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].
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 = 0.4
; 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 = 2.1
; 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 = 7.
; minimum = 0.
)
The scale Q_half in the equation above, defining the
half-way point between hard and soft interactions.
parm
BeamRemnants:halfMassForKT
(default = 2.
; 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.
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 = 2.5
; 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.
Diffractive system
When an incoming hadron beam is diffractively excited, it is modeled
as if either a valence quark or a gluon is kicked out from the hadron.
In the former case this produces a simple string to the leftover
remnant, in the latter it gives a hairpin arrangement where a string
is stretched from one quark in the remnant, via the gluon, back to the
rest of the remnant. The latter ought to dominate at higher mass of
the diffractive system. Therefore an approximate behaviour like
P_q / P_g = N / m^p
is assumed.
parm
BeamRemnants:pickQuarkNorm
(default = 5.0
; minimum = 0.
)
The abovementioned normalization N for the relative quark
rate in diffractive systems.
parm
BeamRemnants:pickQuarkPower
(default = 1.0
; minimum = 0.
)
The abovementioned mass-dependence power p for the relative
quark rate in diffractive systems.
When a gluon is kicked out from the hadron, the longitudinal momentum
sharing between the the two remnant partons is determined by the
same parameters as above. It is plausible that the primordial
kT may be lower than in perturbative processes, however:
parm
BeamRemnants:diffPrimKTwidth
(default = 0.5
; minimum = 0.
)
The width of Gaussian distributions in p_x and p_y
separately that is assigned as a primordial kT to the two
beam remnants when a gluon is kicked out of a diffractive system.
parm
BeamRemnants:diffLargeMassSuppress
(default = 2.
; minimum = 0.
)
The choice of longitudinal and transverse structure of a diffractive
beam remnant for a kicked-out gluon implies a remnant mass
m_rem distribution (i.e. quark plus diquark invariant mass
for a baryon beam) that knows no bounds. A suppression like
(1 - m_rem^2 / m_diff^2)^p is therefore introduced, where
p is the diffLargeMassSuppress
parameter.