WAsymm: WDecay.nb

File WDecay.nb, 14.5 KB (added by (none), 13 years ago)
Line 
1(* Content-type: application/mathematica *)
2
3(*** Wolfram Notebook File ***)
4(* http://www.wolfram.com/nb *)
5
6(* CreatedBy='Mathematica 6.0' *)
7
8(*CacheID: 234*)
9(* Internal cache information:
10NotebookFileLineBreakTest
11NotebookFileLineBreakTest
12NotebookDataPosition[ 145, 7]
13NotebookDataLength[ 14703, 517]
14NotebookOptionsPosition[ 12682, 441]
15NotebookOutlinePosition[ 13040, 457]
16CellTagsIndexPosition[ 12997, 454]
17WindowFrame->Normal
18ContainsDynamic->False*)
19
20(* Beginning of Notebook Content *)
21Notebook[{
22
23Cell[CellGroupData[{
24Cell[TextData[{
25 "Calculation for ",
26 StyleBox[" ", "DisplayFormula"],
27 Cell[BoxData[
28 FormBox[
29 SuperscriptBox["W", "+"], TraditionalForm]]],
30 StyleBox["\[RightArrow] ", "DisplayFormula"],
31 Cell[BoxData[
32 FormBox[
33 SuperscriptBox["e", "+"], TraditionalForm]]],
34 " ",
35 Cell[BoxData[
36 FormBox[
37 SubscriptBox["\[Nu]", "e"], TraditionalForm]]]
38}], "Title",
39 CellChangeTimes->{{3.42892672627883*^9, 3.428926824323577*^9}}],
40
41Cell[CellGroupData[{
42
43Cell["Input FeynCalc", "Subsection"],
44
45Cell[BoxData[
46 RowBox[{
47 RowBox[{"<<", "HighEnergyPhysics`fc`"}], ";"}]], "Input"]
48}, Closed]],
49
50Cell[CellGroupData[{
51
52Cell["Preliminaries", "Subsection"],
53
54Cell[CellGroupData[{
55
56Cell["Kinematics 1->2", "Subsubsection",
57 CellChangeTimes->{{3.428926835790217*^9, 3.428926837886043*^9}}],
58
59Cell[BoxData[
60 RowBox[{"\[IndentingNewLine]",
61 RowBox[{
62 RowBox[{
63 RowBox[{
64 RowBox[{"ScalarProduct", "[",
65 RowBox[{"P", ",", "P"}], "]"}], "=", "mw2"}], ";"}],
66 "\[IndentingNewLine]",
67 RowBox[{
68 RowBox[{
69 RowBox[{"ScalarProduct", "[",
70 RowBox[{"pe", ",", "pe"}], "]"}], "=", "0"}], ";"}],
71 "\[IndentingNewLine]",
72 RowBox[{
73 RowBox[{
74 RowBox[{"ScalarProduct", "[",
75 RowBox[{"p\[Nu]", ",", "p\[Nu]"}], "]"}], "=", "0"}], ";"}],
76 "\[IndentingNewLine]",
77 RowBox[{
78 RowBox[{
79 RowBox[{"ScalarProduct", "[",
80 RowBox[{"P", ",", "pe"}], "]"}], "=",
81 RowBox[{"mw2", "/", "2"}]}], ";"}], "\[IndentingNewLine]",
82 RowBox[{
83 RowBox[{
84 RowBox[{"ScalarProduct", "[",
85 RowBox[{"P", ",", "p\[Nu]"}], "]"}], "=",
86 RowBox[{"mw2", "/", "2"}]}], ";"}], "\[IndentingNewLine]",
87 RowBox[{
88 RowBox[{
89 RowBox[{"ScalarProduct", "[",
90 RowBox[{"pe", ",", "p\[Nu]"}], "]"}], "=",
91 RowBox[{"mw2", "/", "2"}]}], ";"}], "\[IndentingNewLine]"}]}]], "Input",
92 CellChangeTimes->{{3.4289268403011503`*^9, 3.428926946965074*^9}}]
