Standard-Model Parameters

The strong coupling

The AlphaStrong class is used to provide a first- or second-order running alpha_strong (or, trivially, a zeroth-order fixed one). Formulae are the standard ones found in [Yao06]. The second-order expression used, eq. (9.5), may be somewhat different in other approaches (with differences formally of higher order), so do not necessarily expect perfect agreement, especially not at small Q^2 scales. The starting alpha_strong value is defined at the M_Z mass scale. The Lambda values are matched at the b and c flavour thresholds, such that alpha_strong is continuous. For second-order matching an approximate iterative method is used.

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

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

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.

The electromagnetic coupling

The AlphaEM class is used to generate a running alpha_em. The input StandardModel:alphaEMmZ value at the M_Z mass is matched to a low-energy behaviour with running starting at the electron mass threshold. The matching is done by fitting an effective running coefficient in the region betweeen the light-quark treshold and the charm/tau threshold. This procedure is approximate, but good enough for our purposes.

Since we allow alpha_em to vary separately for hard processes, timelike showers, spacelike showers and multiple 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.

parm  StandardModel:alphaEM0   (default = 0.00729735; minimum = 0.0072973; maximum = 0.0072974)
The alpha_em value at vanishing momentum transfer (and also below m_e).

parm  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

There are two degrees of freedom that can be set, related to the electroweak mixing angle:

parm  StandardModel:sin2thetaW   (default = 0.2312; minimum = 0.225; maximum = 0.240)
The sine-squared of the weak mixing angle, as used in all Z^0 and W^+- masses and couplings, except for the vector couplings of fermions to the Z^0, see below. Default is the MSbar value from [Yao06].

parm  StandardModel:sin2thetaWbar   (default = 0.2315; minimum = 0.225; maximum = 0.240)
The sine-squared of the weak mixing angle, as used to derive the vector couplings of fermions to the Z^0, in the relation v_f = a_f - 4 e_f sin^2(theta_W)bar. Default is the effective-angle value from [Yao06].

The quark weak-mixing matrix

The absolute values of the Cabibbo-Kobayashi-Maskawa matrix elements are set by the following nine real values taken from [Yao06] - currently the CP-violating phase is not taken into account in this parametrization. It is up to the user to pick a consistent unitary set of new values whenever changes are made.

parm  StandardModel:Vud   (default = 0.97383; minimum = 0.973; maximum = 0.975)
The V_ud CKM matrix element.

parm  StandardModel:Vus   (default = 0.2272; minimum = 0.224; maximum = 0.230)
The V_us CKM matrix element.

parm  StandardModel:Vub   (default = 0.00396; minimum = 0.0037; maximum = 0.0042)
The V_ub CKM matrix element.

parm  StandardModel:Vcd   (default = 0.2271; minimum = 0.224; maximum = 0.230)
The V_cd CKM matrix element.

parm  StandardModel:Vcs   (default = 0.97296; minimum = 0.972; maximum = 0.974)
The V_cs CKM matrix element.

parm  StandardModel:Vcb   (default = 0.04221; minimum = 0.0418; maximum = 0.0426)
The V_cb CKM matrix element.

parm  StandardModel:Vtd   (default = 0.00814; minimum = 0.006; maximum = 0.010)
The V_td CKM matrix element.

parm  StandardModel:Vts   (default = 0.04161; minimum = 0.039; maximum = 0.043)
The V_ts CKM matrix element.

parm  StandardModel:Vtb   (default = 0.9991; minimum = 0.99907; maximum = 0.9992)
The V_tb CKM matrix element.

The CoupSM class

The Pythia class contains a public instance coupSM of the CoupSM class. This class contains one instance each of the AlphaStrong and AlphaEM classes, and additionally stores the weak couplings and the quark mixing matrix mentioned above. This class is used especially in the calculation of cross sections and resonance widths, but could also be used elsewhere. Specifically, as already mentioned, there are separate AlphaStrong and AlphaEM instances for timelike and spacelike showers and for multiple interactions, while weak couplings and the quark mixing matrix are only stored here. With the exception of the first two methods below, which are for internal use, the subsequent ones could also be used externally.

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::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.