Couplings and Scales

  1. Couplings and K factor
  2. Renormalization scales
  3. Factorization scales
Here is collected some possibilities to modify the scale choices of couplings and parton densities for all internally implemented hard processes. This is based on them all being derived from the SigmaProcess base class. The matrix-element coding is also used by the multiparton-interactions machinery, but there with a separate choice of alpha_strong(M_Z^2) value and running, and separate PDF scale choices. Also, in 2 → 2 and 2 → 3 processes where resonances are produced, their couplings and thereby their Breit-Wigner shapes are always evaluated with the resonance mass as scale, irrespective of the choices below.

We stress that couplings and scales are set separately from the values on this page for multiparton interactions, timelike showers, and spacelike showers. This allows a bigger flexibility, but also requires a bit more work e.g. if you insist on using the same alpha_s everywhere.

Couplings and K factor

The size of QCD cross sections is mainly determined by

parm  SigmaProcess:alphaSvalue   (default = 0.13; minimum = 0.06; maximum = 0.25)
The alpha_strong value at scale M_Z^2.

The actual value is then regulated by the running to the Q^2 renormalization scale, at which alpha_strong is evaluated

mode  SigmaProcess:alphaSorder   (default = 1; minimum = 0; maximum = 3)
Order at which alpha_strong runs,
option 0 : zeroth order, i.e. alpha_strong is kept fixed.
option 1 : first order, which is the normal value.
option 2 : second order. Since other parts of the code do not go to second order there is no strong reason to use this option, but there is also nothing wrong with it.
option 3 : third order, with the same comment as for second order. The expression in the 2006 RPP is used here.

QED interactions are regulated by the alpha_electromagnetic value at the Q^2 renormalization scale of an interaction.

mode  SigmaProcess:alphaEMorder   (default = 1; minimum = -1; maximum = 1)
The running of alpha_em used in hard processes.
option 1 : first-order running, constrained to agree with StandardModel:alphaEMmZ at the Z^0 mass.
option 0 : zeroth order, i.e. alpha_em is kept fixed at its value at vanishing momentum transfer.
option -1 : zeroth order, i.e. alpha_em is kept fixed, but at StandardModel:alphaEMmZ, i.e. its value at the Z^0 mass.

In addition there is the possibility of a global rescaling of cross sections (which could not easily be accommodated by a changed alpha_strong, since alpha_strong runs)

parm  SigmaProcess:Kfactor   (default = 1.0; minimum = 0.5; maximum = 4.0)
Multiply almost all cross sections by this common fix factor. Excluded are only unresolved processes, where cross sections are better set directly, and multiparton interactions, which have a separate K factor of their own. This degree of freedom is primarily intended for hadron colliders, and should not normally be used for e^+e^- annihilation processes.

Renormalization scales

The Q^2 renormalization scale can be chosen among a few different alternatives, separately for 2 → 1, 2 → 2 and two different kinds of 2 → 3 processes. In addition a common multiplicative factor may be imposed.

mode  SigmaProcess:renormScale1   (default = 1; minimum = 1; maximum = 2)
The Q^2 renormalization scale for 2 → 1 processes. The same options also apply for those 2 → 2 and 2 → 3 processes that have been specially marked as proceeding only through an s-channel resonance, by the isSChannel() virtual method of SigmaProcess.
option 1 : the squared invariant mass, i.e. sHat.
option 2 : fix scale set in SigmaProcess:renormFixScale below.

mode  SigmaProcess:renormScale2   (default = 2; minimum = 1; maximum = 6)
The Q^2 renormalization scale for 2 → 2 processes.
option 1 : the smaller of the squared transverse masses of the two outgoing particles, i.e. min(mT_3^2, mT_4^2) = pT^2 + min(m_3^2, m_4^2).
option 2 : the geometric mean of the squared transverse masses of the two outgoing particles, i.e. mT_3 * mT_4 = sqrt((pT^2 + m_3^2) * (pT^2 + m_4^2)).
option 3 : the arithmetic mean of the squared transverse masses of the two outgoing particles, i.e. (mT_3^2 + mT_4^2) / 2 = pT^2 + 0.5 * (m_3^2 + m_4^2). Useful for comparisons with PYTHIA 6, where this is the default.
option 4 : squared invariant mass of the system, i.e. sHat. Useful for processes dominated by s-channel exchange.
option 5 : fix scale set in SigmaProcess:renormFixScale below.
option 6 : Use squared invariant momentum transfer -tHat. This is a common choice for lepton-hadron scattering processes. In that case -tHat=Q^2.

mode  SigmaProcess:renormScale3   (default = 3; minimum = 1; maximum = 6)
The Q^2 renormalization scale for "normal" 2 → 3 processes, i.e excepting the vector-boson-fusion processes below. Here it is assumed that particle masses in the final state either match or are heavier than that of any t-channel propagator particle. (Currently only g g / q qbar → H^0 Q Qbar processes are implemented, where the "match" criterion holds.)
option 1 : the smaller of the squared transverse masses of the three outgoing particles, i.e. min(mT_3^2, mT_4^2, mT_5^2).
option 2 : the geometric mean of the two smallest squared transverse masses of the three outgoing particles, i.e. sqrt( mT_3^2 * mT_4^2 * mT_5^2 / max(mT_3^2, mT_4^2, mT_5^2) ).
option 3 : the geometric mean of the squared transverse masses of the three outgoing particles, i.e. (mT_3^2 * mT_4^2 * mT_5^2)^(1/3).
option 4 : the arithmetic mean of the squared transverse masses of the three outgoing particles, i.e. (mT_3^2 + mT_4^2 + mT_5^2)/3.
option 5 : squared invariant mass of the system, i.e. sHat.
option 6 : fix scale set in SigmaProcess:renormFixScale below.