93}, Open ]]
94}, Closed]],
95
96Cell[CellGroupData[{
97
98Cell["Amplitude (1 diagram)", "Subsection",
99 CellChangeTimes->{{3.4289292421393127`*^9, 3.428929244675111*^9}}],
100
101Cell["The summed and averaged square matrix element", "Text",
102 CellChangeTimes->{{3.4289271420095243`*^9, 3.4289271454722757`*^9}, {
103 3.4289273354903717`*^9, 3.428927351049911*^9}, {3.4289275105378*^9,
104 3.42892751343362*^9}}],
105
106Cell[CellGroupData[{
107
108Cell[BoxData[
109 RowBox[{"M2", "=",
110 RowBox[{
111 RowBox[{"1", "/", "3"}],
112 RowBox[{
113 RowBox[{"(",
114 RowBox[{
115 RowBox[{"gw", "/", "2"}], "/",
116 RowBox[{"Sqrt", "[", "2", "]"}]}], ")"}], "^", "2"}],
117 RowBox[{"Tr", "[",
118 RowBox[{
119 RowBox[{"GSD", "[", "pe", "]"}], ".",
120 RowBox[{"GAD", "[", "mu", "]"}], ".",
121 RowBox[{"(",
122 RowBox[{"1", "-", "GA5"}], ")"}], ".",
123 RowBox[{"GSD", "[", "p\[Nu]", "]"}], ".",
124 RowBox[{"GAD", "[", "nu", "]"}], ".",
125 RowBox[{"(",
126 RowBox[{"1", "-", "GA5"}], ")"}]}], "]"}],
127 RowBox[{"(",
128 RowBox[{
129 RowBox[{"-",
130 RowBox[{"MetricTensor", "[",
131 RowBox[{"mu", ",", "nu"}], "]"}]}], "+",
132 RowBox[{
133 RowBox[{"FV", "[",
134 RowBox[{"P", ",", "mu"}], "]"}],
135 RowBox[{
136 RowBox[{"FV", "[",
137 RowBox[{"P", ",", "nu"}], "]"}], "/", "mw2"}]}]}], ")"}]}]}]], "Input",\
138
139 CellChangeTimes->{{3.428927033246361*^9, 3.4289271401967373`*^9}, {
140 3.42892727787171*^9, 3.428927413135103*^9}, {3.4289274432481937`*^9,
141 3.4289274930463*^9}, {3.428929121395401*^9, 3.428929122088388*^9},
142 3.428929375657975*^9, {3.428929541763723*^9, 3.428929566521463*^9}}],
143
144Cell[BoxData[
145 FormBox[
146 RowBox[{
147 FractionBox["1", "6"], " ",
148 SuperscriptBox["gw", "2"], " ",
149 RowBox[{"(",
150 RowBox[{
151 FractionBox[
152 RowBox[{
153 SuperscriptBox["P", "mu"], " ",
154 SuperscriptBox["P", "nu"]}], "mw2"], "-",
155 SuperscriptBox["g",
156 RowBox[{"mu", "nu"}]]}], ")"}], " ",
157 RowBox[{"(",
158 RowBox[{
159 RowBox[{
160 RowBox[{"-", "2"}], " ", "\[ImaginaryI]", " ",
161 SuperscriptBox["\[Epsilon]",
162 RowBox[{
163 FormBox[
164 FormBox["mu",
165 TraditionalForm],
166 TraditionalForm],
167 FormBox[
168 FormBox["nu",
169 TraditionalForm],
170 TraditionalForm],
171 FormBox["pe",
172 TraditionalForm],
173 FormBox["p\[Nu]",
174 TraditionalForm]}]]}], "-",
175 RowBox[{"mw2", " ",
176 SuperscriptBox["g",
177 RowBox[{"mu", "nu"}]]}], "+",
178 RowBox[{"2", " ",
179 SuperscriptBox[
180 FormBox["p\[Nu]",
181 TraditionalForm],
182 FormBox[
183 FormBox["mu",
184 TraditionalForm],
