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<chapter name="Standard-Model Parameters">
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<h2>Standard-Model Parameters</h2>
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<h3>The strong coupling</h3>
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The <code>AlphaStrong</code> class is used to provide a first- or
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second-order running <ei>alpha_strong</ei> (or, trivially, a
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zeroth-order fixed one). Formulae are the standard ones found in
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<ref>Yao06</ref>. The second-order expression used, eq. (9.5),
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may be somewhat different in other approaches (with differences
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formally of higher order), so do not necessarily expect perfect
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agreement, especially not at small <ei>Q^2</ei> scales. The starting
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<ei>alpha_strong</ei> value is defined at the <ei>M_Z</ei> mass scale.
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The <ei>Lambda</ei> values are matched at the <ei>b</ei> and <ei>c</ei>
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flavour thresholds, such that <ei>alpha_strong</ei> is continuous.
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For second-order matching an approximate iterative method is used.
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Since we allow <ei>alpha_strong</ei> to vary separately for
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hard processes, timelike showers, spacelike showers and multiparton
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interactions, the relevant values can be set in each of these classes.
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The default behaviour is everywhere first-order running.
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The <ei>alpha_strong</ei> calculation is initialized by
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<code>init( value, order)</code>, where <code>value</code>
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is the <ei>alpha_strong</ei> value at <ei>M_Z</ei> and <code>order</code>
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is the order of the running, 0, 1 or 2. Thereafter the value can be
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calculated by <code>alphaS(scale2)</code>, where
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<code>scale2</code> is the <ei>Q^2</ei> scale in GeV^2.
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For applications inside shower programs, a second-order <code>alpha_s</code>
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value can be obtained as the product of the two functions
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<code>alphaS1Ord(scale2)</code> and <code>alphaS2OrdCorr(scale2)</code>,
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where the first gives a simple first-order running (but with the
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second-order <ei>Lambda</ei>) and the second the correction factor,
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below unity, for the second-order terms. This allows a compact handling
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of evolution equations.
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<h3>The electromagnetic coupling</h3>
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The <code>AlphaEM</code> class is used to generate a running
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<ei>alpha_em</ei>. The input <code>StandardModel:alphaEMmZ</code>
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value at the <ei>M_Z</ei> mass is matched to a low-energy behaviour
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with running starting at the electron mass threshold. The matching
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is done by fitting an effective running coefficient in the region
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betweeen the light-quark treshold and the charm/tau threshold. This
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procedure is approximate, but good enough for our purposes.
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Since we allow <ei>alpha_em</ei> to vary separately for
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hard processes, timelike showers, spacelike showers and multiparton
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interactions, the choice between using a fixed or a running
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<ei>alpha_em</ei> can be made in each of these classes.
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The default behaviour is everywhere first-order running.
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The actual values assumed at zero momentum transfer and
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at <ei>M_Z</ei> are only set here, however.
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<parm name="StandardModel:alphaEM0" default="0.00729735"
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min="0.0072973" max="0.0072974">
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The <ei>alpha_em</ei> value at vanishing momentum transfer
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(and also below <ei>m_e</ei>).
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<parm name="StandardModel:alphaEMmZ" default="0.00781751"
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min="0.00780" max="0.00783">
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The <ei>alpha_em</ei> value at the <ei>M_Z</ei> mass scale.
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Default is taken from <ref>Yao06</ref>.
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The <ei>alpha_em</ei> calculation is initialized by
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<code>init(order)</code>, where <code>order</code> is the order of
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the running, 0 or 1, with -1 a special option to use the fix value
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provided at <ei>M_Z</ei>. Thereafter the value can be
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calculated by <code>alphaEM(scale2)</code>, where
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<code>scale2</code> is the <ei>Q^2</ei> scale in GeV^2.
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<h3>The electroweak couplings</h3>
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There are two degrees of freedom that can be set, related to the
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electroweak mixing angle:
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<parm name="StandardModel:sin2thetaW" default="0.2312"
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min="0.225" max="0.240">
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The sine-squared of the weak mixing angle, as used in all <ei>Z^0</ei>
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and <ei>W^+-</ei> masses and couplings, except for the vector couplings
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of fermions to the <ei>Z^0</ei>, see below. Default is the MSbar value
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from <ref>Yao06</ref>.
