Beam Remnants
Introduction
The BeamParticle class contains information on all partons
extracted from a beam (so far). As each consecutive multiparton interaction
defines its respective incoming parton to the hard scattering a
new slot is added to the list. This information is modified when
the backwards evolution of the spacelike shower defines a new
initiator parton. It is used, both for the multiparton interactions
and the spacelike showers, to define rescaled parton densities based
on the x and flavours already extracted, and to distinguish
between valence, sea and companion quarks. Once the perturbative
evolution is finished, further beam remnants are added to obtain a
consistent set of flavours. The current physics framework is further
described in [Sjo04].
The introduction of rescattering
in the multiparton interactions framework further complicates the
processing of events. Specifically, when combined with showers,
the momentum of an individual parton is no longer uniquely associated
with one single subcollision. Nevertheless the parton is classified
with one system, owing to the technical and administrative complications
of more complete classifications. Therefore the addition of primordial
kT to the subsystem initiator partons does not automatically
guarantee overall pT conservation. Various tricks are used to
minimize the mismatch, with a brute force shift of all parton
pT's as a final step.
Much of the above information is stored in a vector of
ResolvedParton objects, which each contains flavour and
momentum information, as well as valence/companion information and more.
The BeamParticle method list() shows the
contents of this vector, mainly for debug purposes.
The BeamRemnants class takes over for the final step
of adding primordial kT to the initiators and remnants,
assigning the relative longitudinal momentum sharing among the
remnants, and constructing the overall kinematics and colour flow.
This step couples the two sides of an event, and could therefore
not be covered in the BeamParticle class, which only
considers one beam at a time.
The methods of these classes are not intended for general use,
and so are not described here.
In addition to the parameters described on this page, note that the
choice of parton densities is made
in the Pythia class. Then pointers to the pdf's are handed
on to BeamParticle at initialization, for all subsequent
usage.
Primordial kT
The primordial kT of initiators of hard-scattering subsystems
are selected according to Gaussian distributions in p_x and
p_y separately. The widths of these distributions are chosen
to be dependent on the hard scale of the central process and on the mass
of the whole subsystem defined by the two initiators:
sigma = (sigma_soft * Q_half + sigma_hard * Q) / (Q_half + Q)
* m / (m + m_half * y_damp)
Here Q is the hard-process renormalization scale for the
hardest process and the pT scale for subsequent multiparton
interactions, m the mass of the system, and
sigma_soft, sigma_hard, Q_half,
m_half and y_damp parameters defined below.
Furthermore each separately defined beam remnant has a distribution
of width sigma_remn, independently of kinematical variables.
flag BeamRemnants:primordialKT
(default = on)
Allow or not selection of primordial kT according to the
parameter values below.
parm BeamRemnants:primordialKTsoft
(default = 0.9; minimum = 0.)
The width sigma_soft in the above equation, assigned as a
primordial kT to initiators in the soft-interaction limit.
parm BeamRemnants:primordialKThard
(default = 1.8; minimum = 0.)
The width sigma_hard in the above equation, assigned as a
primordial kT to initiators in the hard-interaction limit.
parm BeamRemnants:halfScaleForKT
(default = 1.5; minimum = 0.)
The scale Q_half in the equation above, defining the
half-way point between hard and soft interactions.
parm BeamRemnants:halfMassForKT
(default = 1.; minimum = 0.)
The scale m_half in the equation above, defining the
half-way point between low-mass and high-mass subsystems.
(Kinematics construction can easily fail if a system is assigned
a primordial kT value higher than its mass, so the
mass-dampening is intended to reduce some troubles later on.)
parm BeamRemnants:reducedKTatHighY
(default = 0.5; minimum = 0.; maximum = 1.)
For a system of mass m and energy E the
dampening factor y_damp above is defined as
y_damp = pow( E/m, r_red), where r_red is the
current parameter. The effect is to reduce the primordial kT
of low-mass systems extra much if they are at large rapidities (recall
that E/m = cosh(y) before kT is added). The reason
for this dampening is purely technical, and for reasonable values
should not have dramatic consequences overall.
parm BeamRemnants:primordialKTremnant
(default = 0.4; minimum = 0.)