185 TraditionalForm]], " ",
186 SuperscriptBox[
187 FormBox["pe",
188 TraditionalForm],
189 FormBox[
190 FormBox["nu",
191 TraditionalForm],
192 TraditionalForm]]}], "+",
193 RowBox[{"2", " ",
194 SuperscriptBox[
195 FormBox["pe",
196 TraditionalForm],
197 FormBox[
198 FormBox["mu",
199 TraditionalForm],
200 TraditionalForm]], " ",
201 SuperscriptBox[
202 FormBox["p\[Nu]",
203 TraditionalForm],
204 FormBox[
205 FormBox["nu",
206 TraditionalForm],
207 TraditionalForm]]}]}], ")"}]}], TraditionalForm]], "Output",
208 CellChangeTimes->{
209 3.4289272789512367`*^9, 3.428927318445057*^9, {3.4289273534290743`*^9,
210 3.428927404797317*^9}, {3.4289274516566753`*^9, 3.42892749447148*^9},
211 3.4289291227183*^9, {3.428929370895793*^9, 3.428929376751039*^9},
212 3.428929567605669*^9, 3.428938985716309*^9}]
213}, Open ]],
214
215Cell["We contract the Lorentz indices", "Text",
216 CellChangeTimes->{{3.428927497453924*^9, 3.428927503026804*^9}}],
217
218Cell[CellGroupData[{
219
220Cell[BoxData[
221 RowBox[{"M2", "=",
222 RowBox[{"Contract", "[", "M2", "]"}]}]], "Input",
223 CellChangeTimes->{{3.428927540707325*^9, 3.42892754927673*^9}}],
224
225Cell[BoxData[
226 FormBox[
227 FractionBox[
228 RowBox[{
229 SuperscriptBox["gw", "2"], " ", "mw2"}], "3"], TraditionalForm]], "Output",\
230
231 CellChangeTimes->{3.428927549552278*^9, 3.4289291245462008`*^9,
232 3.4289293726684103`*^9, 3.428929569719249*^9, 3.428938985747068*^9}]
233}, Open ]],
234
235Cell["We insert he definition of the Fermi constant", "Text",
236 CellChangeTimes->{{3.428927552171399*^9, 3.428927561923086*^9}}],
237
238Cell[CellGroupData[{
239
240Cell[BoxData[
241 RowBox[{"M2", "=",
242 RowBox[{"M2", "/.",
243 RowBox[{
244 RowBox[{"gw", "^", "2"}], "\[Rule]",
245 RowBox[{"8", " ", "mw2", " ",
246 RowBox[{"Gf", "/",
247 RowBox[{"Sqrt", "[", "2", "]"}]}]}]}]}]}]], "Input",
248 CellChangeTimes->{{3.428927568404044*^9, 3.42892758705121*^9}}],
249
250Cell[BoxData[
251 FormBox[
252 RowBox[{
253 FractionBox["4", "3"], " ",
254 SqrtBox["2"], " ", "Gf", " ",
255 SuperscriptBox["mw2", "2"]}], TraditionalForm]], "Output",
256 CellChangeTimes->{3.428927588058874*^9, 3.4289291256255703`*^9,
257 3.428929571106011*^9, 3.428938985780642*^9}]
258}, Open ]]
259}, Closed]],
260
261Cell[CellGroupData[{
262
263Cell["Phase Space", "Subsection",
264 CellChangeTimes->{{3.4289292202493353`*^9, 3.428929221917066*^9}}],
265
266Cell["The massless two particle phase space is", "Text",
267 CellChangeTimes->{{3.4289286541859837`*^9, 3.428928662992796*^9}}],
268
269Cell[CellGroupData[{
270
