For the other beam remnants, the relative energy-sharing variable
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,
for the energy fraction taken by the
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
remnant is of an intermediate type,
In a photon beam, with a remnant , the variable is chosen the same way it would have been in a corresponding meson remnant.
Before the variable is used to assign remnant momenta, it is also necessary to consider the issue of primordial . The initiator partons are thus assigned each a 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 axis, to one where the initiators have the required primordial values.
The 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 , which is then always chosen as for pairs in the fragmentation description.
The variable is interpreted as a sharing of light-cone energy and
momentum, i.e. for the beam moving in the direction and
for the other one. When the two transverse masses
and of a composite remnant have been
constructed, the total transverse mass can therefore be found as
Whether there is one remnant parton or two, the transverse mass of the remnant is not likely to agree with 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 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 , 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 is conserved, but not energy or momentum separately. If both beam remnants have low energy, but both still exist, then the one with lower is the one that will not be boosted.