Hugonweb | e+e- Cross-section vs. Mass

This page is about the $e^+ e^- \rightarrow \mathrm{Z}/\gamma^*\rightarrow f\bar{f}$ cross section

Most of this work uses PDG RPP 2013 Z Boson Section

The cross section splits into a Z part, a photon part, and an interference part, at LO:

$$ \sigma_Z^0 = \frac{12\pi}{M^2_Z} \frac{\Gamma^{e}\Gamma^{\mu}}{\Gamma_Z^2} \frac{s \Gamma^2_Z}{(s-M^2_Z)^2 + s^2\Gamma^2/M^2_Z} $$

$$ \sigma_\gamma^0 = \frac{4 \pi \alpha^2(s) Q_f^2N_c^f}{3 s} $$

$$ \sigma_{Z \gamma}^0 = - \frac{2\sqrt{2}\alpha(s)}{3} (G_F Q_f N^f_c g_V^f g_V^e) \frac{M^2_Z(s-M_Z^2)}{(s-M^2_Z)^2-s^2\Gamma^2/M_Z^2} $$

Leptons have $N_c$, number of colors, 1, while quarks have 3.

For e e -> mu mu

Using $\alpha(M_Z) = 1/128.91$ and $\sin^2 \theta_W(M_Z) = 0.23126$, $G_F = 1.166 \times 10^{-5}$ GeV$^{-2}$, $M_Z=91.1876$ GeV, $\Gamma_Z = 2.4952$ GeV, $$\frac{\Gamma_\ell}{\Gamma_Z} \approx 3.3658\%$$ from PDG 2013:

$$ \sigma_Z^0 \approx \frac{s}{(s-M^2_Z)^2 + s^2\Gamma^2/M^2_Z} \times 3.20\times 10^{-5} $$

$$ \sigma_\gamma^0 \approx \frac{1}{s} \times 2.52\times 10^{-4} $$

$$ \sigma_{Z \gamma}^0 \approx - \frac{(s-M_Z^2)}{(s-M^2_Z)^2-s^2\Gamma^2/M_Z^2} \times 1.015 \times 10^{-6} $$

For e e -> nu nu

$$ \sigma_Z^0 \approx \frac{s}{(s-M^2_Z)^2 + s^2\Gamma^2/M^2_Z} \times 6.34\times 10^{-3} $$

For e e -> u u

BR of Z -> u-type quarks is 0.116, $g_V^u= 0.25$ from PDG 2013

$$ \sigma_Z^0 \approx \frac{s}{(s-M^2_Z)^2 + s^2\Gamma^2/M^2_Z} \times 1.10 \times 10^{-4} $$

$$ \sigma_\gamma^0 \approx \frac{1}{s} \times 3.36 \times 10^{-4} $$

$$ \sigma_{Z \gamma}^0 \approx - \frac{(s-M_Z^2)}{(s-M^2_Z)^2-s^2\Gamma^2/M_Z^2} \times (-1.34 \times 10^{-5}) $$

For e e -> d d

BR of Z -> d-type quarks is 0.156, $g_V^d = -0.33$ from PDG 2013

$$ \sigma_Z^0 \approx \frac{s}{(s-M^2_Z)^2 + s^2\Gamma^2/M^2_Z} \times 1.48 \times 10^{-4} $$

$$ \sigma_\gamma^0 \approx \frac{1}{s} \times 8.4 \times 10^{-5} $$

$$ \sigma_{Z \gamma}^0 \approx - \frac{(s-M_Z^2)}{(s-M^2_Z)^2-s^2\Gamma^2/M_Z^2} \times 8.85 \times 10^{-6} $$