271Cell[BoxData[
272 RowBox[{"d\[CapitalPhi]2", "=",
273 RowBox[{
274 RowBox[{
275 RowBox[{"(",
276 RowBox[{"2", "Pi"}], ")"}], "^", "4"}],
277 RowBox[{"\[Delta]", "[",
278 RowBox[{"P", "-", "pe", "-", "p\[Nu]"}], "]"}],
279 RowBox[{
280 RowBox[{
281 RowBox[{"d3pe", "/",
282 RowBox[{
283 RowBox[{"(",
284 RowBox[{"2", "Pi"}], ")"}], "^", "3"}]}], "/", "2"}], "/", "Ee"}],
285 " ",
286 RowBox[{
287 RowBox[{
288 RowBox[{"d3p\[Nu]", "/",
289 RowBox[{
290 RowBox[{"(",
291 RowBox[{"2", "Pi"}], ")"}], "^", "3"}]}], "/", "2"}], "/",
292 "E\[Nu]"}]}]}]], "Input",
293 CellChangeTimes->{{3.428928668808182*^9, 3.428928748243292*^9}, {
294 3.4289288594542103`*^9, 3.428928869884198*^9}}],
295
296Cell[BoxData[
297 FormBox[
298 FractionBox[
299 RowBox[{"d3pe", " ", "d3p\[Nu]", " ",
300 RowBox[{"\[Delta]", "(",
301 RowBox[{"P", "-", "pe", "-", "p\[Nu]"}], ")"}]}],
302 RowBox[{"16", " ", "Ee", " ", "E\[Nu]", " ",
303 SuperscriptBox["\[Pi]", "2"]}]], TraditionalForm]], "Output",
304 CellChangeTimes->{{3.428928862716354*^9, 3.428928870072042*^9},
305 3.428929126861431*^9, 3.428929574710586*^9, 3.428938985810316*^9}]
306}, Open ]],
307
308Cell["\<\
309Performing the integration over d3p\[Nu] in the rest frame of the W gives\
310\>", "Text",
311 CellChangeTimes->{{3.428928749821361*^9, 3.428928784904477*^9}, {
312 3.4289288233373938`*^9, 3.428928824472897*^9}}],
313
314Cell[CellGroupData[{
315
316Cell[BoxData[
317 RowBox[{"PS", "=",
318 RowBox[{
319 RowBox[{
320 RowBox[{"1", "/", "4"}], "/",
321 RowBox[{
322 RowBox[{"(",
323 RowBox[{"2", "Pi"}], ")"}], "^", "2"}]}], " ",
324 RowBox[{"\[Delta]", "[",
325 RowBox[{"mw", "-",
326 RowBox[{"2", "Ee"}]}], "]"}],
327 RowBox[{"d3pe", "/",
328 RowBox[{"Ee", "^", "2"}]}]}]}]], "Input",
329 CellChangeTimes->{{3.4289287922794847`*^9, 3.4289288819309*^9}}],
330
331Cell[BoxData[
332 FormBox[
333 FractionBox[
334 RowBox[{"d3pe", " ",
335 RowBox[{"\[Delta]", "(",
336 RowBox[{"mw", "-",
337 RowBox[{"2", " ", "Ee"}]}], ")"}]}],
338 RowBox[{"16", " ",
339 SuperscriptBox["Ee", "2"], " ",
340 SuperscriptBox["\[Pi]", "2"]}]], TraditionalForm]], "Output",
341 CellChangeTimes->{{3.428928849656254*^9, 3.428928882523725*^9},
342 3.428929068231016*^9, 3.428929127845537*^9, 3.428929574829381*^9,
343 3.428938985844248*^9}]
344}, Open ]],
345
346Cell["\<\
347Going to spherical coordinates, and usging pe dpe = Ee dEe, we get\
348\>", "Text",
349 CellChangeTimes->{{3.428928883567082*^9, 3.4289289264749002`*^9}}],
350
351Cell[CellGroupData[{
352
353Cell[BoxData[
354 RowBox[{"PS", "=",
355 RowBox[{
356 RowBox[{"PS", "/.",
