Antenna Functions

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, the non-singular terms of the antenna functions, their color 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.

Laurent Series Representation

AB->arb

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

The generic name format is thus Vincia:IK:x, 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 color tag of I and the anti-color 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 antenna is fully specified by giving the coefficients LaurentC(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 color lines created in the splitting in the large-NC limit (i.e., the limiting value should be 3 for gluon emission and 1 for gluon splitting, modulo 1/NC^2 ambiguities), 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.

Non-Singular Coefficients and Color Factors

Finite terms, i.e., with both 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.

Gluon Emission Antennae

Finite Terms for qqbar→qgqbar

The default coefficients are derived from Z decay and absorb the tree-level matrix element for Z→qgqbar into the shower off Z→qqbar.

flag  Vincia:qQ:emit   (default = on)

parm  Vincia:qQ:emit:chargeFactor   (default = 2.66666667)

parm  Vincia:qQ:emit:LaurentC(0,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentC(1,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentC(0,1)   (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.

flag  Vincia:qg:emit   (default = on)

parm  Vincia:qg:emit:chargeFactor   (default = 3.0)

parm  Vincia:qg:emit:LaurentC(0,0)   (default = 2.0)

parm  Vincia:qg:emit:LaurentC(1,0)   (default = -1.0)

parm  Vincia:qg:emit:LaurentC(0,1)   (default = 0.0)

Finite Terms for gg→ggg

flag  Vincia:gg:emit   (default = on)

parm  Vincia:gg:emit:chargeFactor   (default = 3.0)

parm  Vincia:gg:emit:LaurentC(0,0)   (default = 2.0)

parm  Vincia:gg:emit:LaurentC(1,0)   (default = 0.0)

parm  Vincia:gg:emit:LaurentC(0,1)   (default = 0.0)

Gluon-Splitting Antennae

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).

Finite Terms for qg→qqbar'q'

flag  Vincia:qg:split   (default = on)

parm  Vincia:qg:split:chargeFactor   (default = 1.0)

parm  Vincia:qg:split:LaurentC(0,0)   (default = 0.3)

parm  Vincia:qg:split:LaurentC(1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentC(0,1)   (default = 0.0)

Finite Terms for gg→gqbarq

flag  Vincia:gg:split   (default = on)

parm  Vincia:gg:split: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).

parm  Vincia:gg:split:LaurentC(0,0)   (default = 0.3)

parm  Vincia:gg:split:LaurentC(1,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentC(0,1)   (default = 0.0)

Mass Corrections

For those particles which are treated as massive (see the section on Masses & Couplings), 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]. We also include a flag which can be used to switch off all mass corrections to the antenna functions (including also the hardcoded terms):

flag  Vincia:useMassTerms   (default = on)
This flag is only intended for testing purposes. If the mass terms are turned off completely, the massive soft eikonal factor and the quasi-collinear splitting function will not be reproduced correctly. This flag should therefore always be left on for physics studies.

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

Gluon-Emission Antennae

Finite Mass Terms for qqbar→qgqbar

parm  Vincia:qQ:emit:DenominatorMassPrefactor   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(1,0,0,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(1,0,1,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(2,0,0,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(2,0,1,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(1,1,0,0)   (default = 0.0)

parm  Vincia:qQ:emit:LaurentM(1,1,1,0)   (default = 0.0)

Finite Mass Terms for qg→qgg

parm  Vincia:qg:emit:LaurentM(1,0,0)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(1,1,0)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(1,0,1)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(2,0,0)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(2,1,0)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(2,0,1)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(2,-1,0)   (default = 0.0)

parm  Vincia:qg:emit:LaurentM(2,-1,1)   (default = 0.0)

parm  Vincia:qg:emit:LaurentMtilde(2,-1,-1,2)   (default = 0.0)

parm  Vincia:qg:emit:LaurentMtilde(3,-1,-1,1)   (default = 0.0)

parm  Vincia:qg:emit:LaurentMtilde(4,-1,-1,1)   (default = 0.0)

Gluon-Splitting Antennae

Finite Mass Terms for qg→qqbar'q'

parm  Vincia:qg:split:LaurentM(1,0,0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentM(1,0,1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentM(1,0,0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentM(2,0,0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentM(2,0,0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentM(2,0,1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(0,2,0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(0,2,0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(0,2,1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(1,0,0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(1,0,0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(1,0,1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(2,0,0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(2,0,0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentMtilde(2,0,1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(0,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(0,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(0,2)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(1,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentCtilde(2,0)   (default = 0.0)

