Final-State Antenna Showers

  1. On/Off Switches
  2. The Strong Coupling for FF antennae
  3. Colour and Charge Factors
  4. Evolution Variable(s)
  5. Antenna Functions
  6. Mass Corrections
  7. Recoils and Kinematics
  8. Octet Partitioning

Within the dipole-antenna formalism, antenna functions are the analogs of the splitting functions used in traditional parton showers. The antenna functions are constructed so as to reproduce the Altarelli-Parisi splitting functions P(z) in collinear limits and the eikonal dipole factor in the soft limit.

We here describe the parameters and switches used to control VINCIA's antenna functions, including their colour factors, renormalisation scales, the number of flavours allowed in gluon splittings during the shower evolution, and a few more options.

On/Off Switches

flag  Vincia:FSR   (default = on)
Main switch for final-state radiation on/off.

The number of quark flavours allowed in final-state gluon splittings (in both final-final and initial-final antennae) is given by

mode  Vincia:nGluonToQuarkF   (default = 5; minimum = 0; maximum = 5)
Number of allowed quark flavours in gluon splittings, g → q qbar, during the shower evolution, phase space permitting. E.g., a change to 4 would exclude g → b bbar but would include the lighter quarks, etc. Note that this parameter does not directly affect the running coupling.

The Strong Coupling (for FSR)

Argument of the Running Coupling

When Vincia:alphaSorder is non-zero, the actual value of alphaS used for shower branchings is governed by the choice of scheme (MSbar or CMW, see the section on SM Couplings and then by running to the scale kμR, at which the shower evaluates αs. μR is the transverse momentum pT A (without the normalisation factor) for gluon emission and mQQ for gluon splitting. The scale factor kμ is given by

parm  Vincia:alphaSkMuF   (default = 0.68; minimum = 0.1; maximum = 10.0)
for gluon emission

and

parm  Vincia:alphaSkMuSplitF   (default = 0.6; minimum = 0.1; maximum = 10.0)
for gluon splitting.

Colour Charge Factors

The normalisation of colour factors in VINCIA is chosen such that the coupling factor for all antenna functions is αS/4π. With this normalisation choice, all gluon-emission colour factors tend to NC in the large-NC limit while all gluon-splitting colour factors tend to unity. (Thus, e.g., the default normalisation of the qqbar → qgqbar antenna function is 2CF.)

parm  Vincia:QQEmitFF:chargeFactor   (default = 2.66666667)

parm  Vincia:QGEmitFF:chargeFactor   (default = 2.85)

parm  Vincia:GGEmitFF:chargeFactor   (default = 3.0)

parm  Vincia:QGSplitFF:chargeFactor   (default = 1.0)

parm  Vincia:GGSplitFF:chargeFactor   (default = 1.0)
Note: the two permutations g-g → g-q+qbar and g-g → qbar+q-g are explicitly summed over in the code (with appropriate swapping of invariants in the latter case).

Evolution Variable for Final State Radition

The choices below govern how the shower fills phase space, and hence how the logarithms generated by it are ordered. This does not affect the LL behaviour, but does affect the tower of higher (subleading) logs generated by the shower and can therefore be signficiant in regions where the leading logs are suppressed or absent. Note that, by construction, the dipole formalism automatically ensures an exact treatment of (leading-colour) coherence effects to leading logarithmic order, and hence additional constraints, such as angular ordering, are not required.

Note 1: Gluon splittings, g→qq, always use mqqbar as their evolution variable, which are interleaved with the choice made here.

Note 2: The absolute normalisation factors cancel in sequential gluon emissions, but enter in the relative interleaving between gluon emissions and gluon splitting.

mode  Vincia:evolutionType   (default = 1)
Choice of functional form of the shower evolution variable (a.k.a. ordering variable) for gluon emissions (see illustrations below).
option 1 : Transverse Momentum. This evolution variable is roughly equal to the inverse of the antenna function for gluon emission, and hence is in some sense the most natural evolution variable. We define it as in ARIADNE, modulo a normalisation factor:

with the normalisation factor NT=4.
option 2 : Dipole Virtuality. This mass-like variable represents a fairly moderate variation on the transverse momentum. It will give slightly more priority to soft branchings over collinear branchings, as compared to transverse momentum. We define it as

with the normalisation factor ND=2.

Evolution Variable: Phase-Space Contours

The contours below illustrate the progression of each evolution variable (normalized to maximum value unity) over the dipole-antenna phase space for four fixed values of yE = QE2/sIK:

Type1 Type2

Antenna Functions

AB->arb

The 2→3 (LL) VINCIA antennae have names such as

The generic name format is thus Vincia:IKx, where I and K are the "mothers" and x is either emit or split, depending on whether the process is gluon emission or gluon splitting. The radiating (parent) antenna is interpreted as spanned between the Les Houches colour tag of I and the anti-colour tag of K, see illustration to the right.