357 RowBox[{
358 RowBox[{"\[Delta]", "[",
359 RowBox[{"mw", "-",
360 RowBox[{"2", "Ee"}]}], "]"}], "\[Rule]",
361 RowBox[{
362 RowBox[{"1", "/", "2"}],
363 RowBox[{"\[Delta]", "[",
364 RowBox[{"Ee", "-",
365 RowBox[{"mw", "/", "2"}]}], "]"}]}]}]}], "/.", " ",
366 RowBox[{"d3pe", "\[Rule]",
367 RowBox[{"4", "Pi", " ",
368 RowBox[{"Ee", "^", "2"}], " ", "dEe"}]}]}]}]], "Input",
369 CellChangeTimes->{{3.4289289441444197`*^9, 3.428929022421103*^9},
370 3.428929070151713*^9}],
371
372Cell[BoxData[
373 FormBox[
374 FractionBox[
375 RowBox[{"dEe", " ",
376 RowBox[{"\[Delta]", "(",
377 RowBox[{"Ee", "-",
378 FractionBox["mw", "2"]}], ")"}]}],
379 RowBox[{"8", " ", "\[Pi]"}]], TraditionalForm]], "Output",
380 CellChangeTimes->{3.428929025150517*^9, 3.428929070325218*^9,
381 3.4289291311060667`*^9, 3.428929574875102*^9, 3.4289389858778963`*^9}]
382}, Open ]],
383
384Cell["Performing the integration", "Text",
385 CellChangeTimes->{{3.428929031938621*^9, 3.428929037282918*^9}}],
386
387Cell[CellGroupData[{
388
389Cell[BoxData[
390 RowBox[{"PS", "=",
391 RowBox[{"PS", "/.",
392 RowBox[{
393 RowBox[{"dEe", " ",
394 RowBox[{"\[Delta]", "[",
395 RowBox[{"Ee", "-",
396 RowBox[{"mw", "/", "2"}]}], "]"}]}], "\[Rule]", "1"}]}]}]], "Input",
397 CellChangeTimes->{{3.428929044586605*^9, 3.428929056603554*^9}}],
398
399Cell[BoxData[
400 FormBox[
401 FractionBox["1",
402 RowBox[{"8", " ", "\[Pi]"}]], TraditionalForm]], "Output",
403 CellChangeTimes->{{3.428929056861126*^9, 3.428929071114458*^9},
404 3.4289291321148567`*^9, 3.428929574912538*^9, 3.4289389861836767`*^9}]
405}, Open ]]
406}, Closed]],
407
408Cell[CellGroupData[{
409
410Cell["Result", "Subsection",
411 CellChangeTimes->{{3.4289292627868023`*^9, 3.4289292633475027`*^9}}],
412
413Cell["The decay rate of the W then becomes", "Text",
414 CellChangeTimes->{{3.428929073874954*^9, 3.428929081291203*^9}}],
415
416Cell[CellGroupData[{
417
418Cell[BoxData[
419 RowBox[{"\[CapitalGamma]", "=",
420 RowBox[{
421 RowBox[{
422 RowBox[{
423 RowBox[{"1", "/", "2"}], "/", "mw"}], " ", "PS", " ", "M2"}], "/.",
424 RowBox[{"mw2", "\[Rule]",
425 RowBox[{"mw", "^", "2"}]}]}]}]], "Input",
426 CellChangeTimes->{{3.428929087038114*^9, 3.428929107627933*^9},
427 3.428929346599481*^9, 3.428938975232931*^9}],
428
429Cell[BoxData[
430 FormBox[
431 FractionBox[
432 RowBox[{"Gf", " ",
433 SuperscriptBox["mw", "3"]}],
434 RowBox[{"6", " ",
435 SqrtBox["2"], " ", "\[Pi]"}]], TraditionalForm]], "Output",
436 CellChangeTimes->{{3.428929108658009*^9, 3.428929133003119*^9},
