Electroweak cross sections

In the standard theory, fermions have the following couplings (illustrated here for the first generation):

$e_{\nu} = 0$,	$v_{\nu} = 1$,	$a_{\nu} = 1$,
$e_{\mathrm{e}} = -1$,	$v_{\mathrm{e}} = -1 + 4\sin^2 \! \theta_W$,	$a_{\mathrm{e}} = -1$,
$e_{\u } = 2/3$,	$v_{\u } = 1 - 8\sin^2 \! \theta_W /3$,	$a_{\nu} = 1$,
$e_{\d } = -1/3$,	$v_{\d } = -1 + 4\sin^2 \! \theta_W /3$,	$a_{\d } = -1$,

with $e$ the electric charge, and $v$ and $a$ the vector and axial couplings to the $\mathrm{Z}^0$. The relative energy dependence of the weak neutral current to the electromagnetic one is given by

$$\chi(s) = \frac{1}{16\sin^2 \! \theta_W \cos^2 \! \theta_W ... ...s - m_{\mathrm{Z}}^2 + i m_{\mathrm{Z}}\Gamma_{\mathrm{Z}}} ~,$$ (17)

where $s = E_{\mathrm{cm}}^2$. In this section the electroweak mixing parameter $\sin^2 \! \theta_W$ and the $\mathrm{Z}^0$ mass $m_{\mathrm{Z}}$ and width $\Gamma_{\mathrm{Z}}$ are considered as constants to be given by you (while the full PYTHIA event generation machinery itself calculates an $s$-dependent width).

Although the incoming $\mathrm{e}^+$ and $\mathrm{e}^-$ beams are normally unpolarized, we have included the possibility of polarized beams, following the formalism of [Ols80]. Thus the incoming $\mathrm{e}^+$ and $\mathrm{e}^-$ are characterized by polarizations $\mathbf{P}^{\pm}$ in the rest frame of the particles:

$$\mathbf{P}^{\pm} = P_{\mathrm{T}}^{\pm} \hat{\mathbf{s}}^{\pm} + P_{\mathrm{L}}^{\pm} \hat{\mathbf{p}}^{\pm} ~,$$ (18)

where $0 \leq P_{\mathrm{T}}^{\pm} \leq 1$ and $-1 \leq P_{\mathrm{L}}^{\pm} \leq 1$, with the constraint

$$(\mathbf{P}^{\pm})^2 = (P_{\mathrm{T}}^{\pm})^2 + (P_{\mathrm{L}}^{\pm})^2 \leq 1 ~.$$ (19)

Here $\hat{\mathbf{s}}^{\pm}$ are unit vectors perpendicular to the beam directions $\hat{\mathbf{p}}^{\pm}$. To be specific, we choose a right-handed coordinate frame with $\hat{\mathbf{p}}^{\pm} = (0,0, \mp 1)$, and standard transverse polarization directions (out of the machine plane for storage rings) $\hat{\mathbf{s}}^{\pm} = (0, \pm 1,0)$, the latter corresponding to azimuthal angles $\varphi^{\pm} = \pm \pi /2$. As free parameters in the program we choose $P_{\mathrm{L}}^+$, $P_{\mathrm{L}}^-$, $P_{\mathrm{T}} = \sqrt{P_{\mathrm{T}}^+ P_{\mathrm{T}}^-}$ and $\Delta \varphi = (\varphi^+ + \varphi^-) /2$.

In the massless QED case, the probability to produce a flavour $\mathrm{f}$ is proportional to $e_{\mathrm{f}}^2$, i.e up-type quarks are four times as likely as down-type ones. In lowest-order massless QFD (Quantum Flavour Dynamics; part of the Standard Model) the corresponding relative probabilities are given by [Ols80]

$$h_{\mathrm{f}}(s) = e_{\mathrm{e}}^2 \, (1 - P_{\mathrm{L}}^+ P_{\mathrm{L}}^-) \, e_... ... P_{\mathrm{L}}^+) \right\} \, \Re\chi(s) \, e_{\mathrm{f}} v_{\mathrm{f}} \, +$$

$$+ \, \left\{ (v_{\mathrm{e}}^2 + a_{\mathrm{e}}^2) (1 - P_{\mathr... ...chi(s) \right\vert^2 \, \left\{ v_{\mathrm{f}}^2 + a_{\mathrm{f}}^2 \right\} ~,$$ (20)

where $\Re\chi(s)$ denotes the real part of $\chi(s)$. The $h_{\mathrm{f}}(s)$ expression depends both on the $s$ value and on the longitudinal polarization of the $\mathrm{e}^{\pm}$ beams in a non-trivial way.

The cross section for the process $\mathrm{e}^+\mathrm{e}^-\to \gamma^* / \mathrm{Z}^0\to \mathrm{f}\overline{\mathrm{f}}$ may now be written as

$$\sigma_{\mathrm{f}}(s) = \frac{4 \pi \alpha_{\mathrm{em}}^2}{3 s} R_{\mathrm{f}}(s) ~,$$ (21)

where $R_{\mathrm{f}}$ gives the ratio to the lowest-order QED cross section for the process $\mathrm{e}^+\mathrm{e}^-\to \mu^+ \mu^-$,

$$R_{\mathrm{f}}(s) = N_C \, R_{\mathrm{QCD}} \, h_{\mathrm{f}}(s) ~.$$ (22)

The factor of $N_C = 3$ counts the number of colour states available for the $\mathrm{q}\overline{\mathrm{q}}$ pair. The $R_{\mathrm{QCD}}$ factor takes into account QCD loop corrections to the cross section. For $n_f$ effective flavours (normally $n_f =5$)

$$R_{\mathrm{QCD}} \approx 1 + \frac{\alpha_{\mathrm{s}}}{\pi... ..._f) \left( \frac{\alpha_{\mathrm{s}}}{\pi} \right)^2 + \cdots$$ (23)

in the $\overline{\mathrm{MS}}$ renormalization scheme [Din79]. Note that $R_{\mathrm{QCD}}$ does not affect the relative quark-flavour composition, and so is of peripheral interest here. (For leptons the $N_C$ and $R_{\mathrm{QCD}}$ factors would be absent, i.e. $N_C \, R_{\mathrm{QCD}} = 1$, but leptonic final states are not generated by this routine.)

Neglecting higher-order QCD and QFD effects, the corrections for massive quarks are given in terms of the velocity $\beta_{\mathrm{f}}$ of a fermion with mass $m_{\mathrm{f}}$, $\beta_{\mathrm{f}} = \sqrt{ 1 - 4 m_{\mathrm{f}}^2 /s}$, as follows. The vector quark current terms in $h_{\mathrm{f}}$ (proportional to $e_{\mathrm{f}}^2$, $e_{\mathrm{f}} v_{\mathrm{f}}$, or $v_{\mathrm{f}}^2$) are multiplied by a threshold factor $\beta_{\mathrm{f}} (3 - \beta_{\mathrm{f}}^2) /2$, while the axial vector quark current term (proportional to $a_{\mathrm{f}}^2$) is multiplied by $\beta_{\mathrm{f}}^3$. While inclusion of quark masses in the QFD formulae decreases the total cross section, first-order QCD corrections tend in the opposite direction [Jer81]. Naïvely, one would expect one factor of $\beta_{\mathrm{f}}$ to get cancelled. So far, the available options are either to include threshold factors in full or not at all.

Given that all five quarks are light at the scale of the $\mathrm{Z}^0$, the issue of quark masses is not really of interest at LEP. Here, however, purely weak corrections are important, in particular since they change the $\b$ quark partial width differently from that of the other ones [Küh89]. No such effects are included in the program.