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 c, b
and t flavour thresholds,
such that alpha_strong is continuous.
For second-order matching an approximate iterative method is used.
For backwards compatibility,
the following global switch determines whether 5- or 6-flavour running
will be used above the t threshold:
mode StandardModel:alphaSnfmax
(default = 6; minimum = 5; maximum = 6)
option 5 : Use 5-flavour running for all scales above the
b flavour threshold (old default).
option 6 : Use 6-flavour running above the t threshold
(new default).
Since we allow alpha_strong to vary separately for
hard processes, timelike showers, spacelike showers and multiparton
interactions, all other relevant values are set in each of these classes.
The default behaviour is everywhere first-order running.
The alpha_strong calculation is initialized by
init( value, order, nfmax), where value
is the alpha_strong value at M_Z, order
is the order of the running, 0, 1 or 2, and nfmax
is the highest number of flavours to include in the running. Thereafter
the value can be calculated by alphaS(scale2), where
scale2 is the Q^2 scale in GeV^2.
By default the charm, bottom and top threshold-matching mass values
are chosen to be 1.5, 4.8 and 171 GeV, respectively. The
setThresholds(double mc, double mb, double mt)
method can be invoked to select other values. To take effect, this
must be done before the AlphaStrong::init() method is called,
since this is where the flavour-dependent Lambda_i values are
calculated and stored. If in doubt, better call it once again.
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.
Resummation arguments [Cat91] show that a set of
universal QCD corrections can be absorbed in coherent parton showers by
applying the so-called CMW rescaling of the MSbar value of
Lambda_QCD. This can be accomplished via a fourth (optional)
boolean argument to init( value, order, nfmax, useCMW),
with default value useCMW = false. When set to
true, the translation amounts to an N_F-dependent
rescaling of Lambda_QCD, relative to its MSbar value, by
a factor 1.661 for NF=3, 1.618 for NF=4, 1.569 for NF=5,
and 1.513 for NF=6. When using this option,
be aware that the original CMW arguments were derived using two-loop running
and that the CMW rescaling may need be taken into account in the context of
matrix-element matching. Note also that this option has only been made
available for timelike and spacelike showers, not for hard processes.
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
between the light-quark threshold 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 multiparton
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 Fermi constant is not much used in the currently coded matrix elements,
since it is redundant, but it is available:
parm StandardModel:GF
(default = 1.16637e-5; minimum = 1.0e-5; maximum = 1.3e-5)
The Fermi coupling constant, in units of GeV^-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.
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 multiparton 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::GF()
the Fermi constant of weak decays, in GeV^-2.
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.