437 3.4289293476527367`*^9, 3.4289295771229353`*^9, 3.428938986228635*^9}]
438}, Open ]]
439}, Closed]]
440}, Open ]]
441},
442WindowSize->{710, 706},
443WindowMargins->{{Automatic, 230}, {Automatic, 4}},
444ShowSelection->True,
445FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)",
446StyleDefinitions->"Default.nb"
447]
448(* End of Notebook Content *)
449
450(* Internal cache information *)
451(*CellTagsOutline
452CellTagsIndex->{}
453*)
454(*CellTagsIndex
455CellTagsIndex->{}
456*)
457(*NotebookFileOutline
458Notebook[{
459Cell[CellGroupData[{
460Cell[590, 23, 427, 15, 79, "Title"],
461Cell[CellGroupData[{
462Cell[1042, 42, 36, 0, 34, "Subsection"],
463Cell[1081, 44, 83, 2, 27, "Input"]
464}, Closed]],
465Cell[CellGroupData[{
466Cell[1201, 51, 35, 0, 26, "Subsection"],
467Cell[CellGroupData[{
468Cell[1261, 55, 106, 1, 25, "Subsubsection"],
469Cell[1370, 58, 1112, 33, 133, "Input"]
470}, Open ]]
471}, Closed]],
472Cell[CellGroupData[{
473Cell[2531, 97, 111, 1, 26, "Subsection"],
474Cell[2645, 100, 228, 3, 26, "Text"],
475Cell[CellGroupData[{
476Cell[2898, 107, 1191, 34, 43, "Input"],
477Cell[4092, 143, 1924, 68, 70, "Output"]
478}, Open ]],
479Cell[6031, 214, 113, 1, 26, "Text"],
480Cell[CellGroupData[{
481Cell[6169, 219, 151, 3, 27, "Input"],
482Cell[6323, 224, 269, 7, 70, "Output"]
483}, Open ]],
484Cell[6607, 234, 127, 1, 26, "Text"],
485Cell[CellGroupData[{
486Cell[6759, 239, 295, 8, 27, "Input"],
487Cell[7057, 249, 275, 7, 70, "Output"]
488}, Open ]]
489}, Closed]],
490Cell[CellGroupData[{
491Cell[7381, 262, 101, 1, 26, "Subsection"],
492Cell[7485, 265, 124, 1, 26, "Text"],
493Cell[CellGroupData[{
494Cell[7634, 270, 696, 23, 27, "Input"],
495Cell[8333, 295, 417, 9, 70, "Output"]
496}, Open ]],
497Cell[8765, 307, 214, 4, 26, "Text"],
498Cell[CellGroupData[{
499Cell[9004, 315, 404, 13, 27, "Input"],
500Cell[9411, 330, 450, 12, 70, "Output"]
501}, Open ]],
502Cell[9876, 345, 158, 3, 26, "Text"],
503Cell[CellGroupData[{
504Cell[10059, 352, 569, 17, 27, "Input"],
505Cell[10631, 371, 358, 9, 70, "Output"]
506}, Open ]],
507Cell[11004, 383, 108, 1, 26, "Text"],
508Cell[CellGroupData[{
509Cell[11137, 388, 293, 8, 27, "Input"],
510Cell[11433, 398, 245, 5, 70, "Output"]
511}, Open ]]
512}, Closed]],
513Cell[CellGroupData[{
514Cell[11727, 409, 98, 1, 26, "Subsection"],
515Cell[11828, 412, 118, 1, 26, "Text"],
516Cell[CellGroupData[{
517Cell[11971, 417, 347, 9, 27, "Input"],
518Cell[12321, 428, 321, 8, 52, "Output"]
519}, Open ]]
520}, Closed]]
521}, Open ]]
522}
523]
524*)
525
526(* End of internal cache information *)