The width sigma_remn, assigned as a primordial kT
to beam-remnant partons.
A net kT imbalance is obtained from the vector sum of the
primordial kT values of all initiators and all beam remnants.
This quantity is compensated by a shift shared equally between
all partons, except that the dampening factor m / (m_half + m)
is again used to suppress the role of small-mass systems.
Note that the current sigma definition implies that
<pT^2> = <p_x^2>+ <p_y^2> = 2 sigma^2.
It thus cannot be compared directly with the sigma
of nonperturbative hadronization, where each quark-antiquark
breakup corresponds to <pT^2> = sigma^2 and only
for hadrons it holds that <pT^2> = 2 sigma^2.
The comparison is further complicated by the reduction of
primordial kT values by the overall compensation mechanism.
flag BeamRemnants:rescatterRestoreY
(default = off)
Is only relevant when rescattering
is switched on in the multiparton interactions scenario. For a normal
interaction the rapidity and mass of a system is preserved when
primordial kT is introduced, by appropriate modification of the
incoming parton momenta. Kinematics construction is more complicated for
a rescattering, and two options are offered. Differences between these
can be used to explore systematic uncertainties in the rescattering
framework.
The default behaviour is to keep the incoming rescattered parton as is,
but to modify the unrescattered incoming parton so as to preserve the
invariant mass of the system. Thereby the rapidity of the rescattering
is modified.
The alternative is to retain the rapidity (and mass) of the rescattered
system when primordial kT is introduced. This is made at the
expense of a modified longitudinal momentum of the incoming rescattered
parton, so that it does not agree with the momentum it ought to have had
by the kinematics of the previous interaction.
For a double rescattering, when both incoming partons have already scattered,
there is no obvious way to retain the invariant mass of the system in the
first approach, so the second is always used.
Colour flow
The colour in the separate subproccsses are tied together via the assignment
of colour flow in the beam remnants. The assignment of colour flow is not
known from first principles and therefore it is not an unambiguous procedure.
Thus two different models have been implemented in Pythia. These
will be referred to as new and old, based on the time of the implementation.
The old model tries to reconstruct the colour flow in a way that a LO PS would
produce the beam remnants. The starting point is the junction structure of the
beam particle (if it is a baryon). The gluons are attached to a quark line and
quark-antiquark pairs are added as if coming from a gluon splittings. Thus
this model captures the qualitative behaviour that is expected from leading
colour QCD. The model is described in more detail in [Sjo04].
The new model is built on the full SU(3) colour structure of QCD. The
starting point is the scattered partons from the MPI. Each of these are
initially assumed uncorrelated in colour space, allowing the total outgoing
colour configuration to be calculated as an SU(3) product. Since the beam
particle is a colour singlet, the beam remnant colour configuration has to be
the inverse of the outgoing colour configuration. The minimum amount of gluons
are added to the beam remnant in order to obtain this colour configuration.
The above assumption of uncorrelated MPIs in colour space is a good
assumption for a few well separated hard MPIs. However if the number of MPIs
become large and ISR is included, such that the energy scale becomes lower
(and thus distances becomes larger), the assumption loses its validity. This
is due to saturation effects. The modelling of saturation is done in crude
manner, as an exponential suppresion of high multiplet states.
None of the models above can provide a full description of the colour
flow in an event, however. Therefore additional colour reconfiguration
is needed. This is referred to as colour reconnection. Several different
models for colour reconnection are implemented, see
Colour Reconection.
mode BeamRemnants:remnantMode
(default = 0; minimum = 0; maximum = 1)
Switch to choose between the two different colour models for the beam remnant.
option 0 : The old beam remnant model.
option 1 : The new beam remnant model.
parm BeamRemnants:saturation
(default = 5; minimum = 0.1; maximum = 100000)
Controls the suppresion due to saturation in the new model. The exact formula
used is exp(-M / k), where M is the multiplet size and k is this
parameter. Thus a small number will result in a large saturation.
