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Remnant Kinematics

By the hard scattering and the initial-state radiation machinery, the shower initiator has been assigned some fraction $x$ of the four-momentum of the beam particle, leaving behind $1-x$ to the remnant. If the remnant consists of two objects, this energy and momentum has to be shared, somehow. For an electron in the old photoproduction machinery, the sharing is given from first principles: if, e.g., the initiator is a $\mathrm{q}$, then that $\mathrm{q}$ was produced in the sequence of branchings $\mathrm{e}\to \gamma \to \mathrm{q}$, where $x_{\gamma}$ is distributed according to the convolution in eq. ([*]). Therefore the $\overline{\mathrm{q}}$ remnant takes a fraction $\chi = (x_{\gamma} -x)/(1-x)$ of the total remnant energy, and the $\mathrm{e}$ takes $1 - \chi$.

For the other beam remnants, the relative energy-sharing variable $\chi$ is not known from first principles, but picked according to some suitable parameterization. Normally several different options are available, that can be set separately for baryon and meson beams, and for hadron + quark and quark + diquark (or antiquark) remnants. In one extreme are shapes in agreement with naïve counting rules, i.e. where energy is shared evenly between `valence' partons. For instance, ${\cal P}(\chi) = 2 \, (1-\chi)$ for the energy fraction taken by the $\mathrm{q}$ in a $\mathrm{q}+ \mathrm{q}\mathrm{q}$ remnant. In the other extreme, an uneven distribution could be used, like in parton distributions, where the quark only takes a small fraction and most is retained by the diquark. The default for a $\mathrm{q}+ \mathrm{q}\mathrm{q}$ remnant is of an intermediate type,

{\cal P}(\chi) \propto \frac{(1 - \chi)^3}
{\sqrt[4]{\chi^2 + c_{\mathrm{min}}^2}} ~,
\end{displaymath} (199)

with $c_{\mathrm{min}} = 2 \langle m_{\mathrm{q}} \rangle / E_{\mathrm{cm}} = (0.6$ GeV) $/ E_{\mathrm{cm}}$ providing a lower cut-off. The default when a hadron is split off to leave a quark or diquark remnant is to use the standard Lund symmetric fragmentation function. In general, the more uneven the sharing of the energy, the less the total multiplicity in the beam-remnant fragmentation. If no multiple interactions are allowed, a rather even sharing is needed to come close to the experimental multiplicity (and yet one does not quite make it). With an uneven sharing there is room to generate more of the total multiplicity by multiple interactions [Sjö87a].

In a photon beam, with a remnant $\mathrm{q}+ \overline{\mathrm{q}}$, the $\chi$ variable is chosen the same way it would have been in a corresponding meson remnant.

Before the $\chi$ variable is used to assign remnant momenta, it is also necessary to consider the issue of primordial $k_{\perp}$. The initiator partons are thus assigned each a $k_{\perp}$ value, vanishing for an electron or photon inside an electron, distributed either according to a Gaussian or an exponential shape for a hadron, and according to either of these shapes or a power-like shape for a quark or gluon inside a photon (which may in its turn be inside an electron). The interaction subsystem is boosted and rotated to bring it from the frame assumed so far, with each initiator along the $\pm z$ axis, to one where the initiators have the required primordial $k_{\perp}$ values.

The $p_{\perp}$ recoil is taken by the remnant. If the remnant is composite, the recoil is evenly split between the two. In addition, however, the two beam remnants may be given a relative $p_{\perp}$, which is then always chosen as for $\mathrm{q}_i \overline{\mathrm{q}}_i$ pairs in the fragmentation description.

The $\chi$ variable is interpreted as a sharing of light-cone energy and momentum, i.e. $E + p_z$ for the beam moving in the $+z$ direction and $E - p_z$ for the other one. When the two transverse masses $m_{\perp 1}$ and $m_{\perp 2}$ of a composite remnant have been constructed, the total transverse mass can therefore be found as

m_{\perp}^2 = \frac{m_{\perp 1}^2}{\chi} +
\frac{m_{\perp 2}^2}{1 - \chi} ~,
\end{displaymath} (200)

if remnant 1 is the one that takes the fraction $\chi$. The choice of a light-cone interpretation to $\chi$ means the definition is invariant under longitudinal boosts, and therefore does not depend on the beam energy itself. A $\chi$ value close to the naïve borders 0 or 1 can lead to an unreasonably large remnant $m_{\perp}$. Therefore an additional check is introduced, that the remnant $m_{\perp}$ be smaller than the naïve c.m. frame remnant energy, $(1-x) E_{\mathrm{cm}}/2$. If this is not the case, a new $\chi$ and a new relative transverse momentum is selected.

Whether there is one remnant parton or two, the transverse mass of the remnant is not likely to agree with $1-x$ times the mass of the beam particle, i.e. it is not going to be possible to preserve the energy and momentum in each remnant separately. One therefore allows a shuffling of energy and momentum between the beam remnants from each of the two incoming beams. This may be achieved by performing a (small) longitudinal boost of each remnant system. Since there are two boost degrees of freedom, one for each remnant, and two constraints, one for energy and one for longitudinal momentum, a solution may be found.

Under some circumstances, one beam remnant may be absent or of very low energy, while the other one is more complicated. One example is Deeply Inelastic Scattering in $\mathrm{e}\mathrm{p}$ collisions, where the electron leaves no remnant, or maybe only a low-energy photon. It is clearly then not possible to balance the two beam remnants against each other. Therefore, if one beam remnant has an energy below 0.2 of the beam energy, i.e. if the initiator parton has $x > 0.8$, then the two boosts needed to ensure energy and momentum conservation are instead performed on the other remnant and on the interaction subsystem. If there is a low-energy remnant at all then, before that, energy and momentum are assigned to the remnant constituent(s) so that the appropriate light-cone combination $E \pm p_z$ is conserved, but not energy or momentum separately. If both beam remnants have low energy, but both still exist, then the one with lower $m_{\perp} / E$ is the one that will not be boosted.

next up previous contents
Next: Multiple Interactions Up: Beam Remnants Previous: Primordial   Contents
Stephen Mrenna 2007-10-30