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:alphaEMmZ
at 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.