The Standard Model

1.1 Overview

The Standard Model of particle physics describes the fundamental particles and their interactions Via three of the four known forces: the electromagnetic, weak, and strong interactions. Gravity is Not included.

The fundamental particles are:

Fermions (spin-1/2):

Quarks (6 flavours, 3 colours each): up, down, charm, strange, top, bottom.
Leptons (6 flavours): electron, muon, tau, and their three neutrinos.

Bosons (integer spin):

Gauge bosons: photon ( $\gamma$ ), $W^\pm$ , $Z^0$ Gluons ( $g$ 8 types).
Scalar boson: Higgs ( $H$ ).

Forces and their gauge bosons:

Interaction	Gauge boson	Acts on	Relative strength
Electromagnetic	$\gamma$	Electric charge	$\alpha \approx 1/137$
Weak	$W^\pm, Z^0$	Weak isospin	$G_F \approx 1.166 \times 10^{-5}$ GeV $^{-2}$
Strong	$g$ (8 gluons)	Colour charge	$\alpha_s \approx 0.118$ (at $m_Z$ )

1.2 Gauge Symmetry

The Standard Model is a renormalisable quantum field theory based on the gauge group:

$G_{\mathrm{SM} = \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y}$

Each factor corresponds to a fundamental interaction:

$\mathrm{SU}(3)_C$ : colour symmetry of the strong interaction, generated by eight Gell-Mann matrices. The quark fields transform as triplets ( $\mathbf{3}$ ) under this group.
$\mathrm{SU}(2)_L$ : weak isospin symmetry, acting only on left-handed fermion doublets. The right-handed fermions are singlets under $\mathrm{SU}(2)_L$ .
$\mathrm{U}(1)_Y$ : weak hypercharge symmetry, acting on all fermions. The hypercharge $Y$ is related to the electric charge by $Q = T_3 + Y/2$ .

The gauge principle requires that the Lagrangian be invariant under local gauge transformations. This forces the introduction of the gauge bosons as connections (covariant derivatives) and fixes Their self-interactions. The non-Abelian groups $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ give rise to Self-interacting gauge bosons ( $W^3W^+W^-$ vertices, three-gluon and four-gluon vertices), while The Abelian group $\mathrm{U}(1)$ gives a non-self-interacting photon.

1.3 Electroweak Unification

The electroweak theory, developed by Glashow, Weinberg, and Salam (Nobel Prize 1979), unifies the Electromagnetic and weak interactions into a single $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ Framework. The unification is hidden at low energies because the Higgs mechanism breaks the Symmetry and gives different masses to the $W^\pm$ , $Z^0$ And photon.

At energies well above the electroweak scale ( $E \gg v \approx 246$ GeV), the four gauge bosons $W^1, W^2, W^3, B$ have equal status and the symmetry is manifest. At low energies, the mixing:

$\begin{pmatrix} Z^0 \\ A \end{pmatrix} = \begin{pmatrix} \cos\theta_W & \sin\theta_W \\ -\sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} W^3 \\ B \end{pmatrix}$

Produces the massive $Z^0$ and the massless photon $A$ . The Weinberg angle $\theta_W$ determines The mixing and satisfies $\sin^2\theta_W \approx 0.231$ .

The electromagnetic coupling $e$ and the weak couplings $g$ , $g"$ are related by:

$e = g\sin\theta_W = g'\cos\theta_W$

This relationship is a direct prediction of the unified theory and has been verified experimentally To high precision at LEP and SLC.

1.4 Quarks and Leptons

The fermions are organised into three generations, each with identical quantum numbers but Increasing mass:

Generation	Quarks	Charge	Leptons	Charge
I	up ( $u$ ), down ( $d$ )	$+2/3$ , $-1/3$	$e$ , $\nu_e$	$-1$ , $0$
II	charm ( $c$ ), strange ( $s$ )	$+2/3$ , $-1/3$	$\mu$ , $\nu_\mu$	$-1$ , $0$
III	top ( $t$ ), bottom ( $b$ )	$+2/3$ , $-1/3$	$\tau$ , $\nu_\tau$	$-1$ , $0$

Each quark comes in three colour charges: red, green, blue. Antiquarks carry anticolours.

Hadrons are colour-neutral bound states of quarks:

Baryons: three quarks (qqq), e.g., proton ( $uud$ ), neutron ( $udd$ ).
Mesons: quark-antiquark pairs ( $q\bar{q}$ ), e.g., $\pi^+$ ( $u\bar{d}$ ), $K^-$ ( $s\bar{u}$ ).