For each antenna function, a full set of helicity-dependent antenna function contributions are implemented. For partons without helicity information, the unpolarised forms (summed over post-branching helicities and averaged over pre-branching ones) are used. The detailed forms of both helicity and helicity-summed/averaged antenna functions are given in the VINCIA Authors' Compendium.

Mass Corrections

mode  Vincia:nFlavZeroMassF   (default = 3; minimum = 0; maximum = 5)
Controls the number of flavours that will be treated as strictly massless by VINCIA, ie with massless kinematics and no mass corrections whatsoever. The remaining flavours, up to the b quark, will still be bookkept as having massless kinematics, but a set of minimal approximate mass effects are included (such as thresholds).

Kinematics and Recoils

While the CM momenta of a 2→3 branching are fixed by the generated invariants (and hence by the antenna function), the global orientation of the produced 3-parton system with respect to the rest of the event (or, equivalently, with respect to the original dipole-antenna axis) suffers from an ambiguity outside the LL limits, which can affect the tower of subleading logs generated and can be significant in regions where the leading logs are suppressed or absent.

To illustrate this ambiguity, consider the emissision of a gluon from a qqbar antenna with some finite amount of transverse momentum (meaning transverse to the original dipole-antenna axis, in the CM of the dipole-antenna). The transverse momenta of the qqbar pair after the branching must now add up to an equal, opposite amount, so that total momentum is conserved, i.e., the emission generates a recoil. By an overall rotation of the post-branching 3-parton system, it is possible to align either the q or the qbar with the original axis, such that it becomes the other one that absorbs the entire recoil (the default in showers based on 1→2 branchings such as old-fashioned parton showers and Catani-Seymour showers), or to align both of them slightly off-axis, so that they share the recoil (the default in VINCIA, see illustration below).

Kinematics Map

Note: this setting only applies to massless partons. For partons treated as massive (see below), a generalisation of the Kosower map (option 3 below) will be used regardless of the value of Vincia:kineMapType. 2->3
			      kinematics

mode  Vincia:kineMapType   (default = 3; minimum = 1; maximum = 3)
Selects which method to use for choosing the Euler angle for the global orientation of the post-branching kinematics construction. This setting applies to all massless antennae, irrespective of the branching type.
option 1 : The ARIADNE angle (see illustration). The recoiling mothers share the recoil in proportion to their energy fractions in the CM of the dipole-antenna. Tree-level expansions of the VINCIA shower compared to tree-level matrix elements through third order in alphaS have shown this strategy to give the best overall approximation, followed closely by the KOSOWER map below.
option 2 : LONGITUDINAL. The parton which has the smallest invariant mass together with the radiated parton is taken to be the "radiator". The remaining parton is taken to be the "recoiler". The recoiler remains oriented along the dipole axis in the branching rest frame and recoils longitudinally against the radiator + radiated partons which have equal and opposite transverse momenta (transverse to the original dipole-antenna axis in the dipole-antenna CM). Comparisons to higher-order QCD matrix elements show this to be by far the worst option of the ones so far implemented, hence it could be useful as an extreme case for uncertainty estimates, but should probably not be considered for central tunes. (Note: exploratory attempts at improving the behaviour of this map, e.g., by selecting probabilistically between the radiator and the recoiler according to approximate collinear splitting kernels, only resulted in marginal improvements. Since such variations would introduce additional complications in the VINCIA matching formalism, they have not been retained in the distributed version.)
option 3 : The KOSOWER map. Comparisons to higher-order QCD matrix elements show only very small differences between this and the ARIADNE map above, but since the KOSOWER map is sometimes used in fixed-order contexts, we deem it interesting to include it as a complementary possibility. (Note: the KOSOWER maps in fact represent a whole family of kinematics maps. For experts, the specific choice made here corresponds to using r=sij/(sij+sjk) in the definition of the map.)

Singular Coefficients

Within the antenna formalism, the collinear singularity of two gluons j and k is distributed between two neighboring antennae. One contains the singularity for j becoming soft, one the singularity for k becoming soft. In showers based on so-called global antenna functions (as opposed to sector functions, which are no longer implemented in VINCIA), the two antennae share the collinear singularity, j||k, point by point in phase space, and only after summing over both is the full collinear AP splitting kernel recovered. The parameter below controls the repartition ambiguity and gives the value of "half" the gluon splitting function on its finite end.

parm  Vincia:octetPartitioning   (default = 0.0; minimum = 0.0; maximum = 1)
Gluon-collinear α parameter. Only used for final-final global antennae. Note: only the default value (0) is consistent with the initial-final (and initial-initial) antenna functions in VINCIA. Special values of interest are: α=0, which corresponds to the Gehrmann-Gehrmann-de Ridder-Glover (GGG) partitioning, and α=1, which corresponds to the Gustafson (ARIADNE) partitioning.