Subleading Colour & Colour Reconnections (CR)

  1. Colour Flow and Colour Indices
  2. Antenna Swing
  3. Non-perturbative Swing
  4. Perturbative Swing
  5. Baryon Junctions
  6. Parameters & Switches

This page collects information and parameters related to both the perturbative (subleading-colour) and non-perturbative colour-flow and colour-reconnection scenarios in VINCIA.

During the perturbative shower evolution, the first aspect of subleading colour is simply what colour factors are used for the antenna functions. In a strict leading-colour limit, one would use CA for all antennae, thus overestimating the amount of radiation from quarks (note that we use a normalisation convention in which the colour factor for quarks is 2CF, hence the difference is explicitly subleading in colour). A more realistic starting point is to use 2CF for quark-antiquark antennae, CA for gluon-gluon ones, and something inbetween for quark-gluon ones. Alternatively, a more sophisticated treatment is under development, which exploits QED-like factorisation properties of multi-gluon amplitudes beyond LC. The following switch determines whether and how subleading-colour corrections are treated in the evolution:

mode  Vincia:modeSLC   (default = 2; minimum = 0; maximum = 3)

option 0 : Strict LC evolution. All gluon-emission colour factors are forced equal to CA thus overcounting the radiation from quarks. Note that matrix-element corrections, if applied at full colour, will still generate corrections to the evolution up to the matched number of legs.
option 1 : Simple Colour Factors. The chargeFactor parameters for each of the antenna functions are used to set the colour factor for each type of gluon-emission antenna; see the section on antenna functions. (Typically, 2CF for qqbar antennae, CA for gg antennae, and the average of 2CF and CA for qg antennae.)
option :
option 2 : Sophisticated Colour Factors. The colour factor for quark-antiquark antennae is forced equal to 2CF. Quark-gluon and gluon-gluon antennae both start out normalised to CA, but a phase-space (and helicity-) dependent correction proportional to -1/NC^2 is applied to QG antennae, containing the collinear parts of a corresponding Q-Qbar antenna. This can be viewed as interpolating between a colour factor of 2CF for radiation collinear with the quark and one proportional to CA for radiation collinear with the gluon.
option 3 : Sophisticated Colour Factors and QCD Multipoles. As for option 2, but also quadrupole (and higher multipole) radiation patterns are allowed as well, starting at order 1/NC^2 for radiation from hard-parton configurations containing at least a q-qbar pair plus two gluons. These patterns are based on the soft limits of subleading-colour amplitudes in QCD, combined with colour-algebra arguments. UNDER DEVELOPMENT.

Colour Flow and Colour Indices

Colour flow is traced using Les-Houches style colour tags, augmented by letting the last digit encode the "colour index", running from 1 to 9, described further in the section below on antenna swing. One ambiguity arises in gluon emission as to which of the daughter antennae should inherit the "parent" colour tag/index, and which should be assigned a new one. This is controlled by the following parameter:

mode  Vincia:CRinheritMode   (default = 1; minimum = -2; maximum = 2)

option 0 : Random
option 1 : The daughter with the largest invariant mass has a probability 1/(1 + r) to inherit the parent tag, with r < 1 the ratio of the smallest to the largest daughter invariant masses squared.
option 2 : The daughter with the largest invariant mass always inherits the parent tag (winner-takes-all extreme variant of option 1).
option -1 : (Unphysical, intended for theory-level studies only). Inverted variant of option 1, so that the daughter with the smallest invariant mass preferentially inherits the parent colour tag.
option -2 : (Unphysical, intended for theory-level studies only). Inverted variant of option 2, so that the daughter with the smallest invariant mas always inherits the parent colour tag.

Antenna Swing

In QCD, the probability that a random quark and a random antiquark are in an overall singlet state is given by the colour algebra:

3 ⊗ 3bar = 1 ⊕ 8

From colour counting alone, any pair of random (colour-disconnected) triplet-antitriplet charges should thus have a 1/9 probability to be in an overall singlet state. Following the approach in ARIADNE, this can be represented by assigning to every leading-colour antenna an index running from 1 to 9 and allowing reconnections to occur only between ones having the same such index. To ensure that the octet nature of gluons is respected, the assignment of indices is restricted so that neighbouring antennae, and quark pairs originating directly from g→qq splittings, always have different indices. Next-to-nearest neighbours in colour space are treated as completely colour-disconnected in the context of this model.

The next question to be addressed is what is the dynamics of the reconnections. From coherence and by analogy with QED (having dealt with the colour-factor suppression above), we expect something like a principle of maximal screening. In an angular-ordering context, this can be illustrated by imagining a system of several colour and anti-colour charges all having the same index. Maximal destructive interference would then imply that the radiation cone around each colour-charged parton should be set up so as to extend to the nearest (in angle) same-index anticolour charge, regardless of the leading-colour connections and keeping in mind that each anticolour charge can only cancel one unit of colour charge. We now discuss how to generalize this to an antenna context, for non-perturbative and perturbative reconnections respectively.

