WAsymm: DYNarrowWidth.2.nb

File DYNarrowWidth.2.nb, 14.6 KB (added by trac, 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[ 14721, 497]
14NotebookOptionsPosition[ 12913, 429]
15NotebookOutlinePosition[ 13290, 446]
16CellTagsIndexPosition[ 13247, 443]
17WindowFrame->Normal
18ContainsDynamic->False*)
19
20(* Beginning of Notebook Content *)
21Notebook[{
22
23Cell[CellGroupData[{
24Cell["\<\
25The Drell-Yan cross-section in the narrow width approximation\
26\>", "Title",
27 CellChangeTimes->{{3.42892672627883*^9, 3.428926824323577*^9}, {
28 3.428937703319875*^9, 3.428937712432496*^9}, {3.428941361957245*^9,
29 3.4289413752252607`*^9}}],
30
31Cell[CellGroupData[{
32
33Cell["Input FeynCalc", "Subsection"],
34
35Cell[BoxData[
36 RowBox[{
37 RowBox[{"<<", "HighEnergyPhysics`fc`"}], ";"}]], "Input",
38 CellChangeTimes->{{3.428941446581498*^9, 3.428941449045033*^9}}]
39}, Open ]],
40
41Cell[CellGroupData[{
42
43Cell["The partonic cross - section", "Section",
44 CellChangeTimes->{{3.428941607370674*^9, 3.428941611186956*^9}}],
45
46Cell[BoxData[
47 RowBox[{
48 RowBox[{"Near", " ", "the", " ", "W", " ", "resonance"}], ",", " ",
49 RowBox[{
50 RowBox[{
51 "the", " ", "partonic", " ", "cross", " ", "section", " ", "\[Sigma]p",
52 " ", "for", " ", "u", " ", "dbar"}], " ", "\[RightArrow]", " ",
53 SuperscriptBox["W", "+"], "\[RightArrow]", " ",
54 RowBox[{
55 SuperscriptBox["e", "+"], " ",
56 SubscriptBox["\[Nu]",
57 RowBox[{"e", " "}]], "can", " ", "be", " ", "written"}]}]}]], "Text",
58 CellChangeTimes->{{3.428941462807769*^9, 3.428941586342557*^9}}],
59
60Cell[CellGroupData[{
61
62Cell[BoxData[
63 RowBox[{"\[Sigma]p", "=",
64 RowBox[{"4",
65 RowBox[{"Pi", "/", "3"}], " ", "\[CapitalGamma]ud", " ",
66 RowBox[{"\[CapitalGamma]ev", " ", "/",
67 RowBox[{"(",
68 RowBox[{
69 RowBox[{
70 RowBox[{"(",
71 RowBox[{"sp", " ", "-", " ",
72 RowBox[{"mw", "^", "2"}]}], ")"}], "^", "2"}], " ", "+",
73 RowBox[{
74 RowBox[{"mw", "^", "2"}],
75 RowBox[{"\[CapitalGamma]tot", "^", "2"}]}]}], ")"}]}]}]}]], "Input",
76 CellChangeTimes->{{3.428941591115911*^9, 3.428941670219965*^9}, {
77 3.428941861583827*^9, 3.428941864412772*^9}}],
78
79Cell[BoxData[
80 FormBox[
81 FractionBox[
82 RowBox[{
83 "4", " ", "\[Pi]", " ", "\[CapitalGamma]ev", " ", "\[CapitalGamma]ud"}],
84 RowBox[{"3", " ",
85 RowBox[{"(",
86 RowBox[{
87 SuperscriptBox[
88 RowBox[{"(",
89 RowBox[{"sp", "-",
90 SuperscriptBox["mw", "2"]}], ")"}], "2"], "+",
91 RowBox[{
92 SuperscriptBox["mw", "2"], " ",
93 SuperscriptBox["\[CapitalGamma]tot", "2"]}]}], ")"}]}]],
94 TraditionalForm]], "Output",
95 CellChangeTimes->{3.428941671372353*^9, 3.4289418646862993`*^9,
