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 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 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].

These and various couplings can be read out from the static CoupEW class:
CoupEW::sin2thetaW() gives the weak mixing angle set above.
CoupEW::cos2thetaW() gives 1 minus it.
CoupEW::sin2thetaWbar() gives the weak mixing angle as used in fermion couplings.
CoupEW::ef(idAbs) gives the electrical charge. Note that this and subsequent routines should be called with a positive idAbs.
CoupEW::vf(idAbs) gives the vector coupling to Z^0.
CoupEW::af(idAbs) gives the axial vector coupling.
CoupEW::t3f(idAbs) gives the weak isospin of lefthanded quarks, i.e. a_f/2.
CoupEW::lf(idAbs) gives the lefthanded coupling, i.e. (v_f + a_f/2)/2 (other definitions may differ by a factor of 2).
CoupEW::rf(idAbs) gives the righthanded coupling, i.e. (v_f - a_f/2)/2 (with comment as above).
CoupEW::ef2(idAbs) gives e_f^2.
CoupEW::vf2(idAbs) gives v_f^2.
CoupEW::af2(idAbs) gives a_f^2.

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.

These couplings can be read back out in a few alternative forms:
VCKM::Vgen(genU, genD) gives the CKM mixing element for up-type generation index genU (1, 2 or 3) and down-type generation index genD.
VCKM::V2gen(genU, genD) gives the square of the above.
VCKM::Vid(id1, id2) gives the CKM mixing element between two quark flavours id1 and id2. The sign of the flavours is irrelevant, since the process may be either of the type q qbar' -> W or q g -> W q'. Flavour combinations with no CKM mixing (e.g. u u) are given a vanishing value.
VCKM::V2id(id1, id2) gives the square of the above.
VCKM::V2sum(id) gives the sum of squares that a given flavour can couple to, excluding the top quark. Is close to unity for the first two generations.
VCKM::V2pick(id) picks a CKM partner quark (with the same sign as id) according to the respective squared elements, again excluding the top quark from the list of possibilities.