Further variables
mode BeamRemnants:maxValQuark
(default = 3; minimum = 0; maximum = 5)
The maximum valence quark kind allowed in acceptable incoming beams,
for which multiparton interactions are simulated. Default is that hadrons
may contain u, d and s quarks,
but not c and b ones, since sensible
kinematics has not really been worked out for the latter.
mode BeamRemnants:companionPower
(default = 4; minimum = 0; maximum = 4)
When a sea quark has been found, a companion antisea quark ought to be
nearby in x. The shape of this distribution can be derived
from the gluon mother distribution convoluted with the
g → q qbar splitting kernel. In practice, simple solutions
are only feasible if the gluon shape is assumed to be of the form
g(x) ~ (1 - x)^p / x, where p is an integer power,
the parameter above. Allowed values correspond to the cases programmed.
Since the whole framework is approximate anyway, this should be good
enough. Note that companions typically are found at small Q^2,
if at all, so the form is supposed to represent g(x) at small
Q^2 scales, close to the lower cutoff for multiparton interactions.
When assigning relative momentum fractions to beam-remnant partons,
valence quarks are chosen according to a distribution like
(1 - x)^power / sqrt(x). This power is given below
for quarks in mesons, and separately for u and d
quarks in the proton, based on the approximate shape of low-Q^2
parton densities. The power for other baryons is derived from the
proton ones, by an appropriate mixing. The x of a diquark
is chosen as the sum of its two constituent x values, and can
thus be above unity. (A common rescaling of all remnant partons and
particles will fix that.) An additional enhancement of the diquark
momentum is obtained by its x value being rescaled by the
valenceDiqEnhance factor.
parm BeamRemnants:valencePowerMeson
(default = 0.8; minimum = 0.)
The abovementioned power for valence quarks in mesons.
parm BeamRemnants:valencePowerUinP
(default = 3.5; minimum = 0.)
The abovementioned power for valence u quarks in protons.
parm BeamRemnants:valencePowerDinP
(default = 2.0; minimum = 0.)
The abovementioned power for valence d quarks in protons.
parm BeamRemnants:valenceDiqEnhance
(default = 2.0; minimum = 0.5; maximum = 10.)
Enhancement factor for valence diquarks in baryons, relative to the
simple sum of the two constituent quarks.
parm BeamRemnants:gluonPower
(default = 4.0; minimum = 0.)
The abovementioned power for gluons.
parm BeamRemnants:xGluonCutoff
(default = 1E-7; minimum = 1E-10; maximum = 1)
The gluon PDF is approximated with g(x) ~ (1 - x)^p / x, which
integrates to infinity when integrated from 0 to 1. This cut-off is
introduced as a minimum to avoid the problems with infinities.
flag BeamRemnants:allowJunction
(default = on)
The off option is intended for debug purposes only, as
follows. When more than one valence quark is kicked out of a baryon
beam, as part of the multiparton interactions scenario, the subsequent
hadronization is described in terms of a junction string topology.
This description involves a number of technical complications that
may make the program more unstable. As an alternative, by switching
this option off, junction configurations are rejected (which gives
an error message that the remnant flavour setup failed), and the
multiparton interactions and showers are redone until a
junction-free topology is found.
flag BeamRemnants:beamJunction
(default = off)
This parameter is only relevant if the new colour reconnection scheme is used.
(see colour reconnection)
This parameter tells whether to form a junction or a di-quark if more
than two valence quarks are found in the beam remnants. If off a di-quark is
formed and if on a junction will be formed.
flag BeamRemnants:allowBeamJunction
(default = on)
This parameter is only relevant if the new Beam remnant model is used.
This parameter tells whether to allow the formation of junction structures
in the colour configuration of the scattered partons.
mode BeamRemnants:unresolvedHadron
(default = 0; minimum = 0; maximum = 3)
Switch to to force either or both of the beam remnants to collapse to a
single hadron, namely the original incoming one. Must only be used when this
is physically meaningful, e.g. when a photon can be viewed as emitted from
a proton that does not break up in the process.
option 0 : Both hadronic beams are resolved.
option 1 : Beam A is unresolved, beam B resolved.
option 2 : Beam A is resolved, beam B unresolved.
option 3 : Both hadronic beams are unresolved.