Standard-Model Parameters
 
  - The strong coupling
- The electromagnetic coupling
- The electroweak couplings
- The quark weak-mixing matrix
- The CoupSM class
- Running coupling in an SU(N) group
The strong coupling
 
 
The AlphaStrong class is used to provide a first-, second- 
or third-order running alpha_strong (or, trivially, a 
zeroth-order fixed one). Formulae are the standard ones found in the 
2006 RPP [Yao06]. The higher-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- or third-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, 2 or 3, 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- or third-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- or third-order Lambda) and the second the correction factor, 
below unity, for the second- or third-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. 
   
 
 
Running coupling in an SU(N) group
 
 
Included in the code is also a class for the running of the coupling 
in an arbitrary SU(N) gauge group. This is not part of the 
Standard Model, but is closely related to the running of 
alpha_strong, so is therefore documented and encoded in its 
proximity. Currently it is used in the Hidden Valley scenario. 
 
class   AlphaSUN   
Running couplings in an SU(N) gauge group, to first, second or 
third order. See definitions in [Rit97], [Yao06] and 
[Pro07]. There are assumed to be no flavour thresholds, i.e. 
the number of flavours is a fixed number over the considered range of 
scales. 
   
 
 AlphaSUN::AlphaSUN()   
the constructor does nothing. You need to use one of the following two 
methods to initialize the generation. 
   
 
 void AlphaSUN::initAlpha( int nC, int nF, int order = 1, double alpha = 0.12, double scale = 91.188)   
Set up the required framework for running, given the number of colours, 
number of flavours, order of running (1, 2 or 3) and the coupling 
strength at a reference scale (by default the Z^0 mass). 
   
 
 void AlphaSUN::initLambda( int nC, int nF, int order = 1, double Lambda = 0.2)   
Set up the required framework for running, given the number of colours, 
number of flavours, order of running (1, 2 or 3) and the Lambda 
parameter to the given order. 
   
 
 double AlphaSUN::alpha(double scale2)   
return the coupling value at the input scale-squared. 
   
 
 double AlphaSUN::alpha1Ord(double scale2)   
   
 double AlphaSUN::alpha2OrdCorr(double scale2)   
the first method gives a simple first-order running coupling value 
(but with the second- or third-order Lambda) and the second method 
gives the correction factor, below unity, for the second- or 
third-order terms. This allows a compact handling of shower evolution 
equations. 
   
 
 double AlphaSUN::Lambda()   
return the Lambda determined by a initAlpha 
call. If instead initLambda has been used, the input 
value there will be echoed.