1 <chapter name="Standard-Model Parameters">
3 <h2>Standard-Model Parameters</h2>
5 <h3>The strong coupling</h3>
7 The <code>AlphaStrong</code> class is used to provide a first- or
8 second-order running <ei>alpha_strong</ei> (or, trivially, a
9 zeroth-order fixed one). Formulae are the standard ones found in
10 <ref>Yao06</ref>. The second-order expression used, eq. (9.5),
11 may be somewhat different in other approaches (with differences
12 formally of higher order), so do not necessarily expect perfect
13 agreement, especially not at small <ei>Q^2</ei> scales. The starting
14 <ei>alpha_strong</ei> value is defined at the <ei>M_Z</ei> mass scale.
15 The <ei>Lambda</ei> values are matched at the <ei>b</ei> and <ei>c</ei>
16 flavour thresholds, such that <ei>alpha_strong</ei> is continuous.
17 For second-order matching an approximate iterative method is used.
20 Since we allow <ei>alpha_strong</ei> to vary separately for
21 hard processes, timelike showers, spacelike showers and multiparton
22 interactions, the relevant values can be set in each of these classes.
23 The default behaviour is everywhere first-order running.
26 The <ei>alpha_strong</ei> calculation is initialized by
27 <code>init( value, order)</code>, where <code>value</code>
28 is the <ei>alpha_strong</ei> value at <ei>M_Z</ei> and <code>order</code>
29 is the order of the running, 0, 1 or 2. Thereafter the value can be
30 calculated by <code>alphaS(scale2)</code>, where
31 <code>scale2</code> is the <ei>Q^2</ei> scale in GeV^2.
34 For applications inside shower programs, a second-order <code>alpha_s</code>
35 value can be obtained as the product of the two functions
36 <code>alphaS1Ord(scale2)</code> and <code>alphaS2OrdCorr(scale2)</code>,
37 where the first gives a simple first-order running (but with the
38 second-order <ei>Lambda</ei>) and the second the correction factor,
39 below unity, for the second-order terms. This allows a compact handling
40 of evolution equations.
42 <h3>The electromagnetic coupling</h3>
44 The <code>AlphaEM</code> class is used to generate a running
45 <ei>alpha_em</ei>. The input <code>StandardModel:alphaEMmZ</code>
46 value at the <ei>M_Z</ei> mass is matched to a low-energy behaviour
47 with running starting at the electron mass threshold. The matching
48 is done by fitting an effective running coefficient in the region
49 betweeen the light-quark treshold and the charm/tau threshold. This
50 procedure is approximate, but good enough for our purposes.
53 Since we allow <ei>alpha_em</ei> to vary separately for
54 hard processes, timelike showers, spacelike showers and multiparton
55 interactions, the choice between using a fixed or a running
56 <ei>alpha_em</ei> can be made in each of these classes.
57 The default behaviour is everywhere first-order running.
58 The actual values assumed at zero momentum transfer and
59 at <ei>M_Z</ei> are only set here, however.
61 <parm name="StandardModel:alphaEM0" default="0.00729735"
62 min="0.0072973" max="0.0072974">
63 The <ei>alpha_em</ei> value at vanishing momentum transfer
64 (and also below <ei>m_e</ei>).
67 <parm name="StandardModel:alphaEMmZ" default="0.00781751"
68 min="0.00780" max="0.00783">
69 The <ei>alpha_em</ei> value at the <ei>M_Z</ei> mass scale.
70 Default is taken from <ref>Yao06</ref>.
74 The <ei>alpha_em</ei> calculation is initialized by
75 <code>init(order)</code>, where <code>order</code> is the order of
76 the running, 0 or 1, with -1 a special option to use the fix value
77 provided at <ei>M_Z</ei>. Thereafter the value can be
78 calculated by <code>alphaEM(scale2)</code>, where
79 <code>scale2</code> is the <ei>Q^2</ei> scale in GeV^2.
81 <h3>The electroweak couplings</h3>
83 There are two degrees of freedom that can be set, related to the
84 electroweak mixing angle:
86 <parm name="StandardModel:sin2thetaW" default="0.2312"
87 min="0.225" max="0.240">
88 The sine-squared of the weak mixing angle, as used in all <ei>Z^0</ei>
89 and <ei>W^+-</ei> masses and couplings, except for the vector couplings
90 of fermions to the <ei>Z^0</ei>, see below. Default is the MSbar value
91 from <ref>Yao06</ref>.
