---
title: Brief Review of the Standard Model
...


The Standard Model of Particle Physics (SM) stands as one of the most predictive and insightful scientific theories ever written. It is the culmination of a hundred years of intense theoretical exploration and experimental tests. It can successfully explain phenomena ranging from nuclear decay and the structure of atoms to the behavior of cosmic ray showers. Included in the theory are three fundamental forces. The first and most familiar is the electromagnetic force which is mediated by the photon and which all particles with electric charge participate. The second is the strong nuclear force. The strong force is mediated by the gluon which controls the interaction between all *colored* particles. It is this force that is responsible for binding quarks together into mesons and baryons, as well as binding proton's and neutrons together into atomic nuclei. Finally, the weak nuclear force, which is mediated by the W and Z bosons, and is responsible for nuclear $\beta$-decay as well as more exotic processes such as interactions with the ghostly neutrino particle and the decay of the top quark.

Notably, the Standard Model is completely unable to explain why apples fall from trees or why the earth orbits the sun since it does not incorporate the gravitational force. The current most complete theory of gravity is General Relativity. However, despite large and ongoing efforts to unite General Relativity with the Standard Model, no resulting theories have had the necessary self-consistency and predictive power to be accepted. Luckily for those working on collider experiments, the effect of gravity is so overwhelmed by the other three fundamental forces that it can be largely ignored. The constituent particles of the SM are shown in [@fig:sm_particles]. These particles are divided into several categories based on some fundamental properties.

# Fundamental Particles

The *fermions* are defined by their half-integer spin of $½\hbar$ and are split between the leptons and the quarks. They are fundamentally described as Dirac spinor fields which in the case of non-interacting massive particles are described by the Lagrangian

$$
\mathcal{L} = \bar{\psi}\left(i\partial\!\!\!/ - m\right)\psi
$$

in natural units, $\hbar=c=1$. The $\partial\!\!\!/$ operator is defined as $\gamma^\mu\partial_\mu$ in Einstein notation where $\gamma^\mu$ are the 4x4 $\gamma$-matrices that satisfy the anti-commutation relations $\left\{\gamma^\mu,\gamma^\nu\right\}=2\eta^{\mu\nu}$ and $\eta^{\mu\nu}$ is the Minkowski metric. Other fermions and interactions can be added to the theory with additional terms, as will be discussed in subsequent sections.

The leptons come in three generations with each generation having an electrically charged member and a neutral member. The charged members are the electron, muon, and tauon (or simply tau), while the neutral members are their three corresponding neutrinos. The electrically charged leptons participate in the electromagnetic and weak interaction, while the neutrinos only participate in the weak interaction. Another feature of the leptons is that while the charged leptons have non-trivial masses, the neutrinos are nearly massless. In fact, they were originally believed to be massless, until the observation of neutrino-mixing implied that at least two of them must have mass. The quarks also come in three generations with two members in each generation. In contrast with the leptons, however, the quarks all carry electric charge. The "up-type" quarks carry charge $+2/3$ while the "down-type" quarks carry charge $-1/3$. The quarks are also *colored* which means that in addition to participating in the electromagnetic and weak interactions, they also interact via the *strong* nuclear force. Each of the six quarks has non-trivial mass, but they are wildly different, running from 2.4 $\mathrm{MeV}/\mathrm{c}^2$ for the up quark all the way to 171 $\mathrm{GeV}/\mathrm{c}^2$ for the top quark, the heaviest fundamental particle in the SM. 

The remaining particles are the bosons, four force-mediators and the Higgs. These will be discussed in the following sections in context with their respective forces and contributions to the SM.

![The particles of the SM](figures/theory/sm_particles.png){#fig:sm_particles}

# Quantum Electrodynamics

One of the early triumphs of quantum field theory (QFT) was the description of electromagnetism developed by Tomonaga[@Tomonaga1946], Schwinger[@Schwinger1948], Feynman[@Feynman1949], and Dyson[@Dyson1949] and eventually coming to be known as quantum electrodynamics (QED). This theory describes the interaction of electrically charged particles with photons. In QED photons are described by a 4-vector field, $A^\mu$, which is used to define the field strength tensor $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. This field, in the absence of charged particles, is governed by the Maxwell Lagrangian

$$
\mathcal{L}_{\mathrm{Max}} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}
$$

Fermion interactions are added with 

$$
\mathcal{L}_{\mathrm{Int}} = \bar{\psi}\left(iD\!\!\!\!/ - m\right)\psi + \mathrm{h.c.}
$$

Where $D\!\!\!\!/ \equiv \gamma^{\mu}\left(\partial_\mu + iqA_\mu\right)$ and h.c. is just the hermitian conjugate of the previous term. Putting these together yields the QED Lagrangian.