Non-perturbative Antenna Swing

In a non-perturbative context, and specifically with the Lund string model in mind, the relevant measure for judging which antennae are "closest" to each other is the string length measure, Λ, which is given by the invariant mass of the string piece endpoints:

Λ(pa,pb) = (pa+pb)2 - (ma+mb)2 = 2*(pa.pb) - 2*ma*mb

where the subtraction of the sum of endpoint masses ensures that a "string" spanned by two massive particles at rest is assigned zero length.

Thus, for two same-index antennae spanned between the four parton momenta (p1,p2) and (k1,k2), we may compare the two string lengths:

Λ1 = Λ(p1,p2) * Λ(k1,k2)
Λ2 = Λ(p1,k2) * Λ(k1,p2)

Note that both expressions are linear in each of the participating momenta, so the choice only depends on the directions of motion of the involved partons.

The current implementation of non-perturbative swing in VINCIA is deterministic: among same-index antennae, the configuration with the smallest Λ always wins. The model therefore currently has no free parameters, apart from a minimum mass-squared of order (1 GeV)2 which is imposed when comparing the configurations, see Parameters and Switches below. The actual measure that is minimized by the algorithm is therefore

Λeff(pa.pb) = Λ(pa,pb) + mmin2

with mmin a regulator expressing the fact that infinitely soft and collinear gluons do not lead to zero-mass string pieces but are instead smoothly absorbed into a physical string of hadronic dimensions. We emphasize that the minimal-mass parameter is only added for the purpose of calculating the Λ measure, without touching the parton momenta themselves. If a small-mass reconnection still wins, despite the mmin penalty, then the corresponding topology is accepted. However, the Lund string model then ensures that any small invariant is smoothly absorbed into the string, as desired. Physically, one should in these cases probably instead compute the overall string measure obtained by clustering the small-invariant momenta together, rather than treating them as separate partons, a sophistication that would introduce corrections like

mmin2 → mmin2 ( 1 + min[ Λ(p,kR)/Λ(k,kR) , Λ(pL,k)/Λ(pL,p) ]

with pL and kR the colour neighbours (of p and k, respectively) of the reconnected small-invariant antenna spanned by (p,k). This last level of sophistication, however, has not been implemented in VINCIA's CR algorithm so far.

Though the model presented here thus has no free parameters (beyond the minimal-mass parameter), for completeness, we note that the corresponding implementation in ARIADNE makes a stochastic selection, with a variable strength.

If there are several possible reconnections to choose between (more than two antennae with the same colour index), VINCIA creates a matrix of dot products, in which the diagonal represents the colour-connected invariants. It then iteratively minimizes the product of diagonal entries by successive column swaps, starting from the largest entry on the diagonal and working downwards until no more swaps would result in a reduction of the overall Λ measure. (We neglect the small risk that this procedure could end up trapped in a local minimum; a possibility that could eventually be overcome by applying stochastic methods and/or by considering higher-than-binary column rearrangements).

Note that this model is only intended to reflect the coherence expected from colour screening. The possibility of additional nonperturbative dynamics, for instance that strings or hadrons, once formed, could rescatter, fuse, or cut each other up, is not explicitly addressed.

Perturbative Antenna Swing

At the perturbative level, the same colour-algebra arguments applied above can be used to determine between which antennae reconnections could potentially occur. In VINCIA, this has been implemented by assigning antenna indices (from 1-9, as above) already at the Born level, and keeping track of them during the perturbative evolution. When a gluon is emitted, the larger of the two daughter antennae inherits the colour index of the parent antenna, while the smaller receives a new index, non-identical to that of its neighbours. This presumably gives the correct behaviour in the collinear limits (in which the new colour line is then only resolved inside the smaller system), while it represents an arbitrary choice at large angles. Note again that we here make a deterministic choice (discontinuous across the line of equal daughter masses) while ARIADNE makes a stochastic one. Alternatively, it is possible that neither daughter antenna should inherit the old index in the wide-angle limit, for future studies to determine.

What the relevant dynamical measure is that should be used to suppress or enhance reconnections among the antennae during the perturbative evolution is not quite as obvious. Morever, the potential consequences of making a mistake are more severe, as it will affect the rest of the perturbative evolution, and hence will result in corrections that go beyond mere hadronisation (power) corrections. A further complication is that, in addition to "colour accidents" (two antennae happening to have the same colour index), we know that there are colour-interference diagrams active at the subleading-colour level, which are associated with different (non-planar and less singular) propagator structures than those of their leading-colour (planar leading-log) counterparts. Finally, the antenna-shower evolution is ordered in a measure of resolution scale, QE. That measure is applied to the colour-connected invariants, but not to the colour-unconnected ones. It is therefore nominally possible for arbitrarily small colour-unconnected invariants to appear at any stage during the evolution, simply due to "phase-space accidents". In a proper subleading-colour evolution picture (if such could be formulated), those unconnected invariants should have been created at a much later stage of the evolution, since they correspond to a low resolution scale at the subleading level.