Finite Mass Terms for gg→gqbarq

parm  Vincia:gg:split:LaurentM(2,0,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentM(2,0,1)   (default = 0.0)

parm  Vincia:gg:split:LaurentM(2,1,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(0,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(0,1)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(0,2)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(1,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(1,1)   (default = 0.0)

parm  Vincia:gg:split:LaurentCtilde(2,0)   (default = 0.0)

Singular Coefficients

Important note: The following coefficients pertain to the singular structure of the antenna functions and should not be changed by the user. Moreover, in the current implementation, several of them only act as placeholders, in the base Antenna class implementation, with the singular terms actually used in the code being hardcoded in functions inheriting from the base class.

Non-Eikonal Collinear Singularities

Strictly for theoretical testing purposes, we provide a possibility to switch off all but the Eikonal terms (i.e., removing the additional collinear-singular terms from the antenna functions, regardless of the values of the coefficients below), using the following switch

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.

The collinear singularity of two gluons j and k in a gluon emission antenna is distributed between two antennae, one contains the singularity for j becoming soft, one the singularity for k becoming soft. The parameter below is the value of "half" the gluon splitting function on its finite end.

parm  Vincia:octetPartitioning   (default = 0.0; minimum = 0.0; maximum = 1)

Gluon-Emission Antennae

Singular Terms for qqbar→qgqbar

parm  Vincia:qQ:emit:LaurentC(-1,-1)   (default = 2.0)

parm  Vincia:qQ:emit:LaurentC(-1,0)   (default = -2.0)

parm  Vincia:qQ:emit:LaurentC(0,-1)   (default = -2.0)

parm  Vincia:qQ:emit:LaurentC(-1,1)   (default = 1.0)

parm  Vincia:qQ:emit:LaurentC(1,-1)   (default = 1.0)

Singular Terms for qg→qgg

parm  Vincia:qg:emit:LaurentC(-1,-1)   (default = 2.0)

parm  Vincia:qg:emit:LaurentC(-1,0)   (default = -2.0)

parm  Vincia:qg:emit:LaurentC(0,-1)   (default = -2.0)

parm  Vincia:qg:emit:LaurentC(1,-1)   (default = 1.0)

parm  Vincia:qg:emit:LaurentC(-1,1)   (default = 1.0)

parm  Vincia:qg:emit:LaurentC(2,-1)   (default = -1.0)

Singular Terms for gg→ggg

parm  Vincia:gg:emit:LaurentC(-1,-1)   (default = 2.0)

parm  Vincia:gg:emit:LaurentC(-1,0)   (default = -2.0)

parm  Vincia:gg:emit:LaurentC(0,-1)   (default = -2.0)

parm  Vincia:gg:emit:LaurentC(-1,1)   (default = 1.0)

parm  Vincia:gg:emit:LaurentC(1,-1)   (default = 1.0)

parm  Vincia:gg:emit:LaurentC(2,-1)   (default = -1.0)

parm  Vincia:gg:emit:LaurentC(-1,2)   (default = -1.0)

Gluon-Splitting Antennae

Singular Terms for qg→qqbar'q'

parm  Vincia:qg:split:LaurentC(-1,-1)   (default = 0.0)

parm  Vincia:qg:split:LaurentC(-1,0)   (default = 0.0)

parm  Vincia:qg:split:LaurentC(-1,1)   (default = 0.0)

parm  Vincia:qg:split:LaurentC(0,-1)   (default = 0.5)

parm  Vincia:qg:split:LaurentC(1,-1)   (default = -1.0)

parm  Vincia:qg:split:LaurentC(2,-1)   (default = 1.0)

Singular Terms for gg→gqbarq

parm  Vincia:gg:split:LaurentC(-1,-1)   (default = 0.0)

parm  Vincia:gg:split:LaurentC(-1,0)   (default = 0.0)

parm  Vincia:gg:split:LaurentC(-1,1)   (default = 0.0)

parm  Vincia:gg:split:LaurentC(0,-1)   (default = 0.5)

parm  Vincia:gg:split:LaurentC(1,-1)   (default = -1.0)

parm  Vincia:gg:split:LaurentC(2,-1)   (default = 1.0)