$$
\mathcal{L}_{\mathrm{QED}} = i\bar{\psi}\gamma^\mu\partial_\mu\psi - q\bar{\psi}\gamma^\mu A_\mu\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}
$$

QED is a gauge theory. This comes from the assumption that the observable is not the potential $A^\mu$, but only the fields it defines. This leads one to ask what transformations can be applied to $A^\mu$ that leave the fields invariant. These *gauge transformations* define symmetries of the theory. In the language of group theory, QED obeys $U(1)$ symmetry. The transformation associated with this symmetry that leaves $\mathcal{L}_{\mathrm{QED}}$ unchanged is:

$$
A^\mu \rightarrow A^\mu + \partial^\mu\chi, \psi \rightarrow e^{-iq\chi}\psi
$$

Enforcing some local gauge symmetry upon a Lagrangian severely limits the form of the terms that can appear. For example, a mass term for the photon field of the form $m_\gamma A^\mu A_\mu$ is forbidden. This is acceptable in QED since the photon is massless, but will prove to be an issue for the massive bosons that come with the weak force, as will be discussed shortly.

## Feynman Diagrams and Cross Sections

Processes in perturbative quantum field theory are often represented using Feynman diagrams. These are graphs that encode the interactions of particles in an intuitive way. A illustrative example is Compton scattering. In this process, a photon is incident on a charged fermion. Experimentally, this is normally an X-ray or gamma ray incident upon an electron in some atomic orbital. The electron absorbs the photon and then re-emits it with a smaller momentum (longer wavelength) than it started with. It is technically possible for the photon to be re-emitted with a larger momentum, but this is referred to as anti-Compton scattering. In any case, the process can be represented by the two diagrams in [@fig:compton_scattering]. There is a formalism for translating these diagrams into a precise calculation that can be performed analytically for simple cases, but often require the assistance of numerical methods. A description of the full formalism is beyond the scope of this discussion, but briefly:

  - Each *vertex* in the graph contributes a factor proportional to the electromagnetic coupling constant $g_e$, a small number.
  - Photons and fermions contribute individual *propagators* that involve their momenta and, in the case of the fermions, their rest masses.
  - Performing the calculation gives the *amplitude* of that diagram, which is a function of the incoming and outgoing particles' momenta and spins. The next step is to add together all of the diagrams under consideration, and then take the absolute square.
  - Collider experiments are typically not sensitive to spin, so an extra step involves averaging over the initial spin states and summing over the final spin states.

![Leading order Feynman diagrams for Compton scattering. The topology of the left diagram is conventionally referred to as "s-channel" while the right's is called "t-channel". Initial particles are on the left of each diagram, and final state particles are on the right.](figures/theory/Compton_Scattering.png){#fig:compton_scattering}

The precise rules are derived from the Lagrangian describing the system using the tools of QFT. The result of this full calculation is the *differential cross section*. This can be interpreted as a probability distribution where, given some initial state particles, it can predict the momentum distributions of the final state particles. Integrating the differential cross section over all initial and final states gives the *cross section*. The diagrams of [@fig:compton_scattering] are just the two *leading order* (LO) diagrams, meaning that of all possible diagrams, these have the smallest possible number of vertices. Since each vertex contributes additional factors of $g_e$, diagrams with additional vertices give smaller and smaller individual contributions to the total cross section. However, the issue is complicated by the fact that the total number of diagrams at each order of $g_e$ increases, and the manner of interference between the diagrams is often not clear without doing the full calculation. This is why precise predictions require the explicit inclusion of more complicated diagrams. The set of diagrams that include the next fewest number of vertices are referred to as *next-to-leading order* (NLO), and beyond that are the *next-to-next-to-leading order* (NNLO) diagrams which are sometimes included for very precise calculations. 

The cross section of any particular process depends upon the conditions facilitating that process. In hadron colliders, the proton's *parton distribution function* (PDF) together with the center-of-mass energy (normally referred to as $\sqrt{s}$) are used to perform the aforementioned integral over initial states. A PDF is the probability density that a constituent particle of a proton carries a certain fraction of the protons total energy. This fraction is called the Bjorken $x$. [@Fig:proton_pdf] shows the PDF for protons at different momentum transfer scales.

