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63ba5337 | 1 | <chapter name="Standard-Model Parameters"> |
2 | ||
3 | <h2>Standard-Model Parameters</h2> | |
4 | ||
5 | <h3>The strong coupling</h3> | |
6 | ||
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. | |
18 | ||
19 | <p/> | |
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. | |
24 | ||
25 | <p/> | |
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. | |
32 | ||
33 | <p/> | |
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. | |
41 | ||
42 | <h3>The electromagnetic coupling</h3> | |
43 | ||
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. | |
51 | ||
52 | <p/> | |
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. | |
60 | ||
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>). | |
65 | </parm> | |
66 | ||
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>. | |
71 | </parm> | |
72 | ||
73 | <p/> | |
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. | |
80 | ||
81 | <h3>The electroweak couplings</h3> | |
82 | ||
83 | There are two degrees of freedom that can be set, related to the | |
84 | electroweak mixing angle: | |
85 | ||
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>. | |
92 | </parm> | |
93 | ||
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>. | |
100 | </parm> | |
101 | ||
102 | <p/> | |
103 | The Fermi constant is not much used in the currently coded matrix elements, | |
104 | since it is redundant, but it is available: | |
105 | ||
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>. | |
109 | </parm> | |
110 | ||
111 | <h3>The quark weak-mixing matrix</h3> | |
112 | ||
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. | |
118 | ||
119 | <parm name="StandardModel:Vud" default="0.97383" min="0.973" max="0.975"> | |
120 | The <ei>V_ud</ei> CKM matrix element. | |
121 | </parm> | |
122 | ||
123 | <parm name="StandardModel:Vus" default="0.2272" min="0.224" max="0.230"> | |
124 | The <ei>V_us</ei> CKM matrix element. | |
125 | </parm> | |
126 | ||
127 | <parm name="StandardModel:Vub" default="0.00396" min="0.0037" max="0.0042"> | |
128 | The <ei>V_ub</ei> CKM matrix element. | |
129 | </parm> | |
130 | ||
131 | <parm name="StandardModel:Vcd" default="0.2271" min="0.224" max="0.230"> | |
132 | The <ei>V_cd</ei> CKM matrix element. | |
133 | </parm> | |
134 | ||
135 | <parm name="StandardModel:Vcs" default="0.97296" min="0.972" max="0.974"> | |
136 | The <ei>V_cs</ei> CKM matrix element. | |
137 | </parm> | |
138 | ||
139 | <parm name="StandardModel:Vcb" default="0.04221" min="0.0418" max="0.0426"> | |
140 | The <ei>V_cb</ei> CKM matrix element. | |
141 | </parm> | |
142 | ||
143 | <parm name="StandardModel:Vtd" default="0.00814" min="0.006" max="0.010"> | |
144 | The <ei>V_td</ei> CKM matrix element. | |
145 | </parm> | |
146 | ||
147 | <parm name="StandardModel:Vts" default="0.04161" min="0.039" max="0.043"> | |
148 | The <ei>V_ts</ei> CKM matrix element. | |
149 | </parm> | |
150 | ||
151 | <parm name="StandardModel:Vtb" default="0.9991" min="0.99907" max="0.9992"> | |
152 | The <ei>V_tb</ei> CKM matrix element. | |
153 | </parm> | |
154 | ||
155 | <h3>The CoupSM class</h3> | |
156 | ||
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. | |
169 | ||
170 | <method name="CoupSM::CoupSM()"> | |
171 | the constructor does nothing. Internal. | |
172 | </method> | |
173 | ||
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>. | |
179 | Internal. | |
180 | </method> | |
181 | ||
182 | <method name="double CoupSM::alphaS(double scale2)"> | |
183 | the <ei>alpha_strong</ei> value at the quadratic scale <code>scale2</code>. | |
184 | </method> | |
185 | ||
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>. | |
189 | </method> | |
190 | ||
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>. | |
195 | </method> | |
196 | ||
197 | <method name="double CoupSM::Lambda3()"> | |
198 | </method> | |
199 | <methodmore name="double CoupSM::Lambda4()"> | |
200 | </methodmore> | |
201 | <methodmore name="double CoupSM::Lambda5()"> | |
202 | the three-, four-, and five-flavour <ei>Lambda</ei> scale. | |
203 | </methodmore> | |
204 | ||
205 | <method name="double CoupSM::alphaEM(double scale2)"> | |
206 | the <ei>alpha_em</ei> value at the quadratic scale <code>scale2</code>. | |
207 | </method> | |
208 | ||
209 | <method name="double CoupSM::sin2thetaW()"> | |
210 | </method> | |
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. | |
214 | </methodmore> | |
215 | ||
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>. | |
219 | </method> | |
220 | ||
221 | <method name="double CoupSM::GF()"> | |
222 | the Fermi constant of weak decays, in GeV<ei>^-2</ei>. | |
223 | </method> | |
224 | ||
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. | |
228 | </method> | |
229 | ||
230 | <method name="double CoupSM::vf(int idAbs)"> | |
231 | </method> | |
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>). | |
235 | </methodmore> | |
236 | ||
237 | <method name="double CoupSM::t3f(int idAbs)"> | |
238 | </method> | |
239 | <methodmore name="double CoupSM::lf(int idAbs)"> | |
240 | </methodmore> | |
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). | |
246 | </methodmore> | |
247 | ||
248 | <method name="double CoupSM::ef2(int idAbs)"> | |
249 | </method> | |
250 | <methodmore name="double CoupSM::vf2(int idAbs)"> | |
251 | </methodmore> | |
252 | <methodmore name="double CoupSM::af2(int idAbs)"> | |
253 | </methodmore> | |
254 | <methodmore name="double CoupSM::efvf(int idAbs)"> | |
255 | </methodmore> | |
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>. | |
259 | </methodmore> | |
260 | ||
261 | <method name="double CoupSM::VCKMgen(int genU, int genD)"> | |
262 | </method> | |
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>). | |
269 | </methodmore> | |
270 | ||
271 | <method name="double CoupSM::VCKMid(int id1, int id2)"> | |
272 | </method> | |
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. | |
281 | </methodmore> | |
282 | ||
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 | |
287 | sector. | |
288 | </method> | |
289 | ||
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. | |
295 | </method> | |
296 | ||
297 | </chapter> | |
298 | ||
299 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |