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Flavour and $x$ Correlations

In the new beam-remnants approach, the flavour content of the remnant is bookkept, and is used to determine possible flavours in consecutive interactions. Thus, the standard parton densities are only used to describe the hardest interaction. Already in the old model, the $x$ scale of parton densities is rescaled in subsequent interactions, such that the new $x' = 1$ corresponds to the remaining momentum rather than the original beam momentum, eq. ([*]). But now the distributions are not only squeezed in this manner, their shapes are also changed, as follows:

$\bullet$
Whenever a valence quark is kicked out, the number of remaining valence quarks of that species is reduced accordingly. Thus, for a proton, the valence d distribution is completely removed if the valence $\d $ quark has been kicked out, whereas the valence $\u $ distribution is halved when one of the two is kicked out. In cases where the valence and sea $\u $ and $\d $ quark distributions are not separately provided from the PDF libraries, it is assumed that the sea is flavour-antiflavour symmetric, so that one can write e.g.
\begin{displaymath}
u(x,Q^2) = u_{\mathrm{val}}(x,Q^2) + u_{\mathrm{sea}}(x,Q^2) =
u_{\mathrm{val}}(x,Q^2) + \overline{u}(x,Q^2).
\end{displaymath} (219)

The parametrized $u$ and $\overline{u}$ distributions are then used to find the relative probability for a kicked-out $\u $ quark to be either valence or sea.
$\bullet$
When a sea quark is kicked out, it must leave behind a corresponding antisea parton in the beam remnant, by flavour conservation. We call this a companion quark, and bookkeep it separately from the normal sea. In the perturbative approximation the sea quark $\ensuremath{\mathrm{q}_{\mathrm{s}}}$ and its companion $\ensuremath{\mathrm{q}_{\mathrm{c}}}$ (not to be confused with flavour labels) come from a gluon branching $\mathrm{g}\to \ensuremath{\mathrm{q}_{\mathrm{s}}}+ \ensuremath{\mathrm{q}_{\mathrm{c}}}$, where it is implicitly understood that if \ensuremath{\mathrm{q}_{\mathrm{s}}} is a quark, \ensuremath{\mathrm{q}_{\mathrm{c}}} is its antiquark, and vice versa. This branching often would not be in the perturbative regime, but we choose to make a perturbative ansatz, and also to neglect subsequent perturbative evolution of the $q_{\c }$ distribution. If so, the shape of the companion $q_{\c }$ distribution as a function of $x_{\c }$, given a sea parton at $x_{\mathrm{s}}$, becomes
\begin{displaymath}
q_{\c }(x_{\c }; x_{\mathrm{s}}) \propto \frac{g(x_{\c } + ...
...hrm{q}_{\mathrm{s}}}\ensuremath{\mathrm{q}_{\mathrm{c}}}}(z),
\end{displaymath} (220)

where
$\displaystyle z$ $\textstyle =$ $\displaystyle \frac{x_{\c }}{x_{\c } + x_{\mathrm{s}}},$ (221)
$\displaystyle P_{\mathrm{g}\to \ensuremath{\mathrm{q}_{\mathrm{s}}}\ensuremath{\mathrm{q}_{\mathrm{c}}}}(z)$ $\textstyle =$ $\displaystyle \frac{1}{2} \left( z^2 + (1-z)^2 \right),$ (222)
$\displaystyle g(x)$ $\textstyle \propto$ $\displaystyle \frac{(1-x)^n}{x}~~~~;~(n=\mathrm{MSTP(87)})$ (223)

is chosen. (Remember that this is supposed to occur at some low $Q^2$ scale, so $g(x)$ above should not be confused with the high-$Q^2$ gluon behaviour.)
$\bullet$
The normalization of valence and companion distributions is fixed by the respective number of quarks, i.e. the sum rules
$\displaystyle \int_0^{x_\mathrm{rem}}\! q_{\mathrm{v}}(x) \; \d x$ $\textstyle =$ $\displaystyle n_{\ensuremath{\mathrm{q}_{\mathrm{v}}}},$ (224)
$\displaystyle \int_0^{x_\mathrm{rem}}\! q_{\mathrm{c},i}(x;x_{\mathrm{s},i}) \; \d x$ $\textstyle =$ $\displaystyle 1 ~~~~
(\mathrm{for~each}~i),$ (225)

where $x_{\mathrm{rem}}$ is the longitudinal momentum fraction left after the previous interactions and $n_{\ensuremath{\mathrm{q}_{\mathrm{v}}}}$ is the number of $\mathrm{q}$ valence quarks remaining. Gluon and sea distributions do not have corresponding requirements. Therefore their normalization is adjusted, up or down, so as to obtain overall momentum conservation in the parton densities, i.e. to fulfil the remaining sum rule:
\begin{displaymath}
\int_0^{x_\mathrm{rem}}\! \left(\sum_{\mathrm{q}} q(x) + g(x) \right) \d x =
x_{\mathrm{rem}}
\end{displaymath} (226)

Detailed formulae may be found in [Sjö04]. However, in that article, the sea+gluon rescaling factor, eq. (4.26), was derived assuming the momentum fraction taken by a companion quark scaled like $1/X$, where $X$ is the total remaining momentum. As it happens, the momentum fraction, $x_s$, taken by the sea quark that originally gave rise to the companion quark, is already taken into account in the definition of the companion distribution. Hence a factor $(X+x_s)/X$ should be introduced for each companion distribution in eq. (4.26). With this change, the companions take up more `room' in momentum space. In fact, in extreme cases it is possible to create so many companions in the beam remnant that the sum of their momenta is larger than allowed by momentum conservation, i.e. the rescaling factor becomes negative. Note that this occurs extremely rarely, and only when very many multiple interactions have occurred in an event. Since the companions are now taking up more room than allowed, we address the problem by scaling them (all of them) down by precisely the amount needed to restore momentum conservation.

The above parton-density strategy is not only used to pick a succession of hard interactions. Each interaction may now, unlike before, have initial- and final-state shower activity associated with it. The initial-state shower is constructed by backwards evolution, using the above parton densities. Even if the hard scattering does not involve a valence quark, say, the possibility exists that the shower will reconstruct back to one.


next up previous contents
Next: Colour Topologies Up: Beam Remnants (and Multiple Previous: Beam Remnants (and Multiple   Contents
Stephen Mrenna 2007-10-30