mode  SigmaProcess:renormScale3VV   (default = 3; minimum = 1; maximum = 6)
The Q^2 renormalization scale for 2 → 3 vector-boson-fusion processes, i.e. f_1 f_2 → H^0 f_3 f_4 with Z^0 or W^+- t-channel propagators. Here the transverse masses of the outgoing fermions do not reflect the virtualities of the exchanged bosons. A better estimate is obtained by replacing the final-state fermion masses by the vector-boson ones in the definition of transverse masses. We denote these combinations mT_Vi^2 = m_V^2 + pT_i^2.
option 1 : the squared mass m_V^2 of the exchanged vector boson.
option 2 : the geometric mean of the two propagator virtuality estimates, i.e. sqrt(mT_V3^2 * mT_V4^2).
option 3 : the geometric mean of the three relevant squared transverse masses, i.e. (mT_V3^2 * mT_V4^2 * mT_H^2)^(1/3).
option 4 : the arithmetic mean of the three relevant squared transverse masses, i.e. (mT_V3^2 + mT_V4^2 + mT_H^2)/3.
option 5 : squared invariant mass of the system, i.e. sHat.
option 6 : fix scale set in SigmaProcess:renormFixScale below.

parm  SigmaProcess:renormMultFac   (default = 1.; minimum = 0.1; maximum = 10.)
The Q^2 renormalization scale for 2 → 1, 2 → 2 and 2 → 3 processes is multiplied by this factor relative to the scale described above (except for the options with a fix scale). Should be use sparingly for 2 → 1 processes.

parm  SigmaProcess:renormFixScale   (default = 10000.; minimum = 1.)
A fix Q^2 value used as renormalization scale for 2 → 1, 2 → 2 and 2 → 3 processes in some of the options above.

Factorization scales

Corresponding options exist for the Q^2 factorization scale used as argument in PDF's. Again there is a choice of form for 2 → 1, 2 → 2 and 2 → 3 processes separately. For simplicity we have let the numbering of options agree, for each event class separately, between normalization and factorization scales, and the description has therefore been slightly shortened. The default values are not necessarily the same, however.

mode  SigmaProcess:factorScale1   (default = 1; minimum = 1; maximum = 2)
The Q^2 factorization scale for 2 → 1 processes. The same options also apply for those 2 → 2 and 2 → 3 processes that have been specially marked as proceeding only through an s-channel resonance.
option 1 : the squared invariant mass, i.e. sHat.
option 2 : fix scale set in SigmaProcess:factorFixScale below.

mode  SigmaProcess:factorScale2   (default = 1; minimum = 1; maximum = 6)
The Q^2 factorization scale for 2 → 2 processes.
option 1 : the smaller of the squared transverse masses of the two outgoing particles.
option 2 : the geometric mean of the squared transverse masses of the two outgoing particles.
option 3 : the arithmetic mean of the squared transverse masses of the two outgoing particles. Useful for comparisons with PYTHIA 6, where this is the default.
option 4 : squared invariant mass of the system, i.e. sHat. Useful for processes dominated by s-channel exchange.
option 5 : fix scale set in SigmaProcess:factorFixScale below.
option 6 : Use squared invariant momentum transfer -tHat. This is a common choice for lepton-hadron scattering processes. In that case -tHat=Q^2.

mode  SigmaProcess:factorScale3   (default = 2; minimum = 1; maximum = 6)
The Q^2 factorization scale for "normal" 2 → 3 processes, i.e excepting the vector-boson-fusion processes below.
option 1 : the smaller of the squared transverse masses of the three outgoing particles.
option 2 : the geometric mean of the two smallest squared transverse masses of the three outgoing particles.
option 3 : the geometric mean of the squared transverse masses of the three outgoing particles.
option 4 : the arithmetic mean of the squared transverse masses of the three outgoing particles.
option 5 : squared invariant mass of the system, i.e. sHat.
option 6 : fix scale set in SigmaProcess:factorFixScale below.

mode  SigmaProcess:factorScale3VV   (default = 2; minimum = 1; maximum = 6)
The Q^2 factorization scale for 2 → 3 vector-boson-fusion processes, i.e. f_1 f_2 → H^0 f_3 f_4 with Z^0 or W^+- t-channel propagators. Here we again introduce the combinations mT_Vi^2 = m_V^2 + pT_i^2 as replacements for the normal squared transverse masses of the two outgoing quarks.
option 1 : the squared mass m_V^2 of the exchanged vector boson.
option 2 : the geometric mean of the two propagator virtuality estimates.
option 3 : the geometric mean of the three relevant squared transverse masses.
option 4 : the arithmetic mean of the three relevant squared transverse masses.
option 5 : squared invariant mass of the system, i.e. sHat.
option 6 : fix scale set in SigmaProcess:factorFixScale below.

parm  SigmaProcess:factorMultFac   (default = 1.; minimum = 0.1; maximum = 10.)
The Q^2 factorization scale for 2 → 1, 2 → 2 and 2 → 3 processes is multiplied by this factor relative to the scale described above (except for the options with a fix scale). Should be use sparingly for 2 → 1 processes.

parm  SigmaProcess:factorFixScale   (default = 10000.; minimum = 1.)
A fix Q^2 value used as factorization scale for 2 → 1, 2 → 2 and 2 → 3 processes in some of the options above.