Final-State Antenna Showers

  1. Antennae On/Off Switches
  2. Colour and Charge Factors
  3. Helicity Dependence
  4. Non-Singular Terms (including the ARIADNE factor)
  5. Mass Corrections
  6. Recoils and Kinematics
  7. Singular Coefficients

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, for instance, helicity dependence, non-singular terms of the antenna functions, their colour factors, and the number of flavors allowed in gluon splittings during the shower evolution. Predefined sets of alternative antenna functions are also provided (or can be defined by the user), as described in the section on Antenna Sets.

On/Off Switches for Individual Antenna Functions

The default is to leave all antenna functions switched on. For special theory studies, it may be convenient to check what happens when one or more antenna functions are switched off. The switches below give this functionality:

flag  Vincia:QQemit   (default = on)

flag  Vincia:QGemit   (default = on)

flag  Vincia:GGemit   (default = on)

flag  Vincia:QGsplit   (default = on)

flag  Vincia:GGsplit   (default = on)

The number of quark flavours allowed in gluon splittings, phase space permitting, is given by

mode  Vincia:nGluonToQuark   (default = 5; minimum = 0; maximum = 5)
Number of allowed quark flavours in gluon splittings, g → q qbar, during the shower evolution. 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 (see the section on Couplings).

Colour Charge Factors

The normalization of colour factors in VINCIA is chosen such that the coupling factor for all antenna functions is αS/4π. With this normalization 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 normalization of the qqbar → qgqbar antenna function is 2CF.)

parm  Vincia:QQemit:chargeFactor   (default = 2.66666667)

parm  Vincia:QGemit:chargeFactor   (default = 3.0)

parm  Vincia:GGemit:chargeFactor   (default = 3.0)

parm  Vincia:QGsplit:chargeFactor   (default = 1.0)

parm  Vincia:GGsplit: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).

Helicity Dependence

For massless partons, the default set of antenna functions in VINCIA depend explicitly on parton helicities. This gives a more accurate shower (since contributions from unphysical polarization states are removed) and speeds up the matching to matrix elements (since only a single helicity amplitude at a time needs to be evaluated). Note that, for partons treated as massive, the default is still to use helicity-summed antenna functions (see below).

flag  Vincia:helicityShower   (default = on)
The old helicity-summed behaviour for massless partons can be reobtained by switching this flag to off.

Further parameters controlling the helicity-dependent antenna functions are found below under Laurent Series Representation.

Non-Singular Terms

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.

The functional forms of the antenna functions are specified by giving the coefficients of a double Laurent series in the two branching invariants (the third invariant, specifying a rotation around the dipole axis, is chosen uniformly). The Laurent expansion formally starts at power (-1,-1) corresponding to the double singularity, but in most cases only the non-singular coefficients are modifiable by the user, see below.

Each massless antenna is fully specified by giving the coefficients C(α,β) of the following expansion:

where α and β are implicitly summed over, m_IK is the mass of the mother antenna, chargeFactor should be normalized to tend to NC raised to the number of new colour lines created in the splitting in the large-NC limit (see above), and y_ij,y_jk are the branching invariants scaled by the mass of the mother antenna, y_ab = s_ab/m2_IK.

Finite Terms in the Laurent Series Representation

Finite terms, i.e., with both Laurent indices greater than or equal to zero, are arbitrary and in general the best choice to make will depend on the specific process considered. In VINCIA, they are regarded as an independent way of estimating the uncertainty due to uncalculated higher orders, an uncertainty which can be explicitly reduced by matrix-element matching. The default is therefore to allow these terms to be nonzero, but for special applications it may be convenient to have one global switch that switches them on and off, regardless of the values given below:

flag  Vincia:useFiniteTerms   (default = on)
Global switch for all antenna function finite terms. Should be on for normal runs. Setting it to off will set all finite term coefficients to zero.

To avoid clutter, we first give the coefficients for antennae involving only massless partons. The additional correction terms when one or more partons are massive are collected in a separate subsection below.

Finite Terms for qqbar→qgqbar

The default coefficients are derived from H and Z decays (for S- and P-wave antennae, respectively) and absorb the tree-level matrix element for H→qgqbar and Z→qgqbar into the shower off H→qqbar and Z→qqbar, respectively.

S-wave (++ and --) antennae emitting without spin-flip (MHV):

parm  Vincia:QQemit:Cs00   (default = 0.0)

parm  Vincia:QQemit:Cs10   (default = 0.0)

parm  Vincia:QQemit:Cs01   (default = 0.0)

S-wave (++ and --) antennae emitting with spin flip (NMHV):

parm  Vincia:QQemit:Cf00   (default = 2.0)

parm  Vincia:QQemit:Cf10   (default = 0.0)

parm  Vincia:QQemit:Cf01   (default = 0.0)

P-wave (+- and -+) antennae:

parm  Vincia:QQemit:Cp00   (default = 0.0)

parm  Vincia:QQemit:Cp10   (default = 0.0)

parm  Vincia:QQemit:Cp01   (default = 0.0)

Finite Terms for qg→qgg

Note: by charge conjugation symmetry, the code also uses this function for qbar-g antennae, with appropriate swapping of invariants.