![Parton distribution functions for two values of momentum transfer, $\mu$. The $\mathrm{u}_v$ and $\mathrm{d}_v$ refer to the proton's valence up and down quarks. Other quark flavors exist through the temporary splitting of gluons into quark-antiquark pairs.[@Olive_2016]](figures/theory/proton_pdf.png){#fig:proton_pdf}

# Quantum Chromodynamics

The theory describing the interaction of colored particles is quantum chromodynamics (QCD). The only colored fermions, quarks, were first proposed by Gell-Mann[@GellMann1964] and Zweig[@Zweig:1964jf] in 1964 to aide in the classification of hadrons, and were first observed four years later at SLAC[@Bloom1969]. In contrast with QED, QCD obeys the non-Abelian SU(3) symmetry. However, just as $A^\mu$ was used to define a field strength tensor in QED, the eight *gluon* fields each define their field strength tensor as

$$
G_{\mu\nu}^{a} = \partial_\mu G_\nu^a - \partial_\nu G_\mu^a - g_s f_{abc}G_\mu^b G_nu^c
$$

where $g_s$ is the strong coupling constant and the $f_{abc}$ are the structure constants of the symmetry group defined by the commutation relations

$$
\left[\frac{\lambda_a}{2},\frac{\lambda_b}{2}\right] = i f_{abc} \frac{\lambda_c}{2}
$$

where $\lambda_a$ are the Gell-Mann matrices. The quarks, similar to the fermions in QED, are represented as spinor fields governed by the Lagrangian

$$
\mathcal{L}_q = \bar{q_f}\left(i\gamma^{\mu}\partial_\mu - m\right)q_f
$$

for a single quark. In $\mathcal{L}_q$, $q_f = (\psi_1, \psi_2, \psi_3)$ where each $\psi$ represents a color components and there is an implied sum over all three. Interactions between the quarks and the gluons are introduced by introducing the covariant derivative $D_\mu$ into $\mathcal{L}_q$ in place of $\partial_mu$ where

$$
D_mu = \partial_\mu + ig_s\frac{\lambda_a}{2}G^a_\mu.
$$

Doing this yields the QCD Lagrangian.

$$
\mathcal{L}_\mathrm{QCD} = \sum_f \bar{q}_f\left(i\gamma^\mu\partial_\mu - m\right)q_f - \frac{1}{4}G_a^{\mu\nu}G^a_{mu\nu} - g_s \bar{q}_f \gamma^\mu \frac{\lambda_a}{2}q_f G^1_\mu
$$

The one-loop beta function for QCD in the Standard Model is

$$
\beta\left(g\right) = -\frac{7\alpha_s^2}{2\pi}
$$

The fact that the beta function is negative implies that the coupling decreases with energy scale, or equivalently, increases with larger length scales. This leads to the property of asymptotic freedom, which simply refers to the shrinking coupling as interaction length scales shorten. The coupling constant being small at high energies also means that one can successfully apply perturbation theory as one would in QED in that context. However, at low energies, such as those of most bound states, perturbative approaches fail and other approaches, such as *Lattice QCD* are applied instead. Quarks and gluons are also subject to color confinement, which means that any observable particle must be color neutral. This can be achieved by having equal quantities of the three colors, conventionally called red, blue, and green. Such is the case for *baryons*, with three quarks carrying one of each color charge. Color neutrality can also be achieved via a quark-antiquark pair possessing the same color, e.g. a green quark and an green anti-quark together make up a colorless *meson*. Colorless combinations of more than three quarks are possible, but not normally encountered. Colorless pairings of gluons, or *glueballs*, are also possible, but have not been observed experimentally.

A consequence of these properties is the hadronization and fragmentation of high energy quarks and gluons into *jets*. Fragmentation is the process by which an energetic quark or gluon, generally resulting from a hard collision, radiates a cascade of less energetic particles, and hadronization is the process where these particles become bound together into a jet of hadrons. However, this process does not occur for the top quark for reasons that will be explored in the next section.


# The Electroweak Interaction

In 1934, Fermi proposed an explanation for nuclear beta decay, $n \rightarrow p e^- \bar{\nu_e}$[@Fermi1934]. This involved a new interaction will come to be identified as the weak nuclear force. This was initially proposed to be a point-like interaction associated with a dimensionful constant $G_F$, but over the next few decades experiments at higher energies would show that this was just a low energy approximation. In reality this process is mediated by a new force mediator, the W boson. However, prior to the discovery of this new particle, in 1956 Yang and Lee suggested that *parity* is not conserved in weak interactions, based on kaon decays[@Lee1956]. The next year, this was confirmed in studies of nuclear beta decays by Wu[@Wu1957]. Conservation of parity, roughly speaking, means that physical arrangements that are unchanged by a reflection about some plane will continue to remain unchanged by the reflection as the system evolves. This property holds for QED and QCD, but not for weak interactions.