1.5 Detailed Particle Properties

Quark masses (at the $\overline{\mathrm{MS}}$ scale $\mu = m_q$ ):

Quark	Mass (MeV/ $c^2$ )	$I$	$I_3$	$S$	$C$	$B'$	$T$
$u$	$2.16$	$1/2$	$+1/2$	$0$	$0$	$0$	$0$
$d$	$4.67$	$1/2$	$-1/2$	$0$	$0$	$0$	$0$
$s$	$93$	$0$	$0$	$-1$	$0$	$0$	$0$
$c$	$1.27 \times 10^3$	$0$	$0$	$0$	$+1$	$0$	$0$
$b$	$4.18 \times 10^3$	$0$	$0$	$0$	$0$	$-1$	$0$
$t$	$1.732 \times 10^5$	$0$	$0$	$0$	$0$	$0$	$+1$

Charged lepton masses:

Lepton	Mass (MeV/ $c^2$ )	Mean lifetime
$e$	$0.511$	Stable
$\mu$	$105.66$	$2.197 \times 10^{-6}$ s
$\tau$	$1776.86$	$2.903 \times 10^{-13}$ s

Gauge boson properties:

Boson	Mass (GeV/ $c^2$ )	Width (GeV)	Electric charge
$\gamma$	$0$	Stable	$0$
$g$	$0$	Confined	$0$
$W^+$	$80.379$	$2.085$	$+1$
$Z^0$	$91.1876$	$2.4952$	$0$
$H$	$125.10$	$0.00407$	$0$

1.6 Worked Example: Particle Identification

Example 1.1: Identifying the quark content of the $\Omega^-$ baryon

The $\Omega^-$ has the following quantum numbers: $Q = -1$ , $B = 1$ , $S = -3$ Strangeness $S = -3$ .

Since $B = 1$ It is a baryon, so it consists of three quarks. The strangeness Contributes $-1$ per strange quark, so all three quarks must be strange:

$\Omega^- = sss$

Check the charge: Each strange quark has $Q = -1/3$ So $Q(sss) = 3 \times (-1/3) = -1$ . Check the baryon number: $B(sss) = 3 \times (1/3) = 1$ . Both agree.

Example 1.2: Identifying a meson from its decay

A neutral meson $X^0$ decays via $X^0 \to K^+ + \pi^-$ . Identify $X^0$ .

The kaon $K^+$ has quark content $u\bar{s}$ ( $Q = +1$ , $S = +1$ ). The pion $\pi^-$ has quark content $d\bar{u}$ ( $Q = -1$ , $S = 0$ ).

Since $X^0$ is a neutral meson ( $q\bar{q}$ ), its quark content must combine the quarks From the decay products. The decay conserves strangeness if $X^0$ has $S = +1$ So It contains one $\bar{s}$ .

The quantum numbers of $X^0$ : $Q = 0$ , $S = +1$ . A meson with these properties Containing a $\bar{s}$ quark must be:

$X^0 = K^0 = d\bar{s}$

Verification: $K^0 \to K^+ + \pi^-$ conserves charge ( $0 = +1 + (-1)$ ) And strangeness ( $+1 = +1 + 0$ ). This decay proceeds via the weak interaction.

1.7 Gauge Bosons

Photon ( $\gamma$ ): Massless, mediates the electromagnetic force. Couples to electric charge.
$W^\pm$ and $Z^0$ : Massive ( $m_W \approx 80.4$ GeV/ $c^2$ , $m_Z \approx 91.2$ GeV/ $c^2$ ), mediate the weak force. $W^\pm$ changes flavour (charged current); $Z^0$ does not (neutral current).
Gluons ( $g$ ): Eight massless gluons mediate the strong force. They carry colour charge themselves, leading to self-interaction (non-Abelian gauge theory).
Higgs boson ( $H$ ): Scalar particle ( $m_H \approx 125$ GeV/ $c^2$ ), responsible for giving mass to $W^\pm$ , $Z^0$ And fermions through the Higgs mechanism.

1.8 Worked Example: CKM Matrix and Flavour Mixing

Example 1.3: Using the CKM matrix to predict decay rates

The Cabibbo—Kobayashi—Maskawa (CKM) matrix relates the weak interaction eigenstates to The mass eigenstates of quarks:

$\begin{pmatrix} d' \\ s' \\ b' \end{pmatrix} = V_{\mathrm{CKM} \begin{pmatrix} d \\ s \\ b \end{pmatrix}}$

Where $d'$ , $s'$ , $b'$ are the weak eigenstates that couple to the $W$ boson. The Magnitude of the CKM elements determines the relative rates of flavour-changing weak Decays.

The experimentally measured magnitudes are approximately:

$\lvert V_{\mathrm{CKM}\rvert \approx \begin{pmatrix} 0.974 & 0.225 & 0.0036 \\ 0.225 & 0.973 & 0.041 \\ 0.0086 & 0.040 & 0.999 \end{pmatrix}}$

Application. The decay $b \to c$ proceeds with amplitude proportional to $\lvert V_{cb}\rvert \approx 0.041$ While $b \to u$ proceeds with $\lvert V_{ub}\rvert \approx 0.0036$ . The ratio of partial widths is approximately:

$\frac{\Gamma(b \to u)}{\Gamma(b \to c)} \sim \frac{\lvert V_{ub}\rvert^2}{\lvert V_{cb}\rvert^2} = \frac{(0.0036)^2}{(0.041)^2} \approx 0.0077$

This means the $b \to u$ transition is suppressed by roughly two orders of magnitude Relative to $b \to c$ Which is why the $B$ meson predominantly decays to charm, not Up quarks.

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