Beam Remnants -- Old Model

The initial-state radiation algorithm reconstructs one shower initiator in each beam. (If initial-state radiation is not included, the initiator is nothing but the incoming parton to the hard interaction.) Together the two initiators delineate an interaction subsystem, which contains all the partons that participate in the initial-state showers, in the hard interaction, and in the final-state showers. Left behind are two beam remnants which, to first approximation, just sail through, unaffected by the hard process. (The issue of additional interactions is covered in the next section.)

A description of the beam remnant structure contains a few components. First, given the flavour content of a (colour-singlet) beam particle, and the flavour and colour of the initiator parton, it is possible to reconstruct the flavour and colour of the beam remnant. Sometimes the remnant may be represented by just a single parton or diquark, but often the remnant has to be subdivided into two separate objects. In the latter case it is necessary to share the remnant energy and momentum between the two. Due to Fermi motion inside hadron beams, the initiator parton may have a `primordial $k_{\perp}$' transverse momentum motion, which has to be compensated by the beam remnant. If the remnant is subdivided, there may also be a relative transverse momentum. In the end, total energy and momentum has to be conserved. To first approximation, this is ensured within each remnant separately, but some final global adjustments are necessary to compensate for the primordial $k_{\perp}$ and any effective beam remnant mass.

Consider first a proton (or, with trivial modifications, any other baryon or antibaryon).

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If the initiator parton is a $\u$ or $\d$ quark, it is assumed to be a valence quark, and therefore leaves behind a diquark beam remnant, i.e. either a $\u\d$ or a $\u\u$ diquark, in a colour antitriplet state. Relative probabilities for different diquark spins are derived within the context of the non-relativistic SU(6) model, i.e. flavour SU(3) times spin SU(2). Thus a $\u\d$ is $3/4$ $\u\d _0$ and $1/4$ $\u\d _1$, while a $\u\u$ is always $\u\u _1$.

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An initiator gluon leaves behind a colour octet $\u\u\d$ state, which is subdivided into a colour triplet quark and a colour antitriplet diquark. SU(6) gives the appropriate subdivision, $1/2$ of the time into $\u + \u\d _0$, $1/6$ into $\u + \u\d _1$ and $1/3$ into $\d + \u\u _1$.

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A sea quark initiator, such as an $\mathrm{s}$, leaves behind a $\u\u\d\overline{\mathrm{s}}$ four-quark state. The PDG flavour coding scheme and the fragmentation routines do not foresee such a state, so therefore it is subdivided into a meson plus a diquark, i.e. $1/2$ into $\u\overline{\mathrm{s}}+ \u\d _0$, $1/6$ into $\u\overline{\mathrm{s}}+ \u\d _1$ and $1/3$ into $\d\overline{\mathrm{s}}+ \u\u _1$. Once the flavours of the meson are determined, the choice of meson multiplet is performed as in the standard fragmentation description.

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Finally, an antiquark initiator, such as an $\overline{\mathrm{s}}$, leaves behind a $\u\u\d\mathrm{s}$ four-quark state, which is subdivided into a baryon plus a quark. Since, to first approximation, the $\mathrm{s}\overline{\mathrm{s}}$ pair comes from the branching $\mathrm{g}\to \mathrm{s}\overline{\mathrm{s}}$ of a colour octet gluon, the subdivision $\u\u\d + \mathrm{s}$ is not allowed, since it would correspond to a colour-singlet $\mathrm{s}\overline{\mathrm{s}}$. Therefore the subdivision is $1/2$ into $\u\d _0\mathrm{s}+ \u$, $1/6$ into $\u\d _1\mathrm{s}+ \u$ and $1/3$ into $\u\u _1\mathrm{s}+ \d$. A baryon is formed among the ones possible for the given flavour content and diquark spin, according to the relative probabilities used in the fragmentation. One could argue for an additional weighting to count the number of baryon states available for a given diquark plus quark combination, but this has not been included.

One may note that any $\u$ or $\d$ quark taken out of the proton is automatically assumed to be a valence quark. Clearly this is unrealistic, but not quite as bad as it might seem. In particular, one should remember that the beam remnant scenario is applied to the initial-state shower initiators at a scale of $Q_0 \approx 1$ GeV and at an $x$ value usually much larger than the $x$ at the hard scattering. The sea quark contribution therefore normally is negligible.

For a meson beam remnant, the rules are in the same spirit, but somewhat easier, since no diquark or baryons need be taken into account. Thus a valence quark (antiquark) initiator leaves behind a valence antiquark (quark), a gluon initiator leaves behind a valence quark plus a valence antiquark, and a sea quark (antiquark) leaves behind a meson (which contains the partner to the sea parton) plus a valence antiquark (quark).

