms related with the scalar \(H\) field (and Higgs boson) are discussed later independently.
In addition to providing a mechanism that leads to mass terms for the weak force bosons, additional interactions of the various fermion fields with the scalar field \(\phi\) can explain their masses. These gauge invariant terms are generally referred to as Yukawa interactions, and correspond to the following Lagrangian terms: \[ \begin{aligned} \mathcal{L}_\textrm{Yukawa} = &- \lambda_l (\bar{L}_l \phi R_l + \bar{R}_l \phi^\dagger L_l ) \\ &- \lambda_d (\bar{L}_q \phi R_d + \bar{R}_d \phi^\dagger L_q )\\ &- \lambda_u (\bar{L}_q i \sigma_2 \phi^\dagger R_u + \bar{R}_u i \sigma_2 \phi L_q ) \end{aligned} \qquad(1.29)\] where \(\lambda_l\) , \(\lambda_d\) and \(\lambda_u\) are the Yukawa coupling parameters. A charge-conjugate transformation \(\phi \rightarrow i \sigma_2 \phi^\dagger\) is used to give mass to up-type quarks. For the quark sector, the \(\lambda_u\) and \(\lambda_d\) couplings can be expressed by a single non diagonal matrix in the flavour basis, referred to as Cabibbo-Kobayashi-Maskawa (CKM matrix) [25], [26], which can in turn be parametrised by three angles and a complex phase. The fact that the matrix is not diagonal leads to flavour mixing, due to the mass eigenstates being different from flavour eigenstates. Another relevant property of fermion masses is that after spontaneous symmetry breaking, the fermion mass is effectively proportional to its coupling with the Higgs scalar field, which is useful to intuitively understand the dominant interactions and decays of the Higgs boson.
Figure 1.2: Graphical depiction1 of the mexican hat potential for the scalar field \(\phi\). A local maximum is present at the origin, but lower energy degenerate minima exist arount it.
In addition of giving masses to both weak bosons and fermions, the remaining degree of freedom after electroweak symmetry breaking gives rise to a scalar field \(H\). The terms of the Lagrangian concerning only \(H\) may be obtained substituting Equation 1.26 in Equation 1.21, leading to the following expression: \[ \mathcal{L}_{H} = \frac{1}{2} \partial_\mu H \partial^\mu H - \mu^2 H^2 - \lambda v H^3 - \frac{\lambda}{4} H^4 \qquad(1.30)\] where the second (quadratic term) can be interpreted as a scalar boson with a mass \(\sqrt{2\mu^2}\), which is commonly referred as the Higgs boson. A particle with a mass of \(125.09(24)\ \textrm{GeV}\) [27] and consistent with the expected properties for the Higgs boson was discovered in 2012 by the CMS and ATLAS collaborations [2], [3]. The cubic \(\lambda v\) and quartic \(\lambda\) terms will give rise to self-interaction interaction vertices. The so-called cubic or trilinear Higgs coupling is discussed in a Higgs pair search using data from the CMS experiment in Chapter 5. The direct determination of the Higgs self-coupling is an relevant missing piece, and an important proof of consistency of the spontaneous symmetry breaking mechanism.
The figure was created by adapting the code from this TeX StackExchange answer.↩