![Leading order Feynman diagram for nuclear $\beta$ decay](figures/theory/beta_decay.png){#fig:beta_decay width=50%}

Fundamentally, this follows from the fact that the weak force acts differently on particles of different *chiralities*. Chirality is a Lorentz invariant quantity that corresponds to eigenvalues of the operator $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$. Any field can be divided into chiral states with eigenvalues of +1, known as right-handed, and -1, known as left-handed. Chirality is related to the more familiar quantity of *helicity*. Helicity is the projection of a particle's spin along the direction of its momentum. Helicity, however, is not a Lorentz invariant quantity because a change of reference frame, or *boost*, can reverse the direction of the momentum, and with it the sign of the helicity. However, for massless particles one cannot reverse the momentum with a boost and the helicity and chirality are equivalent.

In the 1960's, work by Glashow[@Glashow1961], Salem, Ward[@Salam1964], and Weinberg[@Weinberg1967] showed that within the unified framework of their *electroweak* theory, a new spin-1 electrically charged particle should exist to mediate the weak force. This new particle was discovered at CERN in 1983 by the UA1[@Arnison1983a] and UA2[@Banner1983] experiments. Its mass is currently measured to be $M_W=80.379\pm 0.012\mathrm{GeV}/c^2$[@PhysRevD.98.030001]. The fields associated with the W boson, $W_\mu^\pm$ couple only to the left components of fermion fields, implying that only left-handed particles and right-handed anti-particles can interact with the $W^\pm$ bosons. The symmetry group obeyed by this electroweak theory that generates these properties is $SU(2)_L\otimes U(1)_Y$.

The weak eigenstates of the fermions are not the same as the mass eigenstates, but rather combinations of them. This mixing is described by the CKM (Cabibbo-Kobayashi-Maskawa) matrix for quarks and the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix for the leptons. The top quark, in particular, due its large mass, can decay weakly via the process $t \rightarrow W^+ b$. This process happens on the timescale of $10^{-25}$ s which is smaller than the timescale necessary for top-flavored hadrons or $t\bar{t}$-quarkonium bound states to form[@Bigi1986]. In addition, because $|V_{tb}| \gg |V_{ts}|,|V_{td}|$ in the CKM matrix, the top quark will predominantly decay into a $W$ with a bottom quark, rather than with a strange or down quark. The bottom quark will form a hadron and then eventually fragment, forming a jet. The W will immediately decay into either a quark anti-quark pair, resulting in two jets, or a charged lepton and a neutrino. The former will be referred to as *hadronically* decaying tops while the latter will be *leptonically* decaying. 

The electroweak theory also predicted the existence of an electrically neutral spin-1 particle. This new particle, called the Z boson, was also observed by the UA1[@Arnison1983b] and UA2[@Bagnaia1983] experiments at CERN, and its mass is currently measured to be $91.1876\pm 0.0021$[@PhysRevD.98.030001].


# Electroweak symmetry breaking and the Higgs field

 As mentioned previously, massive bosons are incompatible with local gauge invariance. This was clearly at odds with the observation of the W and Z bosons. A potential solution to this was proposed by Higgs[@Higgs1964], Brout, Englert[@Englert1964], and Guralnik, Hagen, and Kibble[@Guralnik1964] in 1964. This solution has come to be known as *spontaneous symmetry breaking*, or the *Higgs mechanism*.

 The solution begins by introducing a doublet of two complex scalar fields, known as the Higgs field.

$$
\phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{2}}\left(\phi_1 + i\phi_2\right) \\ \frac{1}{\sqrt{2}}\left(\phi_3 + i\phi_4\right) \end{pmatrix}
$$

Here, the $\phi_i$'s are real scalar fields. The Lagrangian that governs the evolution of these fields is

$$
\mathcal{L_\phi} = \left(D_\mu \phi\right)^\dagger \left(D^\mu \phi\right) - V\left(\phi\right)
$$

Where the covariant derivative $D_\mu$ is here defined as

$$
D_\mu = \partial_\mu + i g_w W^i_\mu \frac{\sigma_i}{2} + i g_Y B_\mu \frac{Y}{2}
$$

and the potential, $V\left(\phi\right)$ is

$$
V\left(\phi\right) = \mu^2 \phi^\dagger \phi + \lambda |\phi|^2, \left(\mu^2 < 0, \lambda > 0 \right)
$$

