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The choice of energy splitting variable

The final-state radiation machinery is always applied in the c.m. frame of the hard scattering, from which normally emerges a pair of evolving partons. Occasionally there may be one evolving parton recoiling against a non-evolving one, as in $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\gamma$, where only the gluon evolves in the final state, but where the energy of the photon is modified by the branching activity of the gluon. (With only one evolving parton and nothing else, it would not be possible to conserve energy and momentum when the parton is assigned a mass.) Thus, before the evolution is performed, the parton pair is boosted to their common c.m. frame, and rotated to sit along the $z$ axis. After the evolution, the full parton shower is rotated and boosted back to the original frame of the parton pair.

The interpretation of the energy and momentum splitting variable $z$ is not unique, and in fact the program allows the possibility to switch between four different alternatives [Ben87a], `local' and `global' $z$ definition combined with `constrained' or `unconstrained' evolution. In all four of them, the $z$ variable is interpreted as an energy fraction, i.e. $E_b = z E_a$ and $E_c = (1-z) E_a$. In the `local' choice of $z$ definition, energy fractions are defined in the rest frame of the grandmother, i.e. the mother of parton $a$. The preferred choice is the `global' one, in which energies are always evaluated in the c.m. frame of the hard scattering. The two definitions agree for the branchings of the partons that emerge directly from the hard scattering, since the hard scattering itself is considered to be the `mother' of the first generation of partons. For instance, in $\mathrm{Z}^0 \to \mathrm{q}\overline{\mathrm{q}}$ the $\mathrm{Z}^0$ is considered the mother of the $\mathrm{q}$ and $\overline{\mathrm{q}}$, even though the branching is not handled by the parton-showering machinery. The `local' and `global' definitions diverge for subsequent branchings, where the `global' tends to allow more shower evolution.

In a branching $a \to bc$ the kinematically allowed range of $z = z_a$ values, $z_{-} < z < z_{+}$, is given by

\begin{displaymath}
z_{\pm} = \frac{1}{2} \left\{ 1 + \frac{m_b^2 - m_c^2}{m_a^2...
...(m_a^2 - m_b^2 - m_c^2)^2
- 4 m_b^2 m_c^2}}{m_a^2} \right\} ~.
\end{displaymath} (168)

With `constrained' evolution, these bounds are respected in the evolution. The cut-off masses $m_{\mathrm{eff},b}$ and $m_{\mathrm{eff},c}$ are used to define the maximum allowed $z$ range, within which $z_a$ is chosen, together with the $m_a$ value. In the subsequent evolution of $b$ and $c$, only pairs of $m_b$ and $m_c$ are allowed for which the already selected $z_a$ fulfils the constraints in eq. ([*]).

For `unconstrained' evolution, which is the preferred alternative, one may start off by assuming the daughters to be massless, so that the allowed $z$ range is

\begin{displaymath}
z_{\pm} = \frac{1}{2} \left\{ 1 \pm \frac{\vert\mathbf{p}_a\vert}{E_a}
\theta(m_a - m_{\mathrm{min},a}) \right\} ~,
\end{displaymath} (169)

where $\theta(x)$ is the step function, $\theta(x) = 1$ for $x > 0$ and $\theta(x) = 0$ for $x < 0$. The decay kinematics into two massless four-vectors $p_b^{(0)}$ and $p_c^{(0)}$ is now straightforward. Once $m_b$ and $m_c$ have been found from the subsequent evolution, subject only to the constraints $m_b < z_a E_a$, $m_c < (1-z_a) E_a$ and $m_b + m_c < m_a$, the actual massive four-vectors may be defined as
\begin{displaymath}
p_{b,c} = p_{b,c}^{(0)} \pm (r_c p_c^{(0)} - r_b p_b^{(0)}) ~,
\end{displaymath} (170)

where
\begin{displaymath}
r_{b,c} = \frac{m_a^2 \pm (m_c^2 -m_b^2) -
\sqrt{ (m_a^2 - m_b^2 - m_c^2)^2 - 4 m_b^2 m_c^2}}{2 m_a^2} ~.
\end{displaymath} (171)

In other words, the meaning of $z_a$ is somewhat reinterpreted post facto. Needless to say, the `unconstrained' option allows more branchings to take place than the `constrained' one. In the following discussion we will only refer to the `global, unconstrained' $z$ choice.


next up previous contents
Next: First branchings and matrix-element Up: Final-State Showers Previous: The choice of evolution   Contents
Stephen_Mrenna 2012-10-24