These complications imply that any attempt at modeling colour reconnections already at the perturbative stage is both a difficult and dangerous proposition. In the absence of a much more rigorous understanding of these effects, it is therefore advisable to take a highly conservative approach: rather do too little than too much. Though we do include a model of perturbative CR, described here, we therefore choose to leave it off by default.

We take the resolution scale as the most crucial parameter. A reconnection involving a very small colour-unconnected invariant should not be allowed to occur until the evolution has reached scales capable of resolving that invariant. Physically, during the evolution at higher scales we should see the coherent sum of those partons, but since we are not able to reformulate the shower algorithm to reflect this, we choose the following strategy. For each possible antenna configuration, the global evolution scale can be calculated by taking the minimum of the evolution scale evaluated for each possible (colour-connected) clustering of that configuration. For the original topology, that scale is just equal to the current evolution scale of the event, QE. For each alternative topology, we require that the resolution scale computed for that topology is either equal to or greater than QE. Otherwise the reconnection is vetoed (for the time being), possibly to be tried again when the evolution scale has reached a lower value.

Modulo this "delaying tactic", we apply the same philosophy as above. Colour-compatible antennae are reconnected if it reduces their invariant masses, again according to a principle of maximal screening. However, since the multiplicative Λ measure is no longer the clear measure to use for comparing two possible antenna configurations, and in the spirit of "least is best", we require that both of the antenna invariants are reduced by the reconnection, otherwise it is vetoed. That is, we only perform a reconnection if we are fairly sure it should really happen.

Here, it could be argued how one should treat reconnections that lead to approximately equal invariant masses, or to only very small changes. Physically, such systems presumably spend their time oscillating between one state and the other, with interferences building up a quadrupole pattern (or higher multipoles, if more antennae are involved). Again, without a much more rigorous understanding of these effects, we do not believe it would make much sense to attempt to treat such cases in detail. Instead, we apply the simple deterministic principle of maximal screening.

As formulated here, the model therefore has no free parameters. Nonetheless, to obtain a smooth continuation to the non-perturbative scenario discussed above, and to provide some additional safety against very small accidental invariants, we add the same mmin parameter to the invariants as above. The effective antenna mass that is minimized is therefore just


i.e., the same measure as is used on the non-perturbative side. Note, however, that while the non-perturbative model allows one string to become larger as long as the product of Λ measures still decreases, the perturbative model still requires that both invariants become smaller. Therefore, even when the perturbative reconnections are switched on, some reconnections may still only be allowed at the stage of the non-perturbative transition: those for which one of the masses increases.

Note that this reconnection scenario only explicitly addresses the case of "colour accidents", when two antennae have compatible indices and hence can screen each other. It does not address non-planar colour-interference diagrams. For instance, the colour factor for a quark-gluon emission antenna is still ambiguous, somewhere between 2CF and CA. To treat such effects, the older model mentioned above, in which all antennae used CA but a negative -1/NC2 one for each qq̅ pair was absorbed into the leading ones, could in principle be used. Thus, the possibility of reviving and combining the old model with this one remains open, of course at the price of further complication. Until we have gained more experience with the current scenario, we therefore defer this possibility to future considerations.

As formulated here, the model leads to perturbative CR in about 2% of LEP events. Thus, the effect is quite strongly suppressed, but still large enough to generate some visible effects in tails of distributions. For completeness, we note that if we did not require both invariants to individually become smaller (minimizing only their product, as in the non-perturbative modeling), the effect would increase to the 10% level. Finally, if we also removed the resolution-scale veto, the effect rises again, to about 30%. Thus, the built-in suppression mechanisms reduces the amount of perturbative CR by about an order of magnitude.

Baryon Junctions

A further intriguing possibility that opens up at subleading colour is the formation of colour-epsilon (and corresponding anticolour-epsilon) structures in the colour field. In the colour algebra, this first appears when considering the combination of two random same-sign colour triplets:

3 ⊗ 3 = 3bar ⊕ 6

where the sextet corresponds to a configuration that can be treated as unreconnected (in analogy with the octet in the formula for triplet-antitriplet combinations above), but the 3bar corresponds to the coherent addition of two colour triplets to form an effective antitriplet (e.g., red + green can look antiblue seen from a distance). In non-perturbative terms, this is represented as a diquark (if the invariant mass is of hadronic size) or, more generally, a string junction, see [Sjö03]. Both carry baryon number, and hence this mechanism provides a new and interesting possibility for baryon formation. This has traditionally been a weak point in the Lund string formalism, so even modest improvements would be welcome. In particular, junctions produce less strong baryon-antibaryon correlations (than the standard mechanism of diquark-pair string breaks), due to separation in both flavour and phase space of the produced baryons and antibaryons.

