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3<title>Standard-Model Parameters</title>
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29
30<h2>Standard-Model Parameters</h2>
31
32<h3>The strong coupling</h3> 
33
34The <code>AlphaStrong</code> class is used to provide a first- or
35second-order running <i>alpha_strong</i> (or, trivially, a
36zeroth-order fixed one). Formulae are the standard ones found in
37[<a href="Bibliography.php" target="page">Yao06</a>]. The second-order expression used, eq. (9.5),
38may be somewhat different in other approaches (with differences
39formally of higher order), so do not necessarily expect perfect
40agreement, especially not at small <i>Q^2</i> scales. The starting
41<i>alpha_strong</i> value is defined at the <i>M_Z</i> mass scale.
42The <i>Lambda</i> values are matched at the <i>b</i> and <i>c</i>
43flavour thresholds, such that <i>alpha_strong</i> is continuous.
44For second-order matching an approximate iterative method is used.
45 
46<p/>
47Since we allow <i>alpha_strong</i> to vary separately for
48hard processes, timelike showers, spacelike showers and  multiparton
49interactions, the relevant values can be set in each of these classes.
50The default behaviour is everywhere first-order running.
51 
52<p/>
53The <i>alpha_strong</i> calculation is initialized by
54<code>init( value, order)</code>, where <code>value</code> 
55is the <i>alpha_strong</i> value at <i>M_Z</i> and <code>order</code>
56is the order of the running, 0, 1 or 2.   Thereafter the value can be
57calculated by <code>alphaS(scale2)</code>, where
58<code>scale2</code> is the <i>Q^2</i> scale in GeV^2.
59
60<p/>
61For applications inside shower programs, a second-order <code>alpha_s</code>
62value can be obtained as the product of the two functions
63<code>alphaS1Ord(scale2)</code> and <code>alphaS2OrdCorr(scale2)</code>,
64where the first gives a simple first-order running (but with the
65second-order <i>Lambda</i>) and the second the correction factor,
66below unity, for the second-order terms. This allows a compact handling
67of evolution equations.
68
69<h3>The electromagnetic coupling</h3> 
70
71The <code>AlphaEM</code> class is used to generate a running
72<i>alpha_em</i>. The input <code>StandardModel:alphaEMmZ</code>
73value at the <i>M_Z</i> mass is matched to a low-energy behaviour
74with running starting at the electron mass threshold. The matching
75is done by fitting an effective running coefficient in the region
76betweeen the light-quark treshold and the charm/tau threshold. This
77procedure is approximate, but good enough for our purposes.
78
79<p/>
80Since we allow <i>alpha_em</i> to vary separately for
81hard processes, timelike showers, spacelike showers and  multiparton
82interactions, the choice between using a fixed or a running
83<i>alpha_em</i> can be made in each of these classes.
84The default behaviour is everywhere first-order running.
85The actual values assumed at zero momentum transfer and
86at <i>M_Z</i> are only set here, however.
87
88<br/><br/><table><tr><td><strong>StandardModel:alphaEM0 </td><td></td><td> <input type="text" name="1" value="0.00729735" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.00729735</strong></code>; <code>minimum = 0.0072973</code>; <code>maximum = 0.0072974</code>)</td></tr></table>
89The <i>alpha_em</i> value at vanishing momentum transfer
90(and also below <i>m_e</i>).
91 
92
93<br/><br/><table><tr><td><strong>StandardModel:alphaEMmZ </td><td></td><td> <input type="text" name="2" value="0.00781751" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.00781751</strong></code>; <code>minimum = 0.00780</code>; <code>maximum = 0.00783</code>)</td></tr></table>
94The <i>alpha_em</i> value at the <i>M_Z</i> mass scale.
95Default is taken from [<a href="Bibliography.php" target="page">Yao06</a>].
