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.