Colour Reconnection
The colour flows in the separate subprocesses defined in the
multiparton-interactions scenario are tied together via the assignment
of colour flow in the beam remnant. This is not an unambiguous
procedure, and currently two different methods are implemented. In the first
model the colour flow is reconstructed by how a PS could have
constructed the configuration. In the second model, the full QCD colour
calculation is taken into account, however the dynamical effects are modeled
loosely, only an overall saturation is taken into account. The idea is to
later account for other dynamical effects through colour reconnections.
A simple "minimal" procedure of colour flow only via the beam remnants
does not result in a scenario in agreement with data, however,
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. Since no final
answer is known, several models are implemented. The different models also
rely on the two different colour assignments in the beam remnant. There are
two, somewhat motivated, models implemented: the original PYTHIA scheme and
a new scheme that tries to incorporate more of the colour knowledge from QCD.
The original PYTHIA scheme relies on the PS-like colour configuration of the
beam remnant. This is combined with an additional step, wherein the gluons
of a lower-pT MPI system are merged with the ones in a higher-pT MPI.
A more detailed description of the merging can be found below.
Relative to the other models it tests fewer reconnection possibilities,
and therefore tends to be reasonably fast.
The new scheme [Chr14a]relies on the full QCD colour configuration
in the beam remnant. This is followed up by a colour reconnection, where the
potential string energy is minimized (ie. the lambda measure is
minimized). The QCD colour rules are also incorporated in the colour
reconnection, and determine the probability that a reconnection is allowed.
The model also allows the creation of junction structures.
In addition to the two models described above, a simple model is implemented,
wherein gluons can be moved from one location to another so as to reduce the
total string length. This is one out of a range of simple models developed
to study potential colour reconnection effects e.g. on top mass
[Arg14], not from the point of view of having the most realistic
description, but in order to probe the potential worst-case spread of
predictions. All of these models are made available separately in
include/Pythia8Plugins/ColourReconnectionHooks.h
, with the
setup illustrated in examples/main29.cc
, but only the
gluon-move one is sufficiently general and realistic that it has been
included among the standard options here.
flag
ColourReconnection:reconnect
(default = on
)
Allow or not a system to be merged with another one.
mode
ColourReconnection:mode
(default = 0
; minimum = 0
; maximum = 2
)
Determine which model is used for colour reconnection. Beware that
different BeamRemnants:remnantMode
should be used for
different reconnection schemes.
option
0 : The MPI-based original Pythia 8 scheme.
option
1 : The new more QCD based scheme.
option
2 : The new gluon-move model.
The MPI-based scheme
In this scheme partons are classified by which MPI system they belong to.
The colour flow of two such systems can be fused, and if so the partons
of the lower-pT system are added to the strings defined by the
higher-pT system in such a way as to give the smallest total
string length. The bulk of these lower-pT partons are gluons,
and this is what the scheme is optimized to handle.
In more detail, an MPI system with a scale pT of the hard
interaction (normally 2 → 2) 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
range
times pT0, the latter being the same
energy-dependent dampening parameter as used for MPIs.
Thus it is easy to merge a low-pT system with any other,
but difficult to merge two high-pT ones with each other.
parm
ColourReconnection:range
(default = 1.8
; minimum = 0.
; maximum = 10.
)
The range
parameter defined above. The higher this number is
the more reconnections can occur. For values above unity the reconnection
rate tends to saturate, since then most systems are already connected with
each other. This is why 10 is set as an effective upper limit, beyond
which it is not meaningful to let the parameter go.
The reconnection procedure is applied 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 by default. Colour inside a resonance therefore is not
reconnected. The
PartonLevel:earlyResDec
can be switched on to perform resonance decays before colour reconnection,
and then the partons from resonance decays could be affected.
Ideally the time scales of resonance decays and of colour reconnection
should be picked dynamically, but this is not yet the case. Notably
the W, Z and t have intermediate decay time
scales, somewhat but not much shorter than typical hadronization times,
whereas the H is much more long-lived.
The newer scheme
parm
ColourReconnection:m0
(default = 0.5
; minimum = 0.1
; maximum = 5.
)
This is the variable used in the lambda measure for the string length.
See the different choices of lambda measure for exact formulaes. This variable
is in the new model also used as a cut for forming pseudo particles that are
not colour reconnected.
mode
ColourReconnection:nColours
(default = 9
; minimum = 1
; maximum = 30
)
The number of reconnection colours, this should not be confused with the
standard number of QCD colours. Each string is given an integer number between
0 and nColours - 1
. Only strings with the same number are allowed
to do a normal string reconnection. The default value provides
the standard QCD probability that a triplet and an anti-triplet is in a
singlet state. The probability for two strings to form a junction structure is
in QCD given by the product of two triplets, which gives a probability of 1/3.
Therefore the number of reconnection colours for junction formation is
iColours % 3
, where iColours refer to the integer of the string.
