Resonance production

We have now covered the simple $2 \to 2$ case. In a $2 \to 1$ process, the $\hat{t}$ integral is absent, and the differential cross section $\d\hat{\sigma}/\d\hat{t}$ is replaced by $\hat{\sigma}(\hat{s})$. The cross section may now be written as

$$\sigma = \int \int \frac{\d\tau}{\tau} \, \d y \, x_1 f_1(x_1, Q^2) \, x_2 f_2(x_2, Q^2) \, \hat{\sigma}(\hat{s})$$

$$= \frac{\pi}{s} \int h_{\tau}(\tau) \, \d\tau \int h_y(y) \, \d y \... ... {\tau^2 h_{\tau}(\tau) \, h_y(y)} \, \frac{\hat{s}}{\pi} \hat{\sigma}(\hat{s})$$

$$= \left\langle \frac{\pi}{s} \, \frac{1}{\tau^2 h_{\tau}(\tau) \, h... ...x_2 f_2(x_2, Q^2) \, \frac{\hat{s}}{\pi} \hat{\sigma}(\hat{s}) \right\rangle ~.$$ (100)

The structure is thus exactly the same, but the $z$-related pieces are absent, and the rôle of the dimensionless cross section is played by $(\hat{s}/\pi) \hat{\sigma}(\hat{s})$.

If the range of allowed decay angles of the resonance is restricted, e.g. by requiring the decay products to have a minimum transverse momentum, effectively this translates into constraints on the $z = \cos\hat{\theta}$ variable of the $2 \to 2$ process. The difference is that the angular dependence of a resonance decay is trivial, and that therefore the $z$-dependent factor can be easily evaluated. For a spin-0 resonance, which decays isotropically, the relevant weight is simply $(z_{-\mathrm{max}} - z_{-\mathrm{min}})/2 + (z_{+\mathrm{max}} - z_{+\mathrm{min}})/2$. For a transversely polarized spin-1 resonance the expression is, instead,

$$\frac{3}{8}(z_{-\mathrm{max}} - z_{-\mathrm{min}}) + \frac{3... ...})^3 + \frac{1}{8}(z_{+\mathrm{max}} - z_{+\mathrm{min}})^3 ~.$$ (101)

Since the allowed $z$ range could depend on $\tau$ and/or $y$ (it does for a $p_{\perp}$ cut), the factor has to be evaluated for each individual phase-space point and included in the expression of eq. ().

For $2 \to 2$ processes where either of the final-state particles is a resonance, or both, an additional choice has to be made for each resonance mass, eq. (). Since the allowed $\tau$, $y$ and $z$ ranges depend on $m_3^2$ and $m_4^2$, the selection of masses have to precede the choice of the other phase-space variables. Just as for the other variables, masses are not selected uniformly over the allowed range, but are rather distributed according to a function $h_m(m^2) \, dm^2$, with a compensating factor $1/h_m(m^2)$ in the Jacobian. The functional form picked is normally

$$h_m(m^2) = \frac{c_1}{{\cal I}_1} \, \frac{1}{\pi} \, \frac{... ... \, \frac{1}{m^2} + \frac{c_4}{{\cal I}_4} \, \frac{1}{m^4} ~.$$ (102)

The definition of the ${\cal I}_i$ integrals is analogous to the one before. The $c_i$ coefficients are not found by optimization, but predetermined, normally to $c_1 = 0.8$, $c_2 = c_3 =0.1$, $c_4 = 0$. Clearly, had the phase space and the cross section been independent of $m_3^2$ and $m_4^2$, the optimal choice would have been to put $c_1 = 1$ and have all other $c_i$ vanishing -- then the $1/h_m$ factor of the Jacobian would exactly have cancelled the Breit-Wigner of eq. () in the cross section. The second and the third terms are there to cover the possibility that the cross section does not die away quite as fast as given by the naïve Breit-Wigner shape. In particular, the third term covers the possibility of a secondary peak at small $m^2$, in a spirit slightly similar to the one discussed for resonance production in $2 \to 1$ processes.

The fourth term is only used for processes involving $\gamma^* / \mathrm{Z}^0$ production, where the $\gamma$ propagator guarantees that the cross section does have a significant secondary peak for $m^2 \to 0$. Therefore here the choice is $c_1 = 0.4$, $c_2 = 0.05$, $c_3 = 0.3$ and $c_4 = 0.25$.

A few special tricks have been included to improve efficiency when the allowed mass range of resonances is constrained by kinematics or by user cuts. For instance, if a pair of equal or charge-conjugate resonances are produced, such as in $\mathrm{e}^+ \mathrm{e}^- \to \mathrm{W}^+ \mathrm{W}^-$, use is made of the constraint that the lighter of the two has to have a mass smaller than half the c.m. energy.