94 <parm name="StandardModel:sin2thetaWbar" default="0.2315"
95 min="0.225" max="0.240">
96 The sine-squared of the weak mixing angle, as used to derive the vector
97 couplings of fermions to the <ei>Z^0</ei>, in the relation
98 <ei>v_f = a_f - 4 e_f sin^2(theta_W)bar</ei>. Default is the
99 effective-angle value from <ref>Yao06</ref>.
103 The Fermi constant is not much used in the currently coded matrix elements,
104 since it is redundant, but it is available:
106 <parm name="StandardModel:GF" default="1.16637e-5"
107 min="1.0e-5" max="1.3e-5">
108 The Fermi coupling constant, in units of GeV<ei>^-2</ei>.
111 <h3>The quark weak-mixing matrix</h3>
113 The absolute values of the Cabibbo-Kobayashi-Maskawa matrix elements are
114 set by the following nine real values taken from <ref>Yao06</ref> -
115 currently the CP-violating phase is not taken into account in this
116 parametrization. It is up to the user to pick a consistent unitary
117 set of new values whenever changes are made.
119 <parm name="StandardModel:Vud" default="0.97383" min="0.973" max="0.975">
120 The <ei>V_ud</ei> CKM matrix element.
123 <parm name="StandardModel:Vus" default="0.2272" min="0.224" max="0.230">
124 The <ei>V_us</ei> CKM matrix element.
127 <parm name="StandardModel:Vub" default="0.00396" min="0.0037" max="0.0042">
128 The <ei>V_ub</ei> CKM matrix element.
131 <parm name="StandardModel:Vcd" default="0.2271" min="0.224" max="0.230">
132 The <ei>V_cd</ei> CKM matrix element.
135 <parm name="StandardModel:Vcs" default="0.97296" min="0.972" max="0.974">
136 The <ei>V_cs</ei> CKM matrix element.
139 <parm name="StandardModel:Vcb" default="0.04221" min="0.0418" max="0.0426">
140 The <ei>V_cb</ei> CKM matrix element.
143 <parm name="StandardModel:Vtd" default="0.00814" min="0.006" max="0.010">
144 The <ei>V_td</ei> CKM matrix element.
147 <parm name="StandardModel:Vts" default="0.04161" min="0.039" max="0.043">
148 The <ei>V_ts</ei> CKM matrix element.
151 <parm name="StandardModel:Vtb" default="0.9991" min="0.99907" max="0.9992">
152 The <ei>V_tb</ei> CKM matrix element.
155 <h3>The CoupSM class</h3>
157 The <code><aloc href="ProgramFlow">Pythia</aloc></code> class contains a
158 public instance <code>coupSM</code> of the <code>CoupSM</code> class.
159 This class contains one instance each of the <code>AlphaStrong</code>
160 and <code>AlphaEM</code> classes, and additionally stores the weak couplings
161 and the quark mixing matrix mentioned above. This class is used especially
162 in the calculation of cross sections and resonance widths, but could also
163 be used elsewhere. Specifically, as already mentioned, there are separate
164 <code>AlphaStrong</code> and <code>AlphaEM</code> instances for timelike
165 and spacelike showers and for multiparton interactions, while weak couplings
166 and the quark mixing matrix are only stored here. With the exception of the
167 first two methods below, which are for internal use, the subsequent ones
168 could also be used externally.
170 <method name="CoupSM::CoupSM()">
171 the constructor does nothing. Internal.
174 <method name="void CoupSM::init(Settings& settings, Rndm* rndmPtr)">
175 this is where the <code>AlphaStrong</code> and <code>AlphaEM</code>
176 instances are initialized, and weak couplings and the quark mixing matrix
177 are read in and set. This is based on the values stored on this page and
178 among the <aloc href="CouplingsAndScales">Couplings and Scales</aloc>.
182 <method name="double CoupSM::alphaS(double scale2)">
183 the <ei>alpha_strong</ei> value at the quadratic scale <code>scale2</code>.
186 <method name="double CoupSM::alphaS1Ord(double scale2)">
187 a first-order overestimate of the full second-order <ei>alpha_strong</ei>
188 value at the quadratic scale <code>scale2</code>.
191 <method name="double CoupSM::alphaS2OrdCorr(double scale2)">
192 a multiplicative correction factor, below unity, that brings the
193 first-order overestimate above into agreement with the full second-order
194 <ei>alpha_strong</ei> value at the quadratic scale <code>scale2</code>.
