Particle Physics
Mandelstam Variable Identities
For $2 \to 2$ scattering with incoming momenta and masses $(p_1, m_1), (p_2, m_2)$ and outgoing $(k_1, m_1'), (k_2, m_2')$:
$$ \begin{aligned} p_1 \cdot p_2 &= \tfrac{1}{2}\left(s - m_1^2 - m_2^2\right) \\ k_1 \cdot k_2 &= \tfrac{1}{2}\left(s - m_1'^2 - m_2'^2\right) \\ p_1 \cdot k_1 &= \tfrac{1}{2}\left(m_1^2 + m_1'^2 - t\right) \\ p_2 \cdot k_2 &= \tfrac{1}{2}\left(m_2^2 + m_2'^2 - t\right) \\ p_1 \cdot k_2 &= \tfrac{1}{2}\left(m_1^2 + m_2'^2 - u\right) \\ p_2 \cdot k_1 &= \tfrac{1}{2}\left(m_2^2 + m_1'^2 - u\right) \end{aligned} $$with $s + t + u = m_1^2 + m_2^2 + m_1'^2 + m_2'^2$.
Standard Model Masses and Degrees of Freedom
| Type | Mass | Spin | $g$ | |
|---|---|---|---|---|
| Quarks | $t, \bar t$ | 173 GeV | $1/2$ | $2 \cdot 2 \cdot 3 = 12$ |
| $b, \bar b$ | 4 GeV | |||
| $c, \bar c$ | 1 GeV | |||
| $s, \bar s$ | 100 MeV | |||
| $d, \bar d$ | 5 MeV | |||
| $u, \bar u$ | 2 MeV | |||
| Gluons | $g_i$ | 0 | 1 | $8 \cdot 2 = 16$ |
| Charged lep. | $\tau^\pm$ | 1777 MeV | $1/2$ | $2 \cdot 2 = 4$ |
| $\mu^\pm$ | 106 MeV | |||
| $e^\pm$ | 511 keV | |||
| Neutrinos | $\nu_\tau, \bar\nu_\tau$ | < 0.6 eV | $1/2$ | $2 \cdot 1 = 2$ |
| $\nu_\mu, \bar\nu_\mu$ | < 0.6 eV | |||
| $\nu_e, \bar\nu_e$ | < 0.6 eV | |||
| Gauge bosons | $W^+$ | 80 GeV | 1 | 3 |
| $W^-$ | 80 GeV | |||
| $Z^0$ | 91 GeV | |||
| $\gamma$ | 0 | 2 | ||
| Higgs | $H^0$ | 125 GeV | 0 | 1 |
Above all SM thresholds: $g_b = 28$, $g_f = 90$, so $g_* = g_b + \tfrac{7}{8} g_f = 106.75$.