A resolved photon is similar in spirit to a meson. A VMD photon is associated with either $\rho^0$, $\omega$, $\phi$ or $\mathrm{J}/\psi$, and so corresponds to a well-defined valence flavour content. Since the $\rho^0$ and $\omega$ are supposed to add coherently, the $\u\overline{\mathrm{u}}: \d\overline{\mathrm{d}}$ mixing is in the ratio $4:1$. Similarly a GVMD state is characterized by its $\mathrm{q}\overline{\mathrm{q}}$ classification, in rates according to $e_{\mathrm{q}}^2$ times a mass suppression for heavier quarks.

In the older photon physics options, where a quark content inside an electron is obtained by a numerical convolution, one does not have to make the distinction between valence and sea flavour. Thus any quark (antiquark) initiator leaves behind the matching antiquark (quark), and a gluon leaves behind a quark + antiquark pair. The relative quark flavour composition in the latter case is assumed proportional to $e_{\mathrm{q}}^2$ among light flavours, i.e. $2/3$ into $\u + \overline{\mathrm{u}}$, $1/6$ into $\d + \overline{\mathrm{d}}$, and $1/6$ into $\mathrm{s}+ \overline{\mathrm{s}}$. If one wanted to, one could also have chosen to represent the remnant by a single gluon.

If no initial-state radiation is assumed, an electron (or, in general, a lepton or a neutrino) leaves behind no beam remnant. Also when radiation is included, one would expect to recover a single electron with the full beam energy when the shower initiator is reconstructed. This does not have to happen, e.g. if the initial-state shower is cut off at a non-vanishing scale, such that some of the emission at low $Q^2$ values is not simulated. Further, for purely technical reasons, the distribution of an electron inside an electron, $f_{\mathrm{e}}^{\mathrm{e}}(x,Q^2)$, is cut off at $x = 1 - 10^{-10}$. This means that always, when initial-state radiation is included, a fraction of at least $10^{-10}$ of the beam energy has to be put into one single photon along the beam direction, to represent this not simulated radiation. The physics is here slightly different from the standard beam remnant concept, but it is handled with the same machinery. Beam remnants can also appear when the electron is resolved with the use of parton distributions, but initial-state radiation is switched off. Conceptually, this is a contradiction, since it is the initial-state radiation that builds up the parton distributions, but sometimes the combination is still useful. Finally, since QED radiation has not yet been included in events with resolved photons inside electrons, also in this case effective beam remnants have to be assigned by the program.

The beam remnant assignments inside an electron, in either of the cases above, is as follows.

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An $\mathrm{e}^-$ initiator leaves behind a $\gamma$ remnant.

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A $\gamma$ initiator leaves behind an $\mathrm{e}^-$ remnant.

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An $\mathrm{e}^+$ initiator leaves behind an $\mathrm{e}^- + \mathrm{e}^-$ remnant.

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A $\mathrm{q}$ ( $\overline{\mathrm{q}}$) initiator leaves behind a $\overline{\mathrm{q}}+ \mathrm{e}^-$ ( $\mathrm{q}+ \mathrm{e}^-$) remnant.

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A $\mathrm{g}$ initiator leaves behind a $\mathrm{g}+ \mathrm{e}^-$ remnant. One could argue that, in agreement with the treatment of photon beams above, the remnant should be $\mathrm{q}+ \overline{\mathrm{q}}+ \mathrm{e}^-$. The program currently does not allow for three beam remnant objects, however.

It is customary to assign a primordial transverse momentum to the shower initiator, to take into account the motion of quarks inside the original hadron, basically as required by the uncertainty principle. A number of the order of $\langle k_{\perp}\rangle \approx m_{\mathrm{p}}/3 \approx 300$ MeV could therefore be expected. However, in hadronic collisions much higher numbers than that are often required to describe data, typically of the order of or even above 1 GeV [EMC87,Bál01] if a Gaussian parameterization is used. (This number is now the default.) Thus, an interpretation as a purely nonperturbative motion inside a hadron is difficult to maintain.

Instead a likely culprit is the initial-state shower algorithm. This is set up to cover the region of hard emissions, but may miss out on some of the softer activity, which inherently borders on nonperturbative physics. By default, the shower does not evolve down to scales below $Q_0 = 1$ GeV. Any shortfall in shower activity around or below this cutoff then has to be compensated by the primordial $k_{\perp}$ source, which thereby largely loses its original meaning. One specific reason for such a shortfall is that the current initial-state shower algorithm does not include non-order emissions in $Q^2$, as is predicted to occur especially at small $x$ and $Q^2$ within the BFKL/CCFM framework [Lip76,Cia87].

By the hard scattering and 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,

$${\cal P}(\chi) \propto \frac{(1 - \chi)^3} {\sqrt[4]{\chi^2 + c_{\mathrm{min}}^2}} ~,$$ (200)

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

$$m_{\perp}^2 = \frac{m_{\perp 1}^2}{\chi} + \frac{m_{\perp 2}^2}{1 - \chi} ~,$$ (201)

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.