that $p_{+i}''/p_{+j}''=r_{jn}/r_{in}$. We have also studied alternative ways to share the recoils but the result does not depend sensitively on the exact formula. When we have generated an emission we always check that $p_+''\leqslant p_+$, which obviously must be satisfied.

To make the analysis simple, transverse momentum, $\bold{p}_\perp$, is only approximately conserved in the program. Exact conservation of $\bold{p}_\perp$ would be possible but is not essential for the purpose of this study. When a new gluon is emitted its transverse momentum is decided by $$p_{\perp n}=\textrm{max}\biggl(\frac{1}{r_{in}},\frac{1}{r_{jn}}\biggr)$$

The recoils on the emitters are given by $$\begin{aligned}
& &\textrm{If} \;  p_{\perp i}\leqslant \frac{1}{r_{in}} \textrm{then} \;
p_{\perp i}'=\frac{1}{r_{in}} \\
& &\textrm{If} \;  p_{\perp j}\leqslant \frac{1}{r_{jn}} \textrm{then} \;
p_{\perp j}'=\frac{1}{r_{jn}} \nonumber\end{aligned}$$ Otherwise we simply set $p_\perp =p_\perp'$, i.e we don’t change the transverse momenta. This simply means that the transverse momentum is decided only by the largest contribution. Hence the transverse momentum of a parton will always be determined by the shortest distance to another parton, with which it has formed a dipole. Since we change $p_+$ and $p_\perp$ we must, to be consistent, also change the rapidity $y$. This is obviously done by setting $$y'=\textrm{ln}\frac{p_\perp'}{p_+'}\geqslant y$$ The inequality is due to the fact 