Approximate Higher-Order Effects

Gluon Splitting Correction

flag  VinciaShower:GluonSplittingCorrection   (default = on)
If two neighbouring dipole-antennae that share a splitting gluon are very unequal in size, then higher-order splitting functions and matrix elements unambiguosly indicate that the total gluon splitting probability is significantly suppressed. This is not taken into account when treating the two antennae as independent radiators. As in Ariadne, we include this effect approximately by applying the additional factor

PAri = 2 sN / (sP + sN)
to gluon splittings. sN is the invariant mass of the dipole-antenna sharing the gluon splitting, sP is the invariant mass of the dipole-antenna in which the gluon is splitting. The additional factor reduces to unity when the two invariants are similar, it suppresses splittings in antenna whose neighbour has a very small invariant mass and it slightly enhances splittings in an antenna whose neighbour has a very large invariant mass. Note: in VINCIA this correction is only applied when the neighbouring antenna is a qg one, not if it is gg.
option on : Use the factor PAri. Default since it greatly improves the shower's approximation to matrix emelents.
option off : Don't use the factor PAri. Included mostly for testing purposes.

Beyond-LL Matching

Matching to beyond-LL splitting kernels is still at a development stage. There are three main components to such a matching

Next-to-Leading Logarithmic (NLL) Corrections

mode  VinciaShower:NLL   (default = 1; minimum = 0; maximum = 2)

option 0 : Off. Scale variations in αs are not compensated at all in the shower evolution.
option 1 : Scale Cancellation Only. Scale variations in αs are compensated up to 1st order by matching to the explicit b0-dependent terms in the second-order (NLL) antenna functions.
option 2 : (Reserved for future use.) Full matching to second-order antenna functions.

Next-to-Leading Color (NLC) Corrections

Note: VINCIA uses a special color-factor normalization in which CF=8/3, and hence the difference CA-CF=1/NC2.

In the soft limit, the full radiation pattern from a color-connected chain of partons, qg...gqbar, can be written as a sum of leading-color dipole-antennae spanned between each of the color-connected partons, plus a negative qqbar antenna spanned directly between the endpoint quarks, proportional to CA/NC2. VINCIA contains a few different options for taking this antenna into account at various levels, as follows:

mode  VinciaShower:NLC   (default = 1; minimum = -1; maximum = 2)

option -1 : Strict LC. All gluon emission color factors forced equal to CA. No NLC correction.
option 0 : Ordinary LC. The color factors are taken from the chosen antenna set. I.e., they will typically be CF for qqbar antennae, CA for gg antennae, and something inbetween for qg antennae. No NLC correction.
option 1 : Eikonal NLC. A negative Eikonal suppressed by 1/NC2 is spanned between the two endpoint quarks of each qg...gqbar chain and is absorbed systematically into the radiation off the LC ones. Has been checked to correctly generate the subleading-color 1/ε2 term of the full 1-loop qqbar→qgqbar splitting function, and also improves the agreement with full-color tree-level Z→4,5,6-parton matrix elements relative to option 0.
option 2 : Full NLC. A negative qqbar antenna (Full = Eikonal + single-log terms) suppressed by 1/N2 is spanned between the two endpoint quarks of each qg...gqbar chain and is absorbed systematically into the radiation off the LC ones. Has been checked to correctly generate both the subleading-color 1/ε2 and 1/ε terms of the full 1-loop qqbar→qgqbar splitting function, and also further improves the agreement with full-color tree-level Z→4,5,6-parton matrix elements relative to option 1. Note: switching these corrections on may reduce the speed of the shower algorithm by up to a factor of 2.

Details on the NLC Correction

Since the subleading-color antenna spanned between the q and qbar is not positive definite, it cannot be treated at the same footing as the leading-color ones. It can, however, be systematically absorbed into the leading-color ones, effectively reducing their color factor from CA towards something closer to CF, with the exact magnitude of the correction depending on the phase space point.

Expressed in dimensionless variables, each LC antenna is modified as follows

aLC( yij , yjk ) → aLC( yij , yjk ) - fj aqq( yij , yjk ) / N2 ,

where N=3 for QCD and the factor fj partitions the full correction among the LC contributions. By default in VINCIA, it is given by

fj = eikonalj / ( Σi eikonali ) .

where again the eikonals must be expressed as dimensionless functions,

eikonalj( yij , yjk ) = 2 ( 1 - yij - yjk ) / ( yij yjk ) .

Practically, the implementation relies on a guess-and-reweight strategy, in which branchings are first generated according to leading-color antennae, but then the resulting configurations are reweighted by vetos, now taking into account the subleading correction which is always negative.

This matching is applied at every branching step (exponentiated) and so is in principle active to all orders. However, since the final state of subleading-color branchings are represented as planar, the approach cannot be used to reach formal 1/NC^4 accuracy.