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<parm name="StandardModel:sin2thetaWbar" default="0.2315"
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min="0.225" max="0.240">
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The sine-squared of the weak mixing angle, as used to derive the vector
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couplings of fermions to the <ei>Z^0</ei>, in the relation
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<ei>v_f = a_f - 4 e_f sin^2(theta_W)bar</ei>. Default is the
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effective-angle value from <ref>Yao06</ref>.
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The Fermi constant is not much used in the currently coded matrix elements,
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since it is redundant, but it is available:
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<parm name="StandardModel:GF" default="1.16637e-5"
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min="1.0e-5" max="1.3e-5">
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The Fermi coupling constant, in units of GeV<ei>^-2</ei>.
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<h3>The quark weak-mixing matrix</h3>
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The absolute values of the Cabibbo-Kobayashi-Maskawa matrix elements are
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set by the following nine real values taken from <ref>Yao06</ref> -
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currently the CP-violating phase is not taken into account in this
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parametrization. It is up to the user to pick a consistent unitary
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set of new values whenever changes are made.
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<parm name="StandardModel:Vud" default="0.97383" min="0.973" max="0.975">
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The <ei>V_ud</ei> CKM matrix element.
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<parm name="StandardModel:Vus" default="0.2272" min="0.224" max="0.230">
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The <ei>V_us</ei> CKM matrix element.
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<parm name="StandardModel:Vub" default="0.00396" min="0.0037" max="0.0042">
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The <ei>V_ub</ei> CKM matrix element.
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<parm name="StandardModel:Vcd" default="0.2271" min="0.224" max="0.230">
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The <ei>V_cd</ei> CKM matrix element.
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<parm name="StandardModel:Vcs" default="0.97296" min="0.972" max="0.974">
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The <ei>V_cs</ei> CKM matrix element.
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<parm name="StandardModel:Vcb" default="0.04221" min="0.0418" max="0.0426">
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The <ei>V_cb</ei> CKM matrix element.
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<parm name="StandardModel:Vtd" default="0.00814" min="0.006" max="0.010">
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The <ei>V_td</ei> CKM matrix element.
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<parm name="StandardModel:Vts" default="0.04161" min="0.039" max="0.043">
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The <ei>V_ts</ei> CKM matrix element.
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<parm name="StandardModel:Vtb" default="0.9991" min="0.99907" max="0.9992">
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The <ei>V_tb</ei> CKM matrix element.
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<h3>The CoupSM class</h3>
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The <code><aloc href="ProgramFlow">Pythia</aloc></code> class contains a
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public instance <code>coupSM</code> of the <code>CoupSM</code> class.
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This class contains one instance each of the <code>AlphaStrong</code>
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and <code>AlphaEM</code> classes, and additionally stores the weak couplings
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and the quark mixing matrix mentioned above. This class is used especially
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in the calculation of cross sections and resonance widths, but could also
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be used elsewhere. Specifically, as already mentioned, there are separate
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<code>AlphaStrong</code> and <code>AlphaEM</code> instances for timelike
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and spacelike showers and for multiparton interactions, while weak couplings
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and the quark mixing matrix are only stored here. With the exception of the
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first two methods below, which are for internal use, the subsequent ones
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could also be used externally.
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<method name="CoupSM::CoupSM()">
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the constructor does nothing. Internal.
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<method name="void CoupSM::init(Settings& settings, Rndm* rndmPtr)">
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this is where the <code>AlphaStrong</code> and <code>AlphaEM</code>
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instances are initialized, and weak couplings and the quark mixing matrix
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are read in and set. This is based on the values stored on this page and
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among the <aloc href="CouplingsAndScales">Couplings and Scales</aloc>.
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<method name="double CoupSM::alphaS(double scale2)">
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the <ei>alpha_strong</ei> value at the quadratic scale <code>scale2</code>.
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<method name="double CoupSM::alphaS1Ord(double scale2)">
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a first-order overestimate of the full second-order <ei>alpha_strong</ei>
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value at the quadratic scale <code>scale2</code>.
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<method name="double CoupSM::alphaS2OrdCorr(double scale2)">
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a multiplicative correction factor, below unity, that brings the
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first-order overestimate above into agreement with the full second-order
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<ei>alpha_strong</ei> value at the quadratic scale <code>scale2</code>.
