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 [Eid04]. 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, using an approximate iterative method for second-order matching. Since we allow alpha_strong to vary separately for hard processes, timelike showers, spacelike showers and multiple interactions, the relevant values are to be set in each of these classes.

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(scale), where scale 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(scale) and alphaS2OrdCorr(scale), 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

Currently alpha_electromagnetic is treated in two rather different ways.

For parton-shower emission of photons, the StandardModel:alphaEMfix fix value is used, as relevant for low scales.

parm name="StandardModel:alphaEMfix" default="0.00729735" min="0.0072973" max="0.0072974"
Fixed alpha_em value used in QED vertices.

For hard processes (also the ones generated in multiple interactions) the AlphaEM class is used, to generate a running alpha_em. Since the main application here is for hard scales, above m_b, no attempt is made to provide an accurate description of the running at low scales. Instead, by default, the value StandardModel:alphaEMmZat the M_Z mass scale is input [Eid04]. It is then run from there to different scales, using the first-order expression for three lepton and five quark flavours. It is frozen at the low-energy value StandardModel:alphaEM0, should the running value exceed this. (With the default values, this happens for Q < 0.1 GeV.)

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

parm name="StandardModel:alphaEM0" default="0.00729735" min="0.0072973" max="0.0072974"
The alpha_em value at vanishing momentum transfer.

parm name="StandardModel:alphaEMmZ" default="0.00781751" min="0.00780" max="0.00783"
The alpha_em value at the m_Z mass scale.

The electroweak couplings

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

parm name="StandardModel:sin2thetaW" default="0.232" min="0.225" max="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.

parm name="StandardModel:sin2thetaWbar" default="0.232" min="0.225" max="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.

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 - 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 name="StandardModel:Vud" default="0.9745" min="0.973" max="0.976"
The V_ud CKM matrix element.

parm name="StandardModel:Vus" default="0.224" min="0.218" max="0.230"
The V_us CKM matrix element.

parm name="StandardModel:Vub" default="0.0037" min="0.002" max="0.006"
The V_ub CKM matrix element.

parm name="StandardModel:Vcd" default="0.224" min="0.218" max="0.230"
The V_cd CKM matrix element.

parm name="StandardModel:Vcs" default="0.9737" min="0.972" max="0.975"
The V_cs CKM matrix element.

parm name="StandardModel:Vcb" default="0.0415" min="0.033" max="0.040"
The V_cb CKM matrix element.

parm name="StandardModel:Vtd" default="0.0094" min="0.004" max="0.020"
The V_td CKM matrix element.

parm name="StandardModel:Vts" default="0.040" min="0.030" max="0.050"
The V_ts CKM matrix element.

parm name="StandardModel:Vtb" default="0.9991" min="0.998" max="0.9995"
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.