Couplings

  1. The Strong Coupling
  2. Running Order
  3. Number of Flavours
  4. Effective Renormalization Scheme
  5. Effective Renormalization Scale
  6. Infrared Behaviour

The Strong Coupling

VINCIA implements its own instance of PYTHIA's AlphaStrong class for the strong coupling. You can find more documentation of the class in the section on Standard-Model Parameters in the PYTHIA documentation. Here, we list the specific parameters and switches governing its use in VINCIA.

Reference Value

The amount of QCD radiation in the shower is determined by

parm  Vincia:alphaSvalue   (default = 0.1215; minimum = 0.06; maximum = 2.0)
The value of αs at the scale mZ, in the MSbar scheme. The default is chosen to obtain a reasonable agreement with LEP event shapes for default shower settings.

Order

mode  Vincia:alphaSorder   (default = 2; minimum = 0; maximum = 2)
Order at which αs runs,
option 0 : zeroth order, i.e. αs is kept fixed.
option 1 : first order, i.e., one-loop running.
option 2 : second order, i.e., two-loop running.

Flavour Thresholds

For both one- and two-loop running, the AlphaStrong class automatically switches from 3-, to 4-, and then to 5-flavour running as one passes the s, c, and b thresholds, respectively, with matching equations imposed at each flavour treshold to ensure continuous values. By default, a change to 6-flavour running is also included above the t threshold, though this can be disabled using the following parameter:

mode  Vincia:alphaSnfmax   (default = 6; minimum = 5; maximum = 6)

option 5 : Use 5-flavour running for all scales above the b flavour threshold (old default).
option 6 : Use 6-flavour running above the t threshold (new default).

Effective Renormalization Scheme

Resummation arguments [Cat91] indicate that a set of universal QCD corrections can be absorbed in coherent parton showers by applying the so-called CMW rescaling of the MSbar value of Lambda_QCD, defined by

αs(CMW) = αs(MSbar) * (1 + K * αs(MSbar) / 2π)
with K = CA * (67/18 - π2/6) - 5/9nf. The translation amounts to an NF-dependent rescaling of Lambda_QCD, relative to its MSbar value, by a factor 1.661 for NF=3, 1.618 for NF=4, 1.569 for NF=5, and 1.513 for NF=6.

By default, the CMW correction is applied in VINCIA. If desired, it can be switched off by using the following switch:

flag  Vincia:alphaScmw   (default = on)

option on : CMW. This option is recommended for the most accurate shower resummation.
option off : MSbar.

Note: The CMW arguments were derived using two-loop running (the default in VINCIA).
Note 2: When this correction is switched on, the rescaling of the coupling away from its MSbar value is properly taken into account by VINCIA's one-loop matching. When switched off, the one-loop corrections generally become larger, to compensate for the missing universal factor.
Note 3: If using VINCIA with an externally defined matching scheme, be aware that the CMW rescaling may need be taken into account in the context of matrix-element matching. Note also that this option has only been made available for timelike and spacelike showers, not for hard processes.
Note 4: tunes using this option need lower values of alpha_strong(m_Z^2) than tunes that do not.

Argument of Running Coupling

When Vincia:alphaSorder is non-zero, the actual value is regulated by running to the scale kμR, at which the shower evaluates αs. The functional form of μR is given by Vincia:alphaSmode and the scale factor kμ is given by Vincia:alphaSkMu.

The functional form of μR is given by

mode  Vincia:alphaSmode   (default = 1; minimum = 0; maximum = 1)

option 0 : The invariant mass of the mother antenna, m(ijk).
option 1 : For gluon emissions: transverse momentum, defined as in ARIADNE, pT = m(ij)*m(jk)/m(ijk). For gluon splittings, g→qq, the qqbar invariant mass, m(qqbar).

For gluon emissions, the scale factor kμ is given by

parm  Vincia:alphaSkMu   (default = 1.0; minimum = 0.1; maximum = 10.0)
If different from unity, αs is evaluated at the scale defined by Vincia:alphaSmode times this scale factor, i.e., it gives the value of kμ in the argument to alphaS(kμR).

parm  Vincia:alphaSkMuSplit   (default = 0.4; minimum = 0.1; maximum = 10.0)
Renormalization-scale prefactor for gluon splittings, multiplying the mQQ scale.

Infrared Freezeout Scale

parm  Vincia:alphaSmuFreeze   (default = 0.5; minimum = 0.0; maximum = 10.0)
In the far infrared, the behaviour of the running coupling is regulated by a shift in the effective renormalization scale, to μeff 2 = μfreeze2 + (kμR)2.

Max Coupling

parm  Vincia:alphaSmax   (default = 2.0; minimum = 0.1; maximum = 10.0)
Largest allowed numerical value for alphaS.