Standard-Model Parameters

The strong coupling

For backwards compatibility, the following global switch determines whether 5- or 6-flavour running will be used above the t threshold:

StandardModel:alphaSnfmax

(default = 6; minimum = 5; maximum = 6)

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

Since we allow alpha_strong to vary separately for hard processes, timelike showers, spacelike showers and multiparton interactions, all other relevant values are set in each of these classes. The default behaviour is everywhere first-order running.

The alpha_strong calculation is initialized by init( value, order, nfmax), where value is the alpha_strong value at M_Z, order is the order of the running, 0, 1 or 2, and nfmax is the highest number of flavours to include in the running. Thereafter the value can be calculated by alphaS(scale2), where scale2 is the Q^2 scale in GeV^2.

By default the charm, bottom and top threshold-matching mass values are chosen to be 1.5, 4.8 and 171 GeV, respectively. The setThresholds(double mc, double mb, double mt) method can be invoked to select other values. To take effect, this must be done before the AlphaStrong::init() method is called, since this is where the flavour-dependent Lambda_i values are calculated and stored. If in doubt, better call it once again.

For applications inside shower programs, a second-order alpha_s value can be obtained as the product of the two functions alphaS1Ord(scale2) and alphaS2OrdCorr(scale2), where the first gives a simple first-order running (but with the second-order Lambda) and the second the correction factor, below unity, for the second-order terms. This allows a compact handling of evolution equations.

Resummation arguments [Cat91] show 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. This can be accomplished via a fourth (optional) boolean argument to init( value, order, nfmax, useCMW), with default value useCMW = false. When set to true, the translation amounts to an N_F-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. When using this option, be aware that the original CMW arguments were derived using two-loop running and 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.

The electromagnetic coupling

Since we allow alpha_em to vary separately for hard processes, timelike showers, spacelike showers and multiparton interactions, the choice between using a fixed or a running alpha_em can be made in each of these classes. The default behaviour is everywhere first-order running. The actual values assumed at zero momentum transfer and at M_Z are only set here, however.

StandardModel:alphaEM0

(default = 0.00729735; minimum = 0.0072973; maximum = 0.0072974)

The alpha_em value at vanishing momentum transfer (and also below m_e).

StandardModel:alphaEMmZ

(default = 0.00781751; minimum = 0.00780; maximum = 0.00783)

The alpha_em value at the M_Z mass scale. Default is taken from [Yao06].

The alpha_em calculation is initialized by init(order), where order is the order of the running, 0 or 1, with -1 a special option to use the fix value provided at M_Z. Thereafter the value can be calculated by alphaEM(scale2), where scale2 is the Q^2 scale in GeV^2.

The electroweak couplings

The Fermi constant is not much used in the currently coded matrix elements, since it is redundant, but it is available:

StandardModel:GF

(default = 1.16637e-5; minimum = 1.0e-5; maximum = 1.3e-5)

The Fermi coupling constant, in units of GeV^-2.

The quark weak-mixing matrix

The CoupSM class

CoupSM::CoupSM()  
the constructor does nothing. Internal.

void CoupSM::init(Settings& settings, Rndm* rndmPtr)  
this is where the AlphaStrong and AlphaEM instances are initialized, and weak couplings and the quark mixing matrix are read in and set. This is based on the values stored on this page and among the ";?>Couplings and Scales. Internal.

double CoupSM::alphaS(double scale2)  
the alpha_strong value at the quadratic scale scale2.

double CoupSM::alphaS1Ord(double scale2)  
a first-order overestimate of the full second-order alpha_strong value at the quadratic scale scale2.

double CoupSM::alphaS2OrdCorr(double scale2)  
a multiplicative correction factor, below unity, that brings the first-order overestimate above into agreement with the full second-order alpha_strong value at the quadratic scale scale2.

double CoupSM::Lambda3()  
double CoupSM::Lambda4()  
double CoupSM::Lambda5()  
the three-, four-, and five-flavour Lambda scale.

double CoupSM::alphaEM(double scale2)  
the alpha_em value at the quadratic scale scale2.

double CoupSM::sin2thetaW()  
double CoupSM::cos2thetaW()  
the sine-squared and cosine-squared of the weak mixing angle, as used in the gauge-boson sector.

double CoupSM::sin2thetaWbar()  
the sine-squared of the weak mixing angle, as used to derive the vector couplings of fermions to the Z^0.

double CoupSM::GF()  
the Fermi constant of weak decays, in GeV^-2.

double CoupSM::ef(int idAbs)  
the electrical charge of a fermion, by the absolute sign of the PDF code, i.e. idAbs must be in the range between 1 and 18.

double CoupSM::vf(int idAbs)  
double CoupSM::af(int idAbs)  
the vector and axial charges of a fermion, by the absolute sign of the PDF code (a_f = +-1, v_f = a_f - 4. * sin2thetaWbar * e_f).

double CoupSM::t3f(int idAbs)  
double CoupSM::lf(int idAbs)  
double CoupSM::rf(int idAbs)  
the weak isospin, left- and righthanded charges of a fermion, by the absolute sign of the PDF code (t^3_f = a_f/2, l_f = (v_f + a_f)/2, r_f = (v_f - a_f)/2; you may find other conventions in the literature that differ by a factor of 2).

double CoupSM::ef2(int idAbs)  
double CoupSM::vf2(int idAbs)  
double CoupSM::af2(int idAbs)  
double CoupSM::efvf(int idAbs)  
double CoupSM::vf2af2(int idAbs)  
common quadratic combinations of the above couplings: e_f^2, v_f^2, a_f^2, e_f * v_f, v_f^2 + a_f^2.

double CoupSM::VCKMgen(int genU, int genD)  
double CoupSM::V2CKMgen(int genU, int genD)  
the CKM mixing element,or the square of it, for up-type generation index genU (1 = u, 2 = c, 3 = t, 4 = t') and down-type generation index genD (1 = d, 2 = s, 3 = b, 4 = b').

double CoupSM::VCKMid(int id1, int id2)  
double CoupSM::V2CKMid(int id1, int id2)  
the CKM mixing element,or the square of it, for flavours id1 and id2, both in the range from -18 to +18. The sign is here not checked (so it can be used both for u + dbar → W+ and u → d + W+, say), but impossible flavour combinations evaluate to zero. The neutrino sector is numbered by flavor eigenstates, so there is no mixing in the lepton-neutrino system.

double CoupSM::V2CKMsum(int id)  
the sum of squared CKM mixing element that a given flavour can couple to, excluding the top quark and fourth generation. Is close to unity for the first two generations. Returns unity for the lepton-neutrino sector.

int CoupSM::V2CKMpick(int id)  
picks a random CKM partner quark or lepton (with the same sign as id) according to the respective squared elements, again excluding the top quark and fourth generation from the list of possibilities. Unambiguous choice for the lepton-neutrino sector. "?>