96 
97
98<p/>
99The <i>alpha_em</i> calculation is initialized by
100<code>init(order)</code>, where <code>order</code> is the order of
101the running, 0 or 1, with -1 a special option to use the fix value
102provided at <i>M_Z</i>.   Thereafter the value can be
103calculated by <code>alphaEM(scale2)</code>, where
104<code>scale2</code> is the <i>Q^2</i> scale in GeV^2.
105
106<h3>The electroweak couplings</h3> 
107
108There are two degrees of freedom that can be set, related to the
109electroweak mixing angle:
110
111<br/><br/><table><tr><td><strong>StandardModel:sin2thetaW </td><td></td><td> <input type="text" name="3" value="0.2312" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.2312</strong></code>; <code>minimum = 0.225</code>; <code>maximum = 0.240</code>)</td></tr></table>
112The sine-squared of the weak mixing angle, as used in all <i>Z^0</i>
113and <i>W^+-</i> masses and couplings, except for the vector couplings
114of fermions to the <i>Z^0</i>, see below. Default is the MSbar value
115from [<a href="Bibliography.php" target="page">Yao06</a>].
116 
117
118<br/><br/><table><tr><td><strong>StandardModel:sin2thetaWbar </td><td></td><td> <input type="text" name="4" value="0.2315" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.2315</strong></code>; <code>minimum = 0.225</code>; <code>maximum = 0.240</code>)</td></tr></table>
119The sine-squared of the weak mixing angle, as used to derive the vector
120couplings of fermions to the <i>Z^0</i>, in the relation
121<i>v_f = a_f - 4 e_f sin^2(theta_W)bar</i>. Default is the
122effective-angle value from [<a href="Bibliography.php" target="page">Yao06</a>].
123 
124
125<p/>
126The Fermi constant is not much used in the currently coded matrix elements,
127since it is redundant, but it is available:
128
129<br/><br/><table><tr><td><strong>StandardModel:GF </td><td></td><td> <input type="text" name="5" value="1.16637e-5" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1.16637e-5</strong></code>; <code>minimum = 1.0e-5</code>; <code>maximum = 1.3e-5</code>)</td></tr></table>
130The Fermi coupling constant, in units of GeV<i>^-2</i>.
131 
132
133<h3>The quark weak-mixing matrix</h3>
134
135The absolute values of the Cabibbo-Kobayashi-Maskawa matrix elements are
136set by the following nine real values taken from [<a href="Bibliography.php" target="page">Yao06</a>] -
137currently the CP-violating phase is not taken into account in this
138parametrization. It is up to the user to pick a consistent unitary
139set of new values whenever changes are made. 
140
141<br/><br/><table><tr><td><strong>StandardModel:Vud </td><td></td><td> <input type="text" name="6" value="0.97383" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.97383</strong></code>; <code>minimum = 0.973</code>; <code>maximum = 0.975</code>)</td></tr></table>
142The <i>V_ud</i> CKM matrix element.
143 
144
145<br/><br/><table><tr><td><strong>StandardModel:Vus </td><td></td><td> <input type="text" name="7" value="0.2272" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.2272</strong></code>; <code>minimum = 0.224</code>; <code>maximum = 0.230</code>)</td></tr></table>
146The <i>V_us</i> CKM matrix element.
147 
148
149<br/><br/><table><tr><td><strong>StandardModel:Vub </td><td></td><td> <input type="text" name="8" value="0.00396" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.00396</strong></code>; <code>minimum = 0.0037</code>; <code>maximum = 0.0042</code>)</td></tr></table>
150The <i>V_ub</i> CKM matrix element.
151 
152
153<br/><br/><table><tr><td><strong>StandardModel:Vcd </td><td></td><td> <input type="text" name="9" value="0.2271" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.2271</strong></code>; <code>minimum = 0.224</code>; <code>maximum = 0.230</code>)</td></tr></table>
154The <i>V_cd</i> CKM matrix element.