197 <method name="double CoupSM::Lambda3()">
199 <methodmore name="double CoupSM::Lambda4()">
201 <methodmore name="double CoupSM::Lambda5()">
202 the three-, four-, and five-flavour <ei>Lambda</ei> scale.
205 <method name="double CoupSM::alphaEM(double scale2)">
206 the <ei>alpha_em</ei> value at the quadratic scale <code>scale2</code>.
209 <method name="double CoupSM::sin2thetaW()">
211 <methodmore name="double CoupSM::cos2thetaW()">
212 the sine-squared and cosine-squared of the weak mixing angle, as used in
213 the gauge-boson sector.
216 <method name="double CoupSM::sin2thetaWbar()">
217 the sine-squared of the weak mixing angle, as used to derive the vector
218 couplings of fermions to the <ei>Z^0</ei>.
221 <method name="double CoupSM::GF()">
222 the Fermi constant of weak decays, in GeV<ei>^-2</ei>.
225 <method name="double CoupSM::ef(int idAbs)">
226 the electrical charge of a fermion, by the absolute sign of the PDF code,
227 i.e. <code>idAbs</code> must be in the range between 1 and 18.
230 <method name="double CoupSM::vf(int idAbs)">
232 <methodmore name="double CoupSM::af(int idAbs)">
233 the vector and axial charges of a fermion, by the absolute sign of the PDF
234 code (<ei>a_f = +-1, v_f = a_f - 4. * sin2thetaWbar * e_f</ei>).
237 <method name="double CoupSM::t3f(int idAbs)">
239 <methodmore name="double CoupSM::lf(int idAbs)">
241 <methodmore name="double CoupSM::rf(int idAbs)">
242 the weak isospin, left- and righthanded charges of a fermion, by the
243 absolute sign of the PDF code (<ei>t^3_f = a_f/2, l_f = (v_f + a_f)/2,
244 r_f = (v_f - a_f)/2</ei>; you may find other conventions in the literature
245 that differ by a factor of 2).
248 <method name="double CoupSM::ef2(int idAbs)">
250 <methodmore name="double CoupSM::vf2(int idAbs)">
252 <methodmore name="double CoupSM::af2(int idAbs)">
254 <methodmore name="double CoupSM::efvf(int idAbs)">
256 <methodmore name="double CoupSM::vf2af2(int idAbs)">
257 common quadratic combinations of the above couplings:
258 <ei>e_f^2, v_f^2, a_f^2, e_f * v_f, v_f^2 + a_f^2</ei>.
261 <method name="double CoupSM::VCKMgen(int genU, int genD)">
263 <methodmore name="double CoupSM::V2CKMgen(int genU, int genD)">
264 the CKM mixing element,or the square of it, for
265 up-type generation index <code>genU</code>
266 (<ei>1 = u, 2 = c, 3 = t, 4 = t'</ei>) and
267 down-type generation index <code>genD</code>
268 (<ei>1 = d, 2 = s, 3 = b, 4 = b'</ei>).
271 <method name="double CoupSM::VCKMid(int id1, int id2)">
273 <methodmore name="double CoupSM::V2CKMid(int id1, int id2)">
274 the CKM mixing element,or the square of it, for
275 flavours <code>id1</code> and <code>id2</code>, both in the
276 range from <ei>-18</ei> to <ei>+18</ei>. The sign is here not
277 checked (so it can be used both for <ei>u + dbar -> W+</ei>
278 and <ei>u -> d + W+</ei>, say), but impossible flavour combinations
279 evaluate to zero. The neutrino sector is numbered by flavor
280 eigenstates, so there is no mixing in the lepton-neutrino system.
283 <method name="double CoupSM::V2CKMsum(int id)">
284 the sum of squared CKM mixing element that a given flavour can couple to,
285 excluding the top quark and fourth generation. Is close to unity
286 for the first two generations. Returns unity for the lepton-neutrino
290 <method name="int CoupSM::V2CKMpick(int id)">
291 picks a random CKM partner quark or lepton (with the same sign as
292 <code>id</code>) according to the respective squared elements, again
293 excluding the top quark and fourth generation from the list of
294 possibilities. Unambiguous choice for the lepton-neutrino sector.
299 <!-- Copyright (C) 2012 Torbjorn Sjostrand -->