The behaviour of junction formation therefore only changes slightly with this
variable.
flag
ColourReconnection:sameNeighbourColours
(default = off
)
In the normal colour reconnection two neighbouring strings are not allowed
to have the same colour. Similar two strings orginating from a gluon split is
not allowed to reconnect. The physics motivation for this is that it would
require colour singlet gluons, and therefore for ordinary physics studies this
should be turned off. But for testing of extreme scenarios (i.e. 1 colour),
this variable needs to be turned on, since it is not possible to have
different neighbouring colours.
flag
ColourReconnection:allowJunctions
(default = on
)
This switch disables the formation of junctions in the colour reconnection.
mode
ColourReconnection:lambdaForm
(default = 0
; minimum = 0
; maximum = 2
)
This allows to switch between different options for what
lambda-measure to use.
The formula shown are how much each end of a dipole or junction contribute to
the total lambda-measure. The energies are defined in respectively
the dipole or junction restframe.
option
0 : lambda = ln (1 + sqrt(2) E/m0)
option
1 : lambda = ln (1 + 2 E/m0)
option
2 : lambda = ln (2 E/m0)
parm
ColourReconnection:minimumGainJun
(default = 1
; minimum = -100
; maximum = 100
)
The minimum lambda has to decrease in order to create a junction
antijunction pair.
flag
ColourReconnection:allowDoubleJunRem
(default = on
)
This parameter tells whether or not to allow a directly connected
junction-antijunction pair to split into two strings. The lambda measure of
the junction system is compared to that of the two possible string
configurations. If the chosen configuration is the junction system, a q-qbar
system is inserted between the junctions by removing some energy/momentum from
the other legs.
The gluon-move scheme
This approach contains two steps, a first "move" one and an optional
second "flip" one. Both are intended to reduce the total "string length"
lambda measure of an event. For multiparton topologies the
correct lambda measure can become quite cumbersome, so here it
is approximated by the sum of lambda contributions from each pair
of partons connected by a colour string piece. For two partons i
and j with invariant mass m_ij this contribution
is defined as lambda_ij = ln(1 + m^2_ij / m2Lambda).
The 1 is added ad hoc to avoid problems in the m_ij → 0
limit, problems which mainly comes from the approximate treatment,
and m2Lambda is a parameter set below.
In the move step all final gluons are identified, alternatively only a
fraction fracGluon of them, and also all colour-connected
parton pairs. For each gluon and each pair it is calculated how the total
lambda would shift if the gluon would be removed from its current
location and inserted in between the pair. The gluon move that gives the
largest negative shift, if any, is then carried out. Alternatively, only
shifts more negative than dLambdaCut are considered. The procedure
is iterated so long as allowed moves can be found.
There is some fine print. If a colour singlet subsystem consists of two
gluons only then it is not allowed to move any of them, since that then
would result in in a colour singlet gluon. Also, at most as many moves
are made as there are gluons, which normally should be enough. A specific
gluon may be moved more than once, however. Finally, a gluon directly
connected to a junction cannot be moved, and also no gluon can be inserted
between it and the junction. This is entirely for practical reasons, but
should not be a problem, since junctions are rare in this model.
The gluon-move steps will not break the connection between string endpoints,
in the sense that a quark and an antiquark that are colour-connected via
a number of gluons will remain so, only the number and identity of the
intermediate gluons may change. Such a scenario may be too restrictive.
Therefore an optional second flip step is introduced. In it all such
colour chains are identified, omitting closed gluon loops. The lambda
change is defined by what happens if the two colour lines are crossed
somewhere, e.g. such that two systems q1 - g1 - qbar1 and
q2 - g2 - qbar2 are flipped to q1 - g1 - g2 - qbar2
and q2 - qbar1. The flip that gives the largest lambda
reduction is carried out, again with dLambdaCut offering a
possibility to restrict the options. As with the move step, the procedure
is repeated so long as it is allowed. An important restriction is
imposed, however, that a given system is only allowed to flip once,
and not with itself. The practical reason is that repeated flips could
split off closed gluon loops quite easily, which tends to result in
bad agreement with data.
As an option, singlet subsystems containing a junction may or may not
be allowed to take part in the flip step. Since the number of junction
systems is limited in this model the differences are not so important.
parm
ColourReconnection:m2Lambda
(default = 1.
; minimum = 0.25
; maximum = 16.
)
The m2Lambda parameter used in the definition of the approximate
lambda measure above. It represents an approximate hadronic
mass-square scale, cf. m0 in the previous model. Its value is
uncertain up to factors of 2, but the lambda change induced by
a potential move or flip is rather insensitive to the precise value,
owing to large cancellations.
parm
ColourReconnection:fracGluon
(default = 1.
; minimum = 0.
; maximum = 1.
)
The probability that a given gluon will be considered for being moved.
It thus gives the average fraction of gluons being considered.
parm
ColourReconnection:dLambdaCut
(default = 0.
; minimum = 0.
; maximum = 10.
)
Restrict gluon moves and colour flips to those that reduce lambda
by more than this amount. The larger this number, the fewer moves and flips
will be performed, but those that remain are the ones most likely to produce
large effects.
mode
ColourReconnection:flipMode
(default = 0
; minimum = 0
; maximum = 2
)
Performing the flip step or not.
option
0 : No flip handling.
option
1 : Allow flips, but not for strings in junction topologies.
option
2 : Allow flips, including for strings in junction topologies.