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First branchings and matrix-element matching

The final-state evolution is normally started from some initial parton pair $1 + 2$, at a $Q_{\mathrm{max}}^2$ scale determined by deliberations already discussed. When the evolution of parton 1 is considered, it is assumed that parton 2 is on-shell, so that the parton 1 energy and momentum are simple functions of its mass (and of the c.m. energy of the pair, which is fixed), and hence also the allowed $z_1$ range for splittings is a function of this mass, eq. ([*]). Correspondingly, parton 2 is evolved under the assumption that parton 1 is on-shell. After both partons have been assigned masses, their correct energies may be found, which are smaller than originally assumed. Therefore the allowed $z$ ranges have shrunk, and it may happen that a branching has been assigned a $z$ value outside this range. If so, the parton is evolved downwards in mass from the rejected mass value; if both $z$ values are rejected, the parton with largest mass is evolved further. It may also happen that the sum of $m_1$ and $m_2$ is larger than the c.m. energy, in which case the one with the larger mass is evolved downwards. The checking and evolution steps are iterated until an acceptable set of $m_1$, $m_2$, $z_1$ and $z_2$ has been found.

The procedure is an extension of the veto algorithm, where an initial overestimation of the allowed $z$ range is compensated by rejection of some branchings. One should note, however, that the veto algorithm is not strictly applicable for the coupled evolution in two variables ($m_1$ and $m_2$), and that therefore some arbitrariness is involved. This is manifest in the choice of which parton will be evolved further if both $z$ values are unacceptable, or if the mass sum is too large.

For a pair of particles which comes from the decay of a resonance within the Standard Model or its MSSM supersymmetric extension, the first branchings are matched to the explicit first-order matrix elements for decays with one additional gluon in the final state, see subsection [*] below. Here we begin by considering in detail how $\gamma^* / \mathrm{Z}^0\to \mathrm{q}\overline{\mathrm{q}}$ is matched to the matrix element for $\gamma^* / \mathrm{Z}^0\to\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ [Ben87a].

The matching is based on a mapping of the parton-shower variables on to the 3-jet phase space. To produce a 3-jet event, $\gamma^* / \mathrm{Z}^0\to \mathrm{q}(p_1) \overline{\mathrm{q}}(p_2) \mathrm{g}(p_3)$, in the shower language, one will pass through an intermediate state, where either the $\mathrm{q}$ or the $\overline{\mathrm{q}}$ is off the mass shell. If the former is the case then

$\displaystyle m^2$ $\textstyle =$ $\displaystyle (p_1 + p_3)^2 = E_{\mathrm{cm}}^2 (1 - x_2) ~,$  
$\displaystyle z$ $\textstyle =$ $\displaystyle \frac{E_1}{E_1 + E_3} = \frac{x_1}{x_1 + x_3} =
\frac{x_1}{2-x_2} ~,$ (172)

where $x_i = 2 E_i / E_{\mathrm{cm}}$. The $\overline{\mathrm{q}}$ emission case is obtained with $1 \leftrightarrow 2$. The parton-shower splitting expression in terms of $m^2$ and $z$, eq. ([*]), can therefore be translated into the following differential 3-jet rate:
$\displaystyle \frac{1}{\sigma} \, \frac{\d\sigma_{\mathrm{PS}}}{\d x_1 \, \d x_2}$ $\textstyle =$ $\displaystyle \frac{\alpha_{\mathrm{s}}}{2 \pi} \, C_F \, \frac{1}{(1-x_1)(1-x_2)}
  $\textstyle \times$ $\displaystyle \left\{ \frac{1-x_1}{x_3} \left( 1 + \left( \frac{x_1}{2-x_2}
...c{1-x_2}{x_3} \left( 1 +
\left( \frac{x_2}{2-x_1} \right)^2 \right) \right\} ~,$ (173)

where the first term inside the curly bracket comes from emission off the quark and the second term from emission off the antiquark. The corresponding expression in matrix-element language is
\frac{1}{\sigma} \, \frac{\d\sigma_{\mathrm{ME}}}{\d x_1 \, ...
...F \, \frac{1}{(1-x_1)(1-x_2)}
\left\{ x_1^2 +x_2^2 \right\} ~.
\end{displaymath} (174)

With the kinematics choice of PYTHIA, the matrix-element expression is always smaller than the parton-shower one. It is therefore possible to run the shower as usual, but to impose an extra weight factor $\d\sigma_{\mathrm{ME}} / \d\sigma_{\mathrm{PS}}$, which is just the ratio of the expressions in curly brackets. If a branching is rejected, the evolution is continued from the rejected $Q^2$ value onwards (the veto algorithm). The weighting procedure is applied to the first branching of both the $\mathrm{q}$ and the $\overline{\mathrm{q}}$, in each case with the (nominal) assumption that none of the other partons branch (neither the sister nor the daughters), so that the relations of eq. ([*]) are applicable.

If a photon is emitted instead of a gluon, the emission rate in parton showers is given by

$\displaystyle \frac{1}{\sigma} \, \frac{\d\sigma_{\mathrm{PS}}}{\d x_1 \, \d x_2}$ $\textstyle =$ $\displaystyle \frac{\alpha_{\mathrm{em}}}{2 \pi} \, \frac{1}{(1-x_1)(1-x_2)}
  $\textstyle \times$ $\displaystyle \left\{ e_{\mathrm{q}}^2 \, \frac{1-x_1}{x_3} \left( 1 + \left(
\left( 1 + \left( \frac{x_2}{2-x_1} \right)^2 \right) \right\} ~,$ (175)

and in matrix elements by [Gro81]
\frac{1}{\sigma} \, \frac{\d\sigma_{\mathrm{ME}}}{\d x_1 \, ...
...{1-x_2}{x_3} \right)^2 \left( x_1^2 +x_2^2 \right) \right\} ~.
\end{displaymath} (176)

As in the gluon emission case, a weighting factor $\d\sigma_{\mathrm{ME}} / \d\sigma_{\mathrm{PS}}$ can therefore be applied when either the original $\mathrm{q}$ ($\ell$) or the original $\overline{\mathrm{q}}$ ( $\overline{\ell}$) emits a photon. For a neutral resonance, such as $\mathrm{Z}^0$, where $e_{\overline{\mathrm{q}}} = - e_{\mathrm{q}}$, the above expressions simplify and one recovers exactly the same ratio $\d\sigma_{\mathrm{ME}} / \d\sigma_{\mathrm{PS}}$ as for gluon emission.

Compared with the standard matrix-element treatment, a few differences remain. The shower one automatically contains the Sudakov form factor and an $\alpha_{\mathrm{s}}$ running as a function of the $p_{\perp}^2$ scale of the branching. The shower also allows all partons to evolve further, which means that the naïve kinematics assumed for a comparison with matrix elements is modified by subsequent branchings, e.g. that the energy of parton 1 is reduced when parton 2 is assigned a mass. All these effects are formally of higher order, and so do not affect a first-order comparison. This does not mean that the corrections need be small, but experimental results are encouraging: the approach outlined does as good as explicit second-order matrix elements for the description of 4-jet production, better in some respects (like overall rate) and worse in others (like some angular distributions).

next up previous contents
Next: Subsequent branches and angular Up: Final-State Showers Previous: The choice of energy   Contents
Stephen Mrenna 2007-10-30