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Subsequent branches and angular ordering

The shower evolution is (almost) always done on a pair of partons, so that energy and momentum can be conserved. In the first step of the evolution, the two original partons thus undergo branchings $1 \to 3 + 4$ and $2 \to 5 + 6$. As described above, the allowed $m_1$, $m_2$, $z_1$ and $z_2$ ranges are coupled by kinematical constraints. In the second step, the pair $3 + 4$ is evolved and, separately, the pair $5 + 6$. Considering only the former (the latter is trivially obtained by symmetry), the partons thus have nominal initial energies $E_3^{(0)} = z_1 E_1$ and $E_4^{(0)} = (1-z_1) E_1$, and maximum allowed virtualities $m_{\mathrm{max},3} = \min(m_1,E_3^{(0)})$ and $m_{\mathrm{max},4} = \min(m_1,E_4^{(0)})$. Initially partons 3 and 4 are evolved separately, giving masses $m_3$ and $m_4$ and splitting variables $z_3$ and $z_4$. If $m_3 + m_4 > m_1$, the parton of 3 and 4 that has the largest ratio of $m_i/m_{\mathrm{max},i}$ is evolved further. Thereafter eq. ([*]) is used to construct corrected energies $E_3$ and $E_4$, and the $z$ values are checked for consistency. If a branching has to be rejected because the change of parton energy puts $z$ outside the allowed range, the parton is evolved further.

This procedure can then be iterated for the evolution of the two daughters of parton 3 and for the two of parton 4, etc., until each parton reaches the cut-off mass $m_{\mathrm{min}}$. Then the parton is put on the mass shell.

The model, as described so far, produces so-called conventional showers, wherein masses are strictly decreasing in the shower evolution. Emission angles are decreasing only in an average sense, however, which means that also fairly `late' branchings can give partons at large angles. Theoretical studies beyond the leading-log level show that this is not correct [Mue81], but that destructive interference effects are large in the region of non-ordered emission angles. To a good first approximation, these so-called coherence effects can be taken into account in parton-shower programs by requiring a strict ordering in terms of decreasing emission angles. (Actually, the fact that the shower described here is already ordered in mass implies that the additional cut on angle will be a bit too restrictive. While effects from this should be small at current energies, some deviations become visible at very high energies.)

The coherence phenomenon is known already from QED. One manifestation is the Chudakov effect [Chu55], discovered in the study of high-energy cosmic $\gamma$ rays impinging on a nuclear target. If a $\gamma$ is converted into a highly collinear $\mathrm{e}^+\mathrm{e}^-$ pair inside the emulsion, the $\mathrm{e}^+$ and $\mathrm{e}^-$ in their travel through the emulsion ionize atoms and thereby produce blackening. However, near the conversion point the blackening is small: the $\mathrm{e}^+$ and $\mathrm{e}^-$ then are still close together, so that an atom traversed by the pair does not resolve the individual charges of the $\mathrm{e}^+$ and the $\mathrm{e}^-$, but only feels a net charge close to zero. Only later, when the $\mathrm{e}^+$ and $\mathrm{e}^-$ are separated by more than a typical atomic radius, are the two able to ionize independently of each other.

The situation is similar in QCD, but is further extended, since now also gluons carry colour. For example, in a branching $\mathrm{q}_0 \to \mathrm{q}\mathrm{g}$ the $\mathrm{q}$ and $\mathrm{g}$ share the newly created pair of opposite colour-anticolour charges, and therefore the $\mathrm{q}$ and $\mathrm{g}$ cannot emit subsequent gluons incoherently. Again the net effect is to reduce the amount of soft gluon emission: since a soft gluon (emitted at large angles) corresponds to a large (transverse) wavelength, the soft gluon is unable to resolve the separate colour charges of the $\mathrm{q}$ and the $\mathrm{g}$, and only feels the net charge carried by the $\mathrm{q}_0$. Such a soft gluon $\mathrm{g}'$ (in the region $\theta_{\mathrm{q}_0 \mathrm{g}'} > \theta_{\mathrm{q}\mathrm{g}}$) could therefore be thought of as being emitted by the $\mathrm{q}_0$ rather than by the $\mathrm{q}$-$\mathrm{g}$ system. If one considers only emission that should be associated with the $\mathrm{q}$ or the $\mathrm{g}$, to a good approximation (in the soft region), there is a complete destructive interference in the regions of non-decreasing opening angles, while partons radiate independently of each other inside the regions of decreasing opening angles ( $\theta_{q g'} < \theta_{q g}$ and $\theta_{g g'} < \theta_{q g}$), once azimuthal angles are averaged over. The details of the colour interference pattern are reflected in non-uniform azimuthal emission probabilities.

The first branchings of the shower are not affected by the angular-ordering requirement -- since the evolution is performed in the c.m. frame of the original parton pair, where the original opening angle is 180$^{\circ}$, any angle would anyway be smaller than this -- but here instead the matrix-element matching procedure is used, where applicable. Subsequently, each opening angle is compared with that of the preceding branching in the shower.

For a branching $a \to bc$ the kinematical approximation

\theta_a \approx \frac{p_{\perp b}}{E_b} + \frac{p_{\perp c}...
..._a) E_a} \right) = \frac{1}{\sqrt{z_a(1-z_a)}}
\end{displaymath} (177)

is used to derive the opening angle (this is anyway to the same level of approximation as the one in which angular ordering is derived). With $\theta_b$ of the $b$ branching calculated similarly, the requirement $\theta_b < \theta_a$ can be reduced to
\frac{z_b (1-z_b)}{m_b^2} > \frac{1-z_a}{z_a m_a^2} ~.
\end{displaymath} (178)

Since photons do not obey angular ordering, the check on angular ordering is not performed when a photon is emitted. When a gluon is emitted in the branching after a photon, its emission angle is restricted by that of the preceding QCD branching in the shower, i.e. the photon emission angle does not enter.

next up previous contents
Next: Other final-state shower aspects Up: Final-State Showers Previous: First branchings and matrix-element   Contents
Stephen Mrenna 2007-10-30