Interestingly, the colour-factor suppression is here only 3/(3+6) = 1/NC, rather than 1/NC2. This is enough to allow a system of two charges and two anticharges to reconnect into a junction-antijunction configuration. (In general, one may consider also cases in which the junction is not connected directly to the antijunction, instead connecting to it via an arbitrary number of gluons, in which case the accident would have to happen twice, once at each end, a higher-order possibility we so far do not consider in this model.)

For the special case of systems that are neighbors in colour space (eg the quarks in Z→qqq̅q̅), the fact that neighbouring colours cannot be identical (due to the octet nature of gluons) leaves only the 6 combinations of non-identical colours our of the total 9 in the formula for triplet-triplet combinations above, of which half are antisymmetric (the 3bar). Thus, in this particular case, the suppression factor is only 1/2.

Though a perturbative formulation could be possible, we shall here restrict our attention to junction formation at the non-perturbative transition stage. This is partly due to the fact that perturbative radiation from colour-epsilon configurations has not yet been implemented in VINCIA. (In a perturbative framework a la angular ordering, one would presumably treat wide-angle radiation coherently, with the 3bar, but small-angle radiation - inside the opening angle of the two triplets - incoherently.)

The relevant string measure for comparing a 0-junction and a 2-junction topology was derived in [Sjö03]. Here, we take a simplified approach, considering the junction-junction system as made up of one string piece between two virtual "diquarks" and one piece inside each of the diquarks, thus representing the the total string length by

Λ2-junction = Λ(p1+p2,k1+k2) * Λ(p1,p2) * Λ(k1,k2)
Λ0-junction = Λ(p1,k1) * Λ(p2,k2)

where (p1,k1) and (p2,k2) are the two original (unreconnected) antennae, with p1 and p2 the momenta of the colour charges and k1 and k2 those of the anticolour charges. Due to the different dimension of the two expressions (one involves three string pieces, the other only two), there is here an explicit dependence on the normalisation of the Λ measure, which we denote m0, expected to be roughly of order 1 GeV. We take this parameter to be independent of the mmin value introduced in Λeff above, so that the actual measures to be compared are:

Λ2-junction = [ Λeff(p1+p2,k1+k2) * Λeff(p1,p2) * Λeff(k1,k2) ] / m0;6
Λ0-junction = [ Λeff(p1,k1) * Λeff(p2,k2) ] / m0;4

Due to the different powers of m0, large m0 values will increase the amount of 2-junction topologies that "win" (i.e., have the smallest Λ measure).

Since we are not technically able to treat systems with more than a single junction-antijunction connection, we impose that only antennae taken from two different q-q̅ chains are able to reconnect to junctions. Physically, this also reflects that it should be likely for the two junctions in a system with two (or even three) strings spanned between the same junction and antijunction to annihilate, thus reverting to the original non-junction colour flow. Likewise, antennae residing inside gluon loops are not given the possibility of participating in junction reconnections.

Simplifying the colour arguments above slightly, we allow reconnections to occur between antennae with different-parity antenna indices (that is, e.g., junction reconnections will be allowed between (1,2) but not between (1,3). In particular, this projects out the same-index case and only keeps half of the remaining 8 possibilities. The generic colour suppression factor for reconnections involving junctions thus comes out to 4/9, instead of 3/9, but we consider this an acceptable margin of error, especially since the restrictions imposed will cause some possibilities to not be tried at all, and the free parameter m0 gives a possibility to further adjust the overall strength of junctions reconnections. We note, however, that for the special case of nearest-neighbours, the probability for different-parity indices is 4/8 = 1/2, as desired (the same-index case never occurs for nearest neighbours).

Parameters and Switches

The level of CR is set by the following switch:

mode  Vincia:CRmode   (default = 0; minimum = 0; maximum = 2)

option 0 : CR off.
option 1 : Non-perturbative CR only.
option 2 : Perturbative and non-perturbative CR.

The mmin parameter in the expression for the effective Λ measure above is given by:

parm  Vincia:CRmMin   (default = 1.0; minimum = 0.0)

The possibility of junction formation at the non-perturbative stage is regulated by the following switch:

flag  Vincia:CRjunctions   (default = off)

The dimensionful scale normalizing the measure for junction reconnections is set by:

parm  Vincia:CRjunctionsM0   (default = 2.0; minimum = 0.1; maximum = 10.0)