Formula Sheet
Introduction to Particle Physics · Mini-course
Natural Units
\[ \hbar=c=1 \]
Useful conversion:
\[ \hbar c \simeq 197.3\ \mathrm{MeV\,fm}. \]
\[ 1\ \mathrm{GeV}^{-1}\simeq 0.197\ \mathrm{fm}, \qquad 1\ \mathrm{fm}\simeq 5.07\ \mathrm{GeV}^{-1}. \]
Momentum and Coordinate
\[ \sigma_x\sigma_p\ge \frac{1}{2}, \qquad \varepsilon_p=\frac{\sigma_p}{p}. \]
\[ \sigma_x\ge \frac{1}{2\sigma_p} = \frac{1}{2\varepsilon_p p} \simeq 10\ \mathrm{fm} \frac{10^{-2}}{\varepsilon_p} \frac{\mathrm{GeV}}{p}. \]
Cross-Sections and Soft Photons
\[ \text{interaction rate}=\sigma\times\text{flux}. \]
For Rutherford-type small-angle scattering:
\[ \frac{d\sigma}{d\Omega}\propto\frac{1}{\sin^4(\theta/2)}. \]
The observable charged-particle cross-section is inclusive:
\[ d\sigma_{\mathrm{obs}} = d\sigma(p\to p') + \sum_{E_\gamma<\Delta E} d\sigma(p\to p'+n\gamma). \]
Real and virtual soft-photon logarithms cancel in the inclusive result:
\[ \frac{d\sigma_{\mathrm{real}}}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_0 \left[ 1+\frac{\alpha}{\pi}L\ell_\mu+O(\alpha^2) \right], \]
\[ \frac{d\sigma_{\mathrm{virt}}}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_0 \left[ 1-\frac{\alpha}{\pi}L\ell_\mu+O(\alpha^2) \right]. \]
Standard Model Gauge Group
\[ SU(3)_c\times SU(2)_L\times U(1)_Y \]
Gauge Principle in QED
Free Dirac field:
\[ \mathcal{L}_0=\overline{\psi}(i\gamma^\mu\partial_\mu-m)\psi. \]
Local phase transformation:
\[ \psi(x)\to e^{i\alpha(x)}\psi(x). \]
Covariant derivative:
\[ D_\mu=\partial_\mu+iqA_\mu. \]
Gauge transformation of the vector potential:
\[ A_\mu(x)\to A_\mu(x)+\frac{1}{q}\partial_\mu\alpha(x). \]
QED Lagrangian:
\[ \mathcal{L} = \overline{\psi}(i\gamma^\mu\partial_\mu-m)\psi -q\overline{\psi}\gamma^\mu\psi A_\mu -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}. \]
Gauge invariance forbids a photon mass term:
\[ m_A^2A_\mu A^\mu. \]
Higgs Vacuum Expectation Value
\[ \langle H\rangle = \frac{v}{\sqrt{2}}, \quad v\simeq 246\ \mathrm{GeV} \]
Vector-Boson Mass from a Condensate
Meissner-type equation:
\[ \left(\partial^2+m_A^2\right)A^\mu=0, \qquad m_A^2=e^2v^2. \]
Weak-interaction range estimate:
\[ R\sim\frac{1}{M_W}. \]
Fermion Masses
\[ m_f = \frac{y_fv}{\sqrt{2}} \]
This parameterizes charged-fermion masses through Yukawa couplings; it does not explain the numerical pattern of those couplings.
Coupled Chiral Waves
\[ H(p)= \begin{pmatrix} p&m\\ m&-p \end{pmatrix}, \qquad \omega_\pm(p)=\pm\sqrt{p^2+m^2}. \]