Antenna Functions

  1. On/Off Switches
  2. Colour and Charge Factors
  3. Helicity Dependence
  4. ARIADNE Factor
  5. Laurent Series Representation
  6. Mass Corrections
  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 behavior 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.

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.

Laurent Series Representation

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, with the invariants defined by s_ab = 2 pa.pb.

For antennae spanned between partons which are treated as massive, additional terms are present, which are described separately below.

Finite Terms

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)

Mass Corrections

For those particles which are treated as massive (see the section on Mass Effects), the terms listed below control the possible non-singular mass corrections that can be added to the antenna functions (within the limits required by positivity). These are included in addition to hard-coded mass corrections to the singular structures that cannot be modified by the user and so are not listed here. See [GRS11] for definitions.

Note 1: the coefficients listed in this section are ignored if useFiniteTerms is switched off, see above.

Note 2: the mass corrections are summed and averaged over helicities.

Finite Mass Terms for qqbar→qgqbar

parm  Vincia:QQemit:DenominatorMassPrefactor   (default = 0.0)
Coefficient x of the mass product term in the denominator which multiplies the collinear and finite antenna terms.

parm  Vincia:QQemit:M1000   (default = 0.0)
The M(1,0,0,0) coefficient.

parm  Vincia:QQemit:M1010   (default = 0.0)
The M(1,0,1,0) coefficient.

parm  Vincia:QQemit:M2000   (default = 0.0)
The M(2,0,0,0) coefficient.

parm  Vincia:QQemit:M2010   (default = 0.0)
The M(2,0,1,0) coefficient.

parm  Vincia:QQemit:M1100   (default = 0.0)
The M(1,1,0,0) coefficient.

parm  Vincia:QQemit:M1110   (default = 0.0)
The M(1,1,1,0) coefficient.

Finite Mass Terms for qg→qgg

parm  Vincia:QGemit:M100   (default = 0.0)
The M(1,0,0) coefficient.

parm  Vincia:QGemit:M110   (default = 0.0)
The M(1,1,0) coefficient.

parm  Vincia:QGemit:M101   (default = 0.0)
The M(1,0,1) coefficient.

parm  Vincia:QGemit:M200   (default = 0.0)
The M(2,0,0) coefficient.

parm  Vincia:QGemit:M210   (default = 0.0)
The M(2,1,0) coefficient.

parm  Vincia:QGemit:M201   (default = 0.0)
The M(2,0,1) coefficient.

parm  Vincia:QGemit:M2m0   (default = 0.0)
The M(2,-1,0) coefficient.

parm  Vincia:QGemit:M2m1   (default = 0.0)
The M(2,-1,1) coefficient.

parm  Vincia:QGemit:T2mm2   (default = 0.0)
The Mtilde(2,-1,-1,2) coefficient.

parm  Vincia:QGemit:T3mm1   (default = 0.0)
The Mtilde(3,-1,-1,1) coefficient.

parm  Vincia:QGemit:T4mm1   (default = 0.0)
The Mtilde(4,-1,-1,1) coefficient.

Finite Mass Terms for qg→qqbar'q'

parm  Vincia:QGsplit:M1000   (default = 0.0)
The M(1,0,0,0) coefficient.

parm  Vincia:QGsplit:M1010   (default = 0.0)
The M(1,0,1,0) coefficient.

parm  Vincia:QGsplit:M1001   (default = 0.0)
The M(1,0,0,1) coefficient.

parm  Vincia:QGsplit:M2000   (default = 0.0)
The M(2,0,0,0) coefficient.

parm  Vincia:QGsplit:M2010   (default = 0.0)
The M(2,0,1,0) coefficient.

parm  Vincia:QGsplit:M2001   (default = 0.0)
The M(2,0,0,1) coefficient.

parm  Vincia:QGsplit:T00   (default = 0.0)
The Ctilde(0,0) coefficient.

parm  Vincia:QGsplit:T10   (default = 0.0)
The Ctilde(1,0) coefficient.

parm  Vincia:QGsplit:T01   (default = 0.0)
The Ctilde(0,1) coefficient.

parm  Vincia:QGsplit:T1000   (default = 0.0)
The Mtilde(1,0,0,0) coefficient.

parm  Vincia:QGsplit:T1010   (default = 0.0)
The Mtilde(1,0,1,0) coefficient.

parm  Vincia:QGsplit:T1001   (default = 0.0)
The Mtilde(1,0,0,1) coefficient.

parm  Vincia:QGsplit:T2000   (default = 0.0)
The Mtilde(2,0,0,0) coefficient.

parm  Vincia:QGsplit:T2010   (default = 0.0)
The Mtilde(2,0,1,0) coefficient.

parm  Vincia:QGsplit:T2001   (default = 0.0)
The Mtilde(2,0,0,1) coefficient.

parm  Vincia:QGsplit:T0200   (default = 0.0)
The Mtilde(0,2,0,0) coefficient.

parm  Vincia:QGsplit:T0210   (default = 0.0)
The Mtilde(0,2,1,0) coefficient.

parm  Vincia:QGsplit:T0201   (default = 0.0)
The Mtilde(0,2,0,1) coefficient.

Finite Mass Terms for gg→gqbarq

parm  Vincia:GGsplit:T00   (default = 0.0)
The Ctilde(0,0) coefficient.

parm  Vincia:GGsplit:T10   (default = 0.0)
The Ctilde(1,0) coefficient.

parm  Vincia:GGsplit:T01   (default = 0.0)
The Ctilde(0,1) coefficient.

parm  Vincia:GGsplit:M200   (default = 0.0)
The M(2,0,0) coefficient.

parm  Vincia:GGsplit:M210   (default = 0.0)
The M(2,1,0) coefficient.

parm  Vincia:GGsplit:M201   (default = 0.0)
The M(2,0,1) coefficient.

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)