155 
156
157<br/><br/><table><tr><td><strong>StandardModel:Vcs </td><td></td><td> <input type="text" name="10" value="0.97296" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.97296</strong></code>; <code>minimum = 0.972</code>; <code>maximum = 0.974</code>)</td></tr></table>
158The <i>V_cs</i> CKM matrix element.
159 
160
161<br/><br/><table><tr><td><strong>StandardModel:Vcb </td><td></td><td> <input type="text" name="11" value="0.04221" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.04221</strong></code>; <code>minimum = 0.0418</code>; <code>maximum = 0.0426</code>)</td></tr></table>
162The <i>V_cb</i> CKM matrix element.
163 
164
165<br/><br/><table><tr><td><strong>StandardModel:Vtd </td><td></td><td> <input type="text" name="12" value="0.00814" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.00814</strong></code>; <code>minimum = 0.006</code>; <code>maximum = 0.010</code>)</td></tr></table>
166The <i>V_td</i> CKM matrix element.
167 
168
169<br/><br/><table><tr><td><strong>StandardModel:Vts </td><td></td><td> <input type="text" name="13" value="0.04161" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.04161</strong></code>; <code>minimum = 0.039</code>; <code>maximum = 0.043</code>)</td></tr></table>
170The <i>V_ts</i> CKM matrix element.
171 
172
173<br/><br/><table><tr><td><strong>StandardModel:Vtb </td><td></td><td> <input type="text" name="14" value="0.9991" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.9991</strong></code>; <code>minimum = 0.99907</code>; <code>maximum = 0.9992</code>)</td></tr></table>
174The <i>V_tb</i> CKM matrix element.
175 
176
177<h3>The CoupSM class</h3> 
178
179The <code><?php $filepath = $_GET["filepath"];
180echo "<a href='ProgramFlow.php?filepath=".$filepath."' target='page'>";?>Pythia</a></code> class contains a
181public instance <code>coupSM</code> of the <code>CoupSM</code> class.
182This class contains one instance each of the <code>AlphaStrong</code>   
183and <code>AlphaEM</code> classes, and additionally stores the weak couplings
184and the quark mixing matrix mentioned above. This class is used especially
185in the calculation of cross sections and resonance widths, but could also
186be used elsewhere. Specifically, as already mentioned, there are separate
187<code>AlphaStrong</code> and <code>AlphaEM</code> instances for timelike
188and spacelike showers and for multiparton interactions, while weak couplings
189and the quark mixing matrix are only stored here. With the exception of the
190first two methods below, which are for internal use, the subsequent ones
191could also be used externally.
192
193<a name="method1"></a>
194<p/><strong>CoupSM::CoupSM() &nbsp;</strong> <br/>
195the constructor does nothing. Internal.
196 
197
198<a name="method2"></a>
199<p/><strong>void CoupSM::init(Settings& settings, Rndm* rndmPtr) &nbsp;</strong> <br/>
200this is where the <code>AlphaStrong</code> and <code>AlphaEM</code>
201instances are initialized, and weak couplings and the quark mixing matrix
202are read in and set. This is based on the values stored on this page and
203among the <?php $filepath = $_GET["filepath"];
204echo "<a href='CouplingsAndScales.php?filepath=".$filepath."' target='page'>";?>Couplings and Scales</a>.
205Internal.
206 
207
208<a name="method3"></a>
209<p/><strong>double CoupSM::alphaS(double scale2) &nbsp;</strong> <br/>
210the <i>alpha_strong</i> value at the quadratic scale <code>scale2</code>.
211 
212
213<a name="method4"></a>
214<p/><strong>double CoupSM::alphaS1Ord(double scale2) &nbsp;</strong> <br/>
215a first-order overestimate of the full second-order <i>alpha_strong</i>
216value at the quadratic scale <code>scale2</code>.
