diff --git a/content/posts/standard-model.md b/content/posts/standard-model.md index a3e9291..b92b257 100644 --- a/content/posts/standard-model.md +++ b/content/posts/standard-model.md @@ -56,7 +56,6 @@ In particle physics, the total energy of a particle is often described by: $$ \begin{equation} E^2 = (pc)^2 + (m_0 c^2)^2 -\label{eq:energy_momentum_mass} \end{equation} $$ @@ -106,7 +105,6 @@ If the color explanation is confusing, you can instead imagine the colors as a v $$ \begin{equation} (1,0)+(0,1)+(-1,-1)=(0,0) -\label{eq:color_neutral_combination} \end{equation} $$ @@ -122,10 +120,10 @@ Quarks are never found in isolation due to a phenomenon called quark confinement ```mermaid graph TB; - id0[q1 ... qn] --> Hadrons; - Hadrons --> Mesons; - Hadrons --> Baryons; - Hadrons --> id1[Exotic Hadrons] + id0[q1 ... qn] --> Hadrons; + Hadrons --> Mesons; + Hadrons --> Baryons; + Hadrons --> id1[Exotic Hadrons] ``` This is a simplified representation of the relationship between quarks and hadrons. Quarks combine to form hadrons, which include mesons and baryons. Exotic hadrons are more complex combinations of quarks. @@ -166,7 +164,6 @@ The probability of a neutrino of flavor $\nu_\alpha$​ transforming into a neut $$ \begin{equation} P(\nu_\alpha\to\nu_\beta)=\sin^2(2\theta)\sin^2(\frac{\Delta m^2L}{4E}) -\label{eq:neutrino_oscillations} \end{equation} $$ @@ -182,7 +179,6 @@ The key to understanding why neutrinos must have mass lies in the $\Delta m^2$ t $$ \begin{equation} \sin^2(\frac{\Delta m^2L}{4E})=\sin^2(0)=0 -\label{eq:neutrino_oscillations_zero_mass} \end{equation} $$ @@ -241,7 +237,6 @@ The QED Lagrangian describes the interactions of electrons, positrons, and photo $$ \begin{equation} \mathcal{L}_{QED} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi -\label{eq:qed_lagrangian} \end{equation} $$ @@ -250,7 +245,6 @@ $$ $$ \begin{equation} F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - \label{eq:em_field_strength_tensor} \end{equation} $$ - **Explanation**: This term describes the field strength of the electromagnetic field. It involves the derivative of the electromagnetic potential $A_\mu$. The field strength tensor is antisymmetric, reflecting the nature of the electromagnetic force. @@ -264,7 +258,6 @@ $$ $$ \begin{equation} D_\mu = \partial_\mu - i e A_\mu - \label{eq:covariant_derivative_for_electrons} \end{equation} $$ - **Explanation**: This accounts for the interaction of electrons with the electromagnetic field. $e$ is the electric charge, and $A_\mu$ is the electromagnetic potential. @@ -289,7 +282,6 @@ The Electroweak Lagrangian describes the interactions of leptons, quarks, and ga $$ \begin{equation} \mathcal{L}_{EW} = -\frac{1}{4} W^a_{\mu\nu} W^{a\mu\nu} - \frac{1}{4} B_{\mu\nu} B^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi -\label{eq:electroweak_lagrangian} \end{equation} $$ @@ -300,7 +292,6 @@ $$ $$ \begin{equation} W^a_{\mu\nu} = \partial_\mu W^a_\nu - \partial_\nu W^a_\mu + g \epsilon^{abc} W^b_\mu W^c_\nu - \label{eq:w_field_strength_tensor} \end{equation} $$ - **$B$ Field Strength Tensor** ($B_{\mu\nu}$): @@ -308,7 +299,6 @@ $$ $$ \begin{equation} B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu - \label{eq:b_field_strength_tensor} \end{equation} $$ - **Explanation**: These terms describe the field strengths of the $W$ and $B$ fields, which are associated with the weak and electromagnetic forces, respectively. The $W$ field strength tensor involves the non-Abelian field strength term, reflecting the non-Abelian nature of the weak force. @@ -322,7 +312,6 @@ $$ $$ \begin{equation} D_\mu = \partial_\mu - i g' Y B_\mu - i g \frac{\tau^a}{2} W^a_\mu - \label{eq:covariant_derivative_for_leptons_quarks} \end{equation} $$ - **Explanation**: This accounts for the interaction of leptons/quarks with the $W$ and $B$ fields. $g'$ and $g$ are the coupling constants for the $U(1)$ and $SU(2)$ gauge groups, respectively. $Y$ is the weak hypercharge, and $\frac{\tau^a}{2}$ are the generators of the $SU(2)$ group. @@ -347,7 +336,6 @@ The QCD Lagrangian describes the interactions of quarks and gluons, the fundamen $$ \begin{equation} \mathcal{L}_{QCD} = -\frac{1}{4} G^a_{\mu\nu} G^{a\mu\nu} + \sum_q \bar{q} (i \gamma^\mu D_\mu - m_q) q -\label{eq:qcd_lagrangian} \end{equation} $$ @@ -356,7 +344,6 @@ $$ $$ \begin{equation} G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g_s f^{abc} G^b_\mu G^c_\nu - \label{eq:gluon_field_strength_tensor} \end{equation} $$ - **Explanation**: This term describes the field strength of gluons. It involves the derivative of the gluon fields $(\partial_\mu G^a_\nu$ and $\partial_\nu G^a_\mu)$ and the non-Abelian field strength term involving the structure constants $f^{abc}$, which accounts for the interaction between gluons themselves. @@ -370,7 +357,6 @@ $$ $$ \begin{equation} D_\mu = \partial_\mu - i g_s \frac{\lambda^a}{2} G^a_\mu - \label{eq:quark_covariant_derivative_qcd} \end{equation} $$ - **Explanation**: This accounts for the interaction of quarks with gluons. $g_s$ is the strong coupling constant, and $\frac{\lambda^a}{2}$ are the generators of the SU(3) color gauge group. $G^a_\mu$ are the gluon fields.