Uncertainty Bands

VINCIA's Uncertainty Bands

A calculation is only as good as the trustworthiness of its uncertainty bands. Although not foolproof, VINCIA attempts to make comprehensive and explicit estimates of the theoretical accuracy of the answers it provides, event by event in phase space. Obviously, VINCIA can only make these estimates for the parts of the calculation that it handles itself - hence the automatic uncertainty estimates are currently limited to the perturbative parts of the calculation.

When switched on, the uncertainty variations are provided as, and can be accessed by the user through, a vector of alternative weights that is provided for each event. If the event is in a phase space region that VINCIA believes to be under good perturbative control (according to the showering and matching criteria specified by the user), the spread in weights will be small, whereas it will be larger for events populating less well controlled phase space regions.

Since the uncertainty estimate is fully differential in phase space, cuts can be imposed post facto without causing any special problems. E.g., if the cuts isolate a poorly controlled region, all the accepted events will have relatively large event weight spreads, and hence the overall relative uncertainty will increase, as expected.

Needless to say, these uncertainties still have to be interpreted with caution, but due to a large amount of flexibility in the VINCIA formalism, VINCIA can perform quite a few variations that are not possible with other generators. We therefore believe that the uncertainties quoted by VINCIA are, at least, reasonably meaningful.

Note also that the "central" set of weights (corresponding to the settings chosen by the user for the current run) is normally unweighted by default in VINCIA. I.e., all the events have identical weights and can be trivially added together statistically. The uncertainty weight sets are computed in such a way that this still holds true on average for each uncertainty variation individually. I.e., although the weights corresponding to uncertainty variation n fluctuate about unity, event to event, their average over many events will still be unity. Formally, this is due to the variations being done in a way that conserves unitarity. Practically, it means that none of the variations change the input total (Born-level) normalization - they only change how it is distributed across phase space. The user should therefore be aware that there can still be an overall "K" factor, not addressed by these uncertainty bands.

The speed of the calculation is not significantly affected by adding uncertainty variations, but the code does run slightly faster without them. We therefore advise to keep them switched off whenever they are not going to be used. See speed below for more on the impact on performance.

Available Variations

The following automatic variations are currently available:

iVar
0Current Settings (user definable).
1DEF antennae (VINCIA's default antenna functions)
2DEF antennae with alphaS(pT2 / kAlphaS), where the value of kAlphaS is user-specifiable (see below)
3DEF antennae with alphaS(kAlphaS * pT2), where the value of kAlphaS is user-specifiable (see below)
4MAX antenna set (large finite terms)
5MIN antenna set (small finite terms)
6NLO-Hi: branching probabilities multiplied by (1+alphaS) to represent unknown (but finite) NLO corrections.
7NLO-Lo: as above, but with division instead of multiplication.
8DEF antennae with stronger ordering in pT
9DEF antennae with ordering in mD
10DEF antennae with all color factors for gluon emission = 3
11DEF antennae with all color factors for gluon emission = 8/3

Note:These are just the automatic variations you can ask VINCIA to do for you. This in no way restricts the user's own power to do further variations, by adjusting individual parameters in VINCIA. Here, we have merely tried to provide some sensible default variations that an inexperienced user might use to get some idea of the trustworthiness of the answers he or she is getting from VINCIA.

Note 2: these variations only represent variations of the perturbative shower parameters. Since VINCIA does not handle the non-perturbative phase of fragmentation, uncertainty estimates for parameters related to that part must still be performed in the traditional way. Especially for infrared sensitive observables, the user is therefore advised to carry out separate variations of the non-perturbative parameters, such as the a and b parameters of the Lund symmetric fragmentation function and of other hadronization and hadron decay parameters that may be of relevance to the study at hand.

Switches and Parameters

VINCIA provides a possibility for evaluating the variations described above automatically. For every event it generates, it will then tell you the effective spread of weights it found for that particular phase space point, which saves you the effort to make separate additional runs, one for each variation. It also saves computing time, since the uncertainty evaluation only mildly affects the overall speed of the generator.