S-wave (++ and --) antennae emitting without spin-flip (MHV):

parm  Vincia:QGemit:Cs00   (default = 0.0)

parm  Vincia:QGemit:Cs10   (default = 0.0)

parm  Vincia:QGemit:Cs01   (default = 0.0)

S-wave (++ and --) antennae emitting with spin flip (NMHV):

parm  Vincia:QGemit:Cf00   (default = 3.0)

parm  Vincia:QGemit:Cf10   (default = 0.0)

parm  Vincia:QGemit:Cf01   (default = 0.0)

P-wave (+- and -+) antennae:

parm  Vincia:QGemit:Cp00   (default = 0.0)

parm  Vincia:QGemit:Cp10   (default = 0.0)

parm  Vincia:QGemit:Cp01   (default = 0.0)

Finite Terms for gg→ggg

S-wave (++ and --) antennae emitting without spin-flip (MHV):

parm  Vincia:GGemit:Cs00   (default = 0.0)

parm  Vincia:GGemit:Cs10   (default = 0.0)

parm  Vincia:GGemit:Cs01   (default = 0.0)

S-wave (++ and --) antennae emitting with spin flip (NMHV):

parm  Vincia:GGemit:Cf00   (default = 3.0)

parm  Vincia:GGemit:Cf10   (default = 1.0)

parm  Vincia:GGemit:Cf01   (default = 1.0)

P-wave (+- and -+) antennae:

parm  Vincia:GGemit:Cp00   (default = 0.0)

parm  Vincia:GGemit:Cp10   (default = 0.0)

parm  Vincia:GGemit:Cp01   (default = 0.0)

Finite Terms for qg→qqbar'q'

Coefficients for q' inheriting the helicity of the splitting gluon:

parm  Vincia:QGsplit:Cs00   (default = 0.0)

parm  Vincia:QGsplit:Cs10   (default = 0.0)

parm  Vincia:QGsplit:Cs01   (default = 0.0)

Coefficients for qbar' inheriting the helicity of the splitting gluon:

parm  Vincia:QGsplit:Cf00   (default = 0.0)

parm  Vincia:QGsplit:Cf10   (default = 0.0)

parm  Vincia:QGsplit:Cf01   (default = 0.0)

Finite Terms for gg→gqbarq

Coefficients for q inheriting the helicity of the splitting gluon:

parm  Vincia:GGsplit:Cs00   (default = 0.0)

parm  Vincia:GGsplit:Cs10   (default = 0.0)

parm  Vincia:GGsplit:Cs01   (default = 0.0)

Coefficients for qbar inheriting the helicity of the splitting gluon:

parm  Vincia:GGsplit:Cf00   (default = 0.0)

parm  Vincia:GGsplit:Cf10   (default = 0.0)

parm  Vincia:GGsplit:Cf01   (default = 0.0)

Gluon Splitting Correction (ARIADNE factor)

flag  Vincia: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 factor

PAri = 2 sN / (sP + sN)
to gluon splittings. sN is the invariant mass of the other dipole-antenna sharing the gluon splitting, sP is the invariant mass of the parent dipole-antenna in which the splitting occurs. The ARIADNE factor reduces to unity when the two invariants are similar, it suppresses splittings in an 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.
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.

Note 1: in VINCIA this correction is only applied when the neighbouring antenna is a qg one, not if it is gg.

Note 2: this correction does not affect the total collinear gluon-splitting singularity, though it does redistribute it; the total size of the gluon-splitting singularity is obtained as a sum over the two sides of the splitting gluon, with the ARIADNE factor summing to give a simple factor 2 on the singular term.

Mass Corrections

mode  Vincia:nFlavZeroMass   (default = 2; 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 treated as having massless kinematics, but approximate mass effects can be included, according to the switches described below. Top quarks will always be treated as with fully massive kinematics (and hence are so far precluded from appearing in PDFs).

mode  Vincia:massCorrectionLevel   (default = 2)
Controls the level of mass corrections that will be applied to all but the lightest Vincia:nFmassless flavours.
option 0 : No mass effects. Equivalent to using Vincia:nFmassless=5.
option 1 : Massive Phase-Space Effects. Take the restricted size of phase-space into account for processes involving massive flavours.
option 2 : Phase-Space and Antenna Mass Effects. As for option 1 but also include the negative corrections to the eikonal emission probabilities for gluon emissions from massive flavours, and the positive corrections to the probabilities for gluon splitting to massive flavours.

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 generalization 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

Important note: The following parameters give limited possibilities to modify the singular structure of individual antenna functions. These should not be changed in ordinary physics runs, but can be used to test, e.g., the effects of soft-eikonal vs collinear terms and of the choice of partitioning of gluon-collinear singularities.

flag  Vincia:useCollinearTerms   (default = on)
Global switch for all antenna function collinear-singular non-Eikonal terms. Should be on for normal runs. Setting it to off will set all collinear-singular non-Eikonal term coefficients to zero.

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 global showers, 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. (In sector showers, instead, each antenna contains the full AP kernel, but only one of them is allowed to contribute to each phase-space point.) In global showers, 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 global showers. 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.

The following parameters adjust the collinear partitioning of helicity-dependent antennae.

parm  Vincia:polGlbAntPartitioning1   (default = 0.0)

parm  Vincia:polGlbAntPartitioning2   (default = 0.0)