Critically, because $\mu^2 < 0$ the ground state is not at $\phi=0$, but instead has to satisfy $|\phi|^2 = -\frac{\mu^2}{2\lambda} \equiv \frac{v^2}{2}$. This is satisfied by an infinite number of ground states. Picking one breaks the $SU(2)_L\otimes U(1)_Y$ symmetry obeyed by the original Lagrangian, $\mathcal{L}_\phi$. The particular ground state can be chosen freely, but a convenient choice is

$$
\phi_{gnd} = \begin{pmatrix} 0 \\ \frac{v}{\sqrt{2}} \end{pmatrix}
$$

![Illustration of the spontaneous symmetry breaking mechanism. The nail, when subjected to a small force, keeps its rotational symmetry, but if the force grows too large it buckles under the strain. The *spontaneous* bend in the nail has broken rotational symmetry. This can also be understood in terms of the nail finding its potential minimum, a minimum that shifts from a single point to a circular collection of points as the force passes its critical value.[@Bellaiche2012]](figures/theory/symmetry_breaking.png){#fig:symmetry_breaking}

Combining $\mathcal{L}_\phi$ with the symmetry-broken electroweak Lagrangian and performing some simplification, we are left with:

\begin{align*}
\mathcal{L} = &\frac{1}{2} \partial_\mu h \partial^\mu h - |\mu|^2 h^2 \\
              -&\frac{1}{4} \left(W^-_{\mu\nu}\right)^\dagger W^{-\mu\nu} + \frac{1}{2}\left(\frac{g_w v}{2}\right)^2\left(W_\mu^-\right)^\dagger W^{-\mu} \\
              -&\frac{1}{4} \left(W^+_{\mu\nu}\right)^\dagger W^{+\mu\nu} + \frac{1}{2}\left(\frac{g_w v}{2}\right)^2\left(W+\mu^-\right)^\dagger W^{+\mu} \\
              -&\frac{1}{4} Z_{\mu\nu}Z^{\mu\nu} + \frac{1}{2}\left(\frac{g_w v}{2\cos{\theta_W}}\right)^2 Z_\mu Z^\mu \\
              -&\frac{1}{4} A_{\mu\nu}A^{\mu\nu} \\
              +&\frac{g_w^2 v}{2} h W_\mu^- W^{+\mu} + \frac{g_w^2}{4}h^2 W_\mu^- W^{+\mu} + \frac{g_w^2 v}{4\cos^2{\theta_W}}h Z_\mu Z^\mu + \frac{g_w^2}{8\cos^2{\theta_W}}h^2 Z_\mu Z^\mu \\
              +&\frac{\mu^2}{v}h^3 + \frac{\mu^2}{4v^2}h^4
\end{align*}

In the above equation, $V_{\mu\nu} \equiv \partial_\mu V_\nu - \partial_\nu V_\mu$ for the W and Z vector bosons. This is best understood by looking through the terms line by line. The first line describes a spin-0 field, $h$, associated with a particle of mass $m_h = \sqrt{2}|\mu|$. This particle is the *Higgs boson*.

Lines 2-4 describe the spin-1 fields $W^\pm_\mu$ and $Z_\mu$. These fields are associated with the *massive* W and Z bosons that have been previously discussed. Their masses are related to fundamental constants as

$$
m_{W^\pm} = \frac{g_w v}{2} \quad \mathrm{and} \quad m_Z = \frac{m_{W^\pm}}{\cos\theta_W}
$$

where $\theta_W$ is the Weinberg angle or *weak mixing angle*. It defines the angle spontaneous symmetry breaking rotates the original neutral fields to obtain the Z boson and the photon. It is also relates the coupling strengths of the two electroweak sub-interactions

$$
e = g_w \cos\theta_W = g_Y\sin\theta_W
$$

Line 5 describes the massless spin-1 field associated with the photon. Line 6 encapsulates the interaction between the vector bosons and the Higgs, and finally line 7 describes the Higgs self-interaction.

The higgs mechanism itself only directly accounts for the masses of the vector bosons. While a significant victory, the masses of the fermions are still not included in the theory. This can be remedied by introducing *Yukawa* couplings between the fermions and the Higgs field. After spontaneous symmetry breaking, these contributions take the form

$$
\mathcal{L}_\mathrm{yukawa} = \sum -m_f \bar{\psi}\psi - \frac{m_f}{v}\bar{\psi}\psi h
$$

The masses in the above equation are related to the *yukawa coupling* which is unique for each particle and represents how strongly that particle interacts with the Higgs field. This relation is simply 

$$
m_f = \frac{y_f}{2} v.
$$

For most fermions, this coupling, $y_f$, is very small. For electrons it is roughly $2\times 10^{-6}$, for example. However, the top quark is unique in that its Yukawa coupling is ~1. This makes the top quark a fine candidate for studying the behaviour of the Higgs. In particular, the production cross section of four top quarks is highly dependent upon the top Yukawa coupling. This is explored more deeply in chapter 5.