217 
218
219<a name="method5"></a>
220<p/><strong>double CoupSM::alphaS2OrdCorr(double scale2) &nbsp;</strong> <br/>
221a multiplicative correction factor, below unity, that brings the
222first-order overestimate above into agreement with the full second-order
223<i>alpha_strong</i> value at the quadratic scale <code>scale2</code>.
224 
225
226<a name="method6"></a>
227<p/><strong>double CoupSM::Lambda3() &nbsp;</strong> <br/>
228 
229<strong>double CoupSM::Lambda4() &nbsp;</strong> <br/>
230 
231<strong>double CoupSM::Lambda5() &nbsp;</strong> <br/>
232the three-, four-, and five-flavour <i>Lambda</i> scale.
233 
234
235<a name="method7"></a>
236<p/><strong>double CoupSM::alphaEM(double scale2) &nbsp;</strong> <br/>
237the <i>alpha_em</i> value at the quadratic scale <code>scale2</code>.
238 
239
240<a name="method8"></a>
241<p/><strong>double CoupSM::sin2thetaW() &nbsp;</strong> <br/>
242 
243<strong>double CoupSM::cos2thetaW() &nbsp;</strong> <br/>
244the sine-squared and cosine-squared of the weak mixing angle, as used in
245the gauge-boson sector.
246 
247
248<a name="method9"></a>
249<p/><strong>double CoupSM::sin2thetaWbar() &nbsp;</strong> <br/>
250the sine-squared of the weak mixing angle, as used to derive the vector
251couplings of fermions to the <i>Z^0</i>.
252 
253
254<a name="method10"></a>
255<p/><strong>double CoupSM::GF() &nbsp;</strong> <br/>
256the Fermi constant of weak decays, in GeV<i>^-2</i>.
257 
258
259<a name="method11"></a>
260<p/><strong>double CoupSM::ef(int idAbs) &nbsp;</strong> <br/>
261the electrical charge of a fermion, by the absolute sign of the PDF code,
262i.e. <code>idAbs</code> must be in the range between 1 and 18.
263 
264
265<a name="method12"></a>
266<p/><strong>double CoupSM::vf(int idAbs) &nbsp;</strong> <br/>
267 
268<strong>double CoupSM::af(int idAbs) &nbsp;</strong> <br/>
269the vector and axial charges of a fermion, by the absolute sign of the PDF
270code (<i>a_f = +-1, v_f = a_f - 4. * sin2thetaWbar * e_f</i>).
271 
272
273<a name="method13"></a>
274<p/><strong>double CoupSM::t3f(int idAbs) &nbsp;</strong> <br/>
275 
276<strong>double CoupSM::lf(int idAbs) &nbsp;</strong> <br/>
277 
278<strong>double CoupSM::rf(int idAbs) &nbsp;</strong> <br/>
279the weak isospin, left- and righthanded charges of a fermion, by the
280absolute sign of the PDF code (<i>t^3_f = a_f/2, l_f = (v_f + a_f)/2,
281r_f = (v_f - a_f)/2</i>; you may find other conventions in the literature
282that differ by a factor of 2).
283 
284
285<a name="method14"></a>
286<p/><strong>double CoupSM::ef2(int idAbs) &nbsp;</strong> <br/>
287 
288<strong>double CoupSM::vf2(int idAbs) &nbsp;</strong> <br/>
289 
290<strong>double CoupSM::af2(int idAbs) &nbsp;</strong> <br/>
291 
292<strong>double CoupSM::efvf(int idAbs) &nbsp;</strong> <br/>
293 
294<strong>double CoupSM::vf2af2(int idAbs) &nbsp;</strong> <br/>
295common quadratic combinations of the above couplings:
296<i>e_f^2, v_f^2, a_f^2, e_f * v_f, v_f^2 + a_f^2</i>.