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<method name="double CoupSM::Lambda3()">
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<methodmore name="double CoupSM::Lambda4()">
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<methodmore name="double CoupSM::Lambda5()">
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the three-, four-, and five-flavour <ei>Lambda</ei> scale.
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<method name="double CoupSM::alphaEM(double scale2)">
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the <ei>alpha_em</ei> value at the quadratic scale <code>scale2</code>.
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<method name="double CoupSM::sin2thetaW()">
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<methodmore name="double CoupSM::cos2thetaW()">
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the sine-squared and cosine-squared of the weak mixing angle, as used in
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the gauge-boson sector.
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<method name="double CoupSM::sin2thetaWbar()">
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the sine-squared of the weak mixing angle, as used to derive the vector
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couplings of fermions to the <ei>Z^0</ei>.
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<method name="double CoupSM::GF()">
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the Fermi constant of weak decays, in GeV<ei>^-2</ei>.
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<method name="double CoupSM::ef(int idAbs)">
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the electrical charge of a fermion, by the absolute sign of the PDF code,
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i.e. <code>idAbs</code> must be in the range between 1 and 18.
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<method name="double CoupSM::vf(int idAbs)">
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<methodmore name="double CoupSM::af(int idAbs)">
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the vector and axial charges of a fermion, by the absolute sign of the PDF
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code (<ei>a_f = +-1, v_f = a_f - 4. * sin2thetaWbar * e_f</ei>).
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<method name="double CoupSM::t3f(int idAbs)">
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<methodmore name="double CoupSM::lf(int idAbs)">
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<methodmore name="double CoupSM::rf(int idAbs)">
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the weak isospin, left- and righthanded charges of a fermion, by the
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absolute sign of the PDF code (<ei>t^3_f = a_f/2, l_f = (v_f + a_f)/2,
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r_f = (v_f - a_f)/2</ei>; you may find other conventions in the literature
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that differ by a factor of 2).
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<method name="double CoupSM::ef2(int idAbs)">
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<methodmore name="double CoupSM::vf2(int idAbs)">
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<methodmore name="double CoupSM::af2(int idAbs)">
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<methodmore name="double CoupSM::efvf(int idAbs)">
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<methodmore name="double CoupSM::vf2af2(int idAbs)">
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common quadratic combinations of the above couplings:
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<ei>e_f^2, v_f^2, a_f^2, e_f * v_f, v_f^2 + a_f^2</ei>.
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<method name="double CoupSM::VCKMgen(int genU, int genD)">
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<methodmore name="double CoupSM::V2CKMgen(int genU, int genD)">
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the CKM mixing element,or the square of it, for
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up-type generation index <code>genU</code>
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(<ei>1 = u, 2 = c, 3 = t, 4 = t'</ei>) and
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down-type generation index <code>genD</code>
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(<ei>1 = d, 2 = s, 3 = b, 4 = b'</ei>).
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<method name="double CoupSM::VCKMid(int id1, int id2)">
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<methodmore name="double CoupSM::V2CKMid(int id1, int id2)">
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the CKM mixing element,or the square of it, for
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flavours <code>id1</code> and <code>id2</code>, both in the
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range from <ei>-18</ei> to <ei>+18</ei>. The sign is here not
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checked (so it can be used both for <ei>u + dbar -> W+</ei>
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and <ei>u -> d + W+</ei>, say), but impossible flavour combinations
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evaluate to zero. The neutrino sector is numbered by flavor
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eigenstates, so there is no mixing in the lepton-neutrino system.
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<method name="double CoupSM::V2CKMsum(int id)">
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the sum of squared CKM mixing element that a given flavour can couple to,
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excluding the top quark and fourth generation. Is close to unity
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for the first two generations. Returns unity for the lepton-neutrino
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<method name="int CoupSM::V2CKMpick(int id)">
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picks a random CKM partner quark or lepton (with the same sign as
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<code>id</code>) according to the respective squared elements, again
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excluding the top quark and fourth generation from the list of
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possibilities. Unambiguous choice for the lepton-neutrino sector.
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<!-- Copyright (C) 2012 Torbjorn Sjostrand -->
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