On/Off

flag  Vincia:uncertaintyBands   (default = off)
Main switch for VINCIA's automatic evaluation of theoretical uncertainty bands. When set to on VINCIA internally keeps track of several variations of the shower approximation and outputs a vector of weights for each event.

Scale Variations

One of the theory uncertainty variations automatically performed by VINCIA when uncertainty bands are switched on is to vary the scale argument used in the strong coupling. The central scale is normally taken to be pT, which absorbs universal 1-loop corrections to the gluon-emission antenna functions into the effective tree-level functions. According to LEP tunings, there isn't a lot of freedom to modify this choice if one desires to remain at least approximately consistent with data. The amount of variation used for the scale uncertainty can be set using the following parameter:

parm  Vincia:uncertaintyBandsKAlphaS   (default = 4.0; minimum = 1.0; maximum = 16.0)
Desired factor of scale variation in the alphaS factors appearing in branching probabilities for uncertainty estimates. Note that this factor is applied to the renormalization scale squared. Hence, the renormalization scale will be varied between pT2/kAlphaS < muR2 < pT2*kAlphaS. The default is to vary the scale choice in GeV by a factor of 2 in either direction, hence a factor 4 on the squared quantity.

Accessing the Uncertainty Bands

The first (zero'th) entry in the vector of event weights always corresponds to the settings chosen by the user, and will normally have weights equal to unity (if showering an unweighted set of events), see the page on weights. If showering a weighted set of events, the nominal (user) weights are propagated through VINCIA and can be obtained through the method

vincia.weight();

The uncertainty bands are represented by alternative sets of weights, where the spread of weights for each indidivual event gives an estimate of how "sure" VINCIA is about the weight for that particular event. Regions of phase space where the theoretical uncertainty is large are thus reflected by large weight spreads, while regions where the shower approximations work well have smaller differences. After showering, the weights corresponding to the uncertainty bands are accessible via the same method as that used for the central weight set, by giving a non-zero index to the weight() method,

vincia.weight(int iVar);

where iVar is an integer code specifying the particular variation you want the weight for, and the number of available variations you can access is given by

vincia.nVariations();

Uncertainty Band Limits

In the uncertainty band evaluation, the maximum deviation allowed for each branching variation, relative to the user settings, is constrained by

parm  Vincia:uncertaintyBandsMaxDev   (default = 2.0; minimum = 1.0; maximum = 10.0)
Maximum allowed value for the ratio of an uncertainty variation to the nominal (user) branching probability, for each branching. Should normally be left in its default position, unless you are an experienced user and know what you're doing. Note: this works in both directions, i.e., the relative branching probability is forced to be in the range

1 / Vincia:uncertaintyBandsMaxDev < P'/P < Vincia:uncertaintyBandsMaxDev

where P' is the branching probability according to the variation, and P is the probability with the settings chosen by the user. Note also that the total shower weight is built up by products of many branchings, so that the final answer may differ by more than Vincia:uncertaintyBandsMaxDev.

parm  Vincia:uncertaintyLimit   (default = 100.0; minimum = 1.0; maximum = 100000000.0)
Upper limit on the total amount of modification to the incoming weight allowed to VINCIA's uncertainty estimates. The default value allows these estimates to change the weight of the event by up to a factor 100 from its original value, for each event, which should be more than sufficient for ordinary applications. Note that, if the uncertainty on the phase space point in question is really greater than a factor of 100, it is doubtful whether VINCIA's estimates would be meaningful anyway, even if given an unrestricted range.

Speed of Uncertainty Evaluation

Numerical tests indicate that the additional computational overhead caused by activating the automatic uncertainty evaluation is not very large, generally increasing the runtime for a fully unweighted event sample by less than ten percent, for order of 10 different variations.

Moreover, since the events generated are the same for all the variations, only one event set needs to be passed, e.g., to models for hadronization and/or detector corrections. The bands are correctly propagated through these models by simply keeping track of the different weight sets and applying these at any stage after the shower evolution has finished. Hence substantial overall speed gains compared to a full-fledged traditional uncertainty estimation should be possible.