Beam Remnants
Introduction
The BeamParticle class contains information on all partons
extracted from a beam (so far). As each consecutive multiple 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 multiple 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 multiple 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_half + m)
Here Q is the hard-process renormalization scale for the
hardest process and the pT scale for subsequent multiple
interactions, m the mass of the system, and
sigma_soft, sigma_hard, Q_half and
m_half 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 = 0.5; 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 = 2.0; 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.; 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: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 multiple 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 flows in the separate subprocesses defined in the
multiple-interactions scenario are tied together via the assignment
of colour flow in the beam remnant. This is not an unambiguous
procedure, but currently no parameters are directly associated with it.
However, a simple "minimal" procedure of colour flow only via the beam
remnants does not result in a scenario in
agreement with data, notably not a sufficiently steep rise of
<pT>(n_ch). The true origin of this behaviour and the
correct mechanism to reproduce it remains one of the big unsolved issues
at the borderline between perturbative and nonperturbative QCD.
As a simple attempt, an additional step is introduced, wherein the gluons
of a lower-pT system are merged with the ones in a higher-pT one.
flag BeamRemnants:reconnectColours
(default = on)
Allow or not a system to be merged with another one.
parm BeamRemnants:reconnectRange
(default = 10.0; minimum = 0.; maximum = 10.)
A system with a hard scale pT can be merged with one of a
harder scale with a probability that is
pT0_Rec^2 / (pT0_Rec^2 + pT^2), where
pT0_Rec is reconnectRange times pT0,
the latter being the same energy-dependent dampening parameter as
used for multiple interactions.
Thus it is easy to merge a low-pT system with any other,
but difficult to merge two high-pT ones with each other.
The procedure is used iteratively. Thus first the reconnection probability
P = pT0_Rec^2 / (pT0_Rec^2 + pT^2) of the lowest-pT
system is found, and gives the probability for merger with the
second-lowest one. If not merged, it is tested with the third-lowest one,
and so on. For the m'th higher system the reconnection
probability thus becomes (1 - P)^(m-1) P. That is, there is
no explicit dependence on the higher pT scale, but implicitly
there is via the survival probability of not already having been merged
with a lower-pT system. Also note that the total reconnection
probability for the lowest-pT system in an event with n
systems becomes 1 - (1 - P)^(n-1). Once the fate of the
lowest-pT system has been decided, the second-lowest is considered
with respect to the ones above it, then the third-lowest, and so on.
Once it has been decided which systems should be joined, the actual merging
is carried out in the opposite direction. That is, first the hardest
system is studied, and all colour dipoles in it are found (including to
the beam remnants, as defined by the holes of the incoming partons).
Next each softer system to be merged is studied in turn. Its gluons are,
in decreasing pT order, inserted on the colour dipole i,j
that gives the smallest (p_g p_i)(p_g p_j)/(p_i p_j), i.e.
minimizes the "disturbance" on the existing dipole, in terms of
pT^2 or Lambda measure (string length). The insertion
of the gluon means that the old dipole is replaced by two new ones.
Also the (rather few) quark-antiquark pairs that can be traced back to
a gluon splitting are treated in close analogy with the gluon case.
Quark lines that attach directly to the beam remnants cannot be merged
but are left behind.
The joining procedure can be viewed as a more sophisticated variant of
the one introduced already in [Sjo87]. Clearly it is ad hoc.
It hopefully captures some elements of truth. The lower pT scale
a system has the larger its spatial extent and therefore the larger its
overlap with other systems. It could be argued that one should classify
individual initial-state partons by pT rather than the system
as a whole. However, for final-state radiation, a soft gluon radiated off
a hard parton is actually produced at late times and therefore probably
less likely to reconnect. In the balance, a classification by system
pT scale appears sensible as a first try.
Note that the reconnection is carried out before resonance decays are
considered. Colour inside a resonance therefore is not reconnected.
This is a deliberate choice, but certainly open to discussion and
extensions at a later stage, as is the rest of this procedure.
Further variables
mode BeamRemnants:maxValQuark
(default = 3; minimum = 0; maximum = 5)
The maximum valence quark kind allowed in acceptable incoming beams,
for which multiple 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 multiple 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 diqaurks in baryons, relative to the
simple sum of the two constituent quarks.
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 multiple 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
multiple interactions and showers are redone until a
junction-free topology is found.