297 
298
299<a name="method15"></a>
300<p/><strong>double CoupSM::VCKMgen(int genU, int genD) &nbsp;</strong> <br/>
301 
302<strong>double CoupSM::V2CKMgen(int genU, int genD) &nbsp;</strong> <br/>
303the CKM mixing element,or the square of it, for
304up-type generation index <code>genU</code>
305(<i>1 = u, 2 = c, 3 = t, 4 = t'</i>) and
306down-type generation index <code>genD</code>
307(<i>1 = d, 2 = s, 3 = b, 4 = b'</i>).
308 
309
310<a name="method16"></a>
311<p/><strong>double CoupSM::VCKMid(int id1, int id2) &nbsp;</strong> <br/>
312 
313<strong>double CoupSM::V2CKMid(int id1, int id2) &nbsp;</strong> <br/>
314the CKM mixing element,or the square of it, for
315flavours <code>id1</code> and <code>id2</code>, both in the
316range from <i>-18</i> to <i>+18</i>. The sign is here not
317checked (so it can be used both for <i>u + dbar -> W+</i>
318and <i>u -> d + W+</i>, say), but impossible flavour combinations
319evaluate to zero. The neutrino sector is numbered by flavor
320eigenstates, so there is no mixing in the lepton-neutrino system.
321 
322
323<a name="method17"></a>
324<p/><strong>double CoupSM::V2CKMsum(int id) &nbsp;</strong> <br/>
325the sum of squared CKM mixing element that a given flavour can couple to,
326excluding the top quark and fourth generation. Is close to unity
327for the first two generations. Returns unity for the lepton-neutrino
328sector.
329 
330
331<a name="method18"></a>
332<p/><strong>int CoupSM::V2CKMpick(int id) &nbsp;</strong> <br/>
333picks a random CKM partner quark or lepton (with the same sign as
334<code>id</code>) according to the respective squared elements, again
335excluding the top quark and fourth generation from the list of
336possibilities. Unambiguous choice for the lepton-neutrino sector.
337 
338
339<input type="hidden" name="saved" value="1"/>
340
341<?php
342echo "<input type='hidden' name='filepath' value='".$_GET["filepath"]."'/>"?>
343
344<table width="100%"><tr><td align="right"><input type="submit" value="Save Settings" /></td></tr></table>
345</form>
346
347<?php
348
349if($_POST["saved"] == 1)
350{
351$filepath = $_POST["filepath"];
352$handle = fopen($filepath, 'a');
353
354if($_POST["1"] != "0.00729735")
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372fwrite($handle,$data);
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374if($_POST["5"] != "1.16637e-5")
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377fwrite($handle,$data);
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382fwrite($handle,$data);
383}
384if($_POST["7"] != "0.2272")
385{
386$data = "StandardModel:Vus = ".$_POST["7"]."\n";
387fwrite($handle,$data);
388}
389if($_POST["8"] != "0.00396")
390{
391$data = "StandardModel:Vub = ".$_POST["8"]."\n";
392fwrite($handle,$data);
393}
394if($_POST["9"] != "0.2271")
395{
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397fwrite($handle,$data);
398}
399if($_POST["10"] != "0.97296")
400{
401$data = "StandardModel:Vcs = ".$_POST["10"]."\n";
402fwrite($handle,$data);
403}
404if($_POST["11"] != "0.04221")
405{
406$data = "StandardModel:Vcb = ".$_POST["11"]."\n";
407fwrite($handle,$data);
408}
409if($_POST["12"] != "0.00814")
410{
411$data = "StandardModel:Vtd = ".$_POST["12"]."\n";
412fwrite($handle,$data);
413}
414if($_POST["13"] != "0.04161")
415{
416$data = "StandardModel:Vts = ".$_POST["13"]."\n";
417fwrite($handle,$data);
418}
419if($_POST["14"] != "0.9991")
420{
421$data = "StandardModel:Vtb = ".$_POST["14"]."\n";
422fwrite($handle,$data);
423}
424fclose($handle);
425}
426
427?>
428</body>
429</html>
430
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