2-to-3 and 2-to-4 Processes

The PYTHIA machinery to handle 2-to-1 and 2-to-2 processes is fairly sophisticated and generic. The same cannot be said about the generation of hard scattering processes with more than two final-state particles. The number of phase-space variables is larger, and it is therefore more difficult to find and transform away all possible peaks in the cross section by a suitably biased choice of phase-space points. In addition, matrix-element expressions for 2-to-3 processes are typically fairly lengthy. Therefore PYTHIA only contains a very limited number of 2-to-3 and 2-to-4 processes, and almost each process is a special case of its own. It is therefore less interesting to discuss details, and we only give a very generic overview.

If the Higgs mass is not light, interactions among longitudinal $\mathrm{W}$ and $\mathrm{Z}$ gauge bosons are of interest. In the program, 2-to-1 processes such as $\W_{\mathrm{L}}^+ \W_{\mathrm{L}}^- \to \mathrm{h}^0$ and 2-to-2 ones such as $\W_{\mathrm{L}}^+ \W_{\mathrm{L}}^- \to \mathrm{Z}_{\mathrm{L}}^0 \mathrm{Z}_{\mathrm{L}}^0$ are included. The former are for use when the $\mathrm{h}^0$ still is reasonably narrow, such that a resonance description is applicable, while the latter are intended for high energies, where different contributions have to be added up. Since the program does not contain $\W_{\mathrm{L}}$ or $\mathrm{Z}_{\mathrm{L}}$ distributions inside hadrons, the basic hard scattering has to be convoluted with the $\mathrm{q}\to \mathrm{q}' \W_{\mathrm{L}}$ and $\mathrm{q}\to \mathrm{q}\mathrm{Z}_{\mathrm{L}}$ branchings, to yield effective 2-to-3 and 2-to-4 processes. However, it is possible to integrate out the scattering angles of the quarks analytically, as well as one energy-sharing variable [Cha85]. Only after an event has been accepted are these other kinematical variables selected. This involves further choices of random variables, according to a separate selection loop.

In total, it is therefore only necessary to introduce one additional variable in the basic phase-space selection, which is chosen to be $\hat{s}'$, the squared invariant mass of the full 2-to-3 or 2-to-4 process, while $\hat{s}$ is used for the squared invariant mass of the inner 2-to-1 or 2-to-2 process. The $y$ variable is coupled to the full process, since parton-distribution weights have to be given for the original quarks at $x_{1,2} = \sqrt{\tau'} \exp{(\pm y)}$. The $\hat{t}$ variable is related to the inner process, and thus not needed for the 2-to-3 processes. The selection of the $\tau' = \hat{s}'/s$ variable is done after $\tau$, but before $y$ has been chosen. To improve the efficiency, the selection is made according to a weighted phase space of the form $\int h_{\tau'}(\tau') \d\tau'$, where

$$h_{\tau'}(\tau') = \frac{c_1}{{\cal I}_1} \frac{1}{\tau'} + ... ...^2} + \frac{c_3}{{\cal I}_3} \frac{1}{\tau' (1 - \tau')}  ,$$ (103)

in conventional notation. The $c_i$ coefficients are optimized at initialization. The $c_3$ term, peaked at $\tau' \approx 1$, is only used for $\mathrm{e}^+\mathrm{e}^-$ collisions. The choice of $h_{\tau'}$ is roughly matched to the longitudinal gauge-boson flux factor, which is of the form

$$\left( 1 + \frac{\tau}{\tau'} \right) \ln \left( \frac{\tau}{\tau'} \right) - 2 \left( 1 - \frac{\tau}{\tau'} \right)  .$$ (104)

For a light $\mathrm{h}$ the effective $\mathrm{W}$ approximation above breaks down, and it is necessary to include the full structure of the $\mathrm{q}\mathrm{q}' \to \mathrm{q}\mathrm{q}' \mathrm{h}^0$ (i.e. $\mathrm{Z}\mathrm{Z}$ fusion) and $\mathrm{q}\mathrm{q}' \to \mathrm{q}'' \mathrm{q}''' \mathrm{h}^0$ (i.e. $\mathrm{W}\mathrm{W}$ fusion) matrix elements. The $\tau'$, $\tau$ and $y$ variables are here retained, and selected according to standard procedures. The Higgs mass is represented by the $\tau$ choice; normally the $\mathrm{h}^0$ is so narrow that the $\tau$ distribution effectively collapses to a $\delta$ function. In addition, the three-body final-state phase space is rewritten as

$$\left( \prod_{i=3}^5 \frac{1}{(2 \pi)^3} \frac{\d ^3 p_i}{2 ... ... \d p_{\perp 4}^2 \frac{\d\varphi_4}{2 \pi} \d y_5  ,$$ (105)

where $\lambda_{\perp 34} = (m_{\perp 34}^2 - m_{\perp 3}^2 - m_{\perp 4}^2)^2 - 4 m_{\perp 3}^2 m_{\perp 4}^2$. The outgoing quarks are labelled 3 and 4, and the outgoing Higgs 5. The $\varphi$ angles are selected isotropically, while the two transverse momenta are picked, with some foreknowledge of the shape of the $\mathrm{W}/ \mathrm{Z}$ propagators in the cross sections, according to $h_{\perp} (p_{\perp}^2) \d p_{\perp}^2$, where

$$h_{\perp}(p_{\perp}^2) = \frac{c_1}{{\cal I}_1} + \frac{c_2}... ...\frac{c_3}{{\cal I}_3} \frac{1}{(m_R^2 + p_{\perp}^2)^2}  ,$$ (106)

with $m_R$ the $\mathrm{W}$ or $\mathrm{Z}$ mass, depending on process, and $c_1 = c_2 = 0.05$, $c_3 = 0.9$. Within the limits given by the other variable choices, the rapidity $y_5$ is chosen uniformly. A final choice remains to be made, which comes from a twofold ambiguity of exchanging the longitudinal momenta of partons 3 and 4 (with minor modifications if they are massive). Here the relative weight can be obtained exactly from the form of the matrix element itself.