96 3.428942151070161*^9, 3.428942597586359*^9}]
97}, Open ]],
98
99Cell["\<\
100where
101
102* \[CapitalGamma]ud is the partial width of the W into u d
103* \[CapitalGamma]ev is the partial width of the W into e \[Nu]
104*\[CapitalGamma]tot is the totel width
105
106At tree-level, we can write approximatively
107
108\[CapitalGamma]ev = \[CapitalGamma]
109\[CapitalGamma]ud = 3 \[CapitalGamma]
110\[CapitalGamma]tot = 9 \[CapitalGamma]
111\
112\>", "Text",
113 CellChangeTimes->{{3.428941673641809*^9, 3.428941779762138*^9}, {
114 3.4289418146980762`*^9, 3.428941823801876*^9}}],
115
116Cell[CellGroupData[{
117
118Cell[BoxData[
119 RowBox[{"\[Sigma]p", "=",
120 RowBox[{"\[Sigma]p", "/.",
121 RowBox[{"{",
122 RowBox[{
123 RowBox[{"\[CapitalGamma]ud", "\[Rule]",
124 RowBox[{"3", "\[CapitalGamma]"}]}], ",",
125 RowBox[{"\[CapitalGamma]ev", "->", "\[CapitalGamma]"}], ",",
126 RowBox[{"\[CapitalGamma]tot", "\[Rule]",
127 RowBox[{"9", "\[CapitalGamma]"}]}]}], "}"}]}]}]], "Input",
128 CellChangeTimes->{{3.4289418266514587`*^9, 3.4289418551565723`*^9}}],
129
130Cell[BoxData[
131 FormBox[
132 FractionBox[
133 RowBox[{"4", " ", "\[Pi]", " ",
134 SuperscriptBox["\[CapitalGamma]", "2"]}],
135 RowBox[{
136 SuperscriptBox[
137 RowBox[{"(",
138 RowBox[{"sp", "-",
139 SuperscriptBox["mw", "2"]}], ")"}], "2"], "+",
140 RowBox[{"81", " ",
141 SuperscriptBox["mw", "2"], " ",
142 SuperscriptBox["\[CapitalGamma]", "2"]}]}]], TraditionalForm]], "Output",\
143
144 CellChangeTimes->{{3.428941855531948*^9, 3.428941867874544*^9},
145 3.4289421541207457`*^9, 3.42894260090372*^9}]
146}, Open ]],
147
148Cell[TextData[{
149 "In the narrow width approximation, using\n\n\[Delta](x) = lim ",
150 Cell[BoxData[
151 FormBox[
152 FractionBox["1", "\[Pi]"], TraditionalForm]]],
153 " ",
154 Cell[BoxData[
155 FormBox[
156 FractionBox["\[Epsilon]",
157 RowBox[{
158 SuperscriptBox["x", "2"], "+",
159 SuperscriptBox["\[Epsilon]", "2"]}]], TraditionalForm]]],
160 "\n \[Epsilon]\[RightArrow]0\nwe have"
161}], "Text",
162 CellChangeTimes->{{3.428941885053216*^9, 3.4289420421627827`*^9}}],
163
164Cell[CellGroupData[{
165
166Cell[BoxData[
167 RowBox[{"\[Sigma]pNW", "=",
168 RowBox[{"9", "mw", " ", "\[CapitalGamma]", " ", "\[Sigma]pw", " ",
169 RowBox[{"DiracDelta", "[",
170 RowBox[{"sp", " ", "-", " ",
171 SuperscriptBox["mw", "2"]}], "]"}]}]}]], "Input",
172 CellChangeTimes->{{3.428942043694252*^9, 3.4289421458530283`*^9}, {
173 3.4289423809661303`*^9, 3.428942388280025*^9}, {3.4289426098473177`*^9,
174 3.428942609966979*^9}}],
175
176Cell[BoxData[
177 FormBox[
178 RowBox[{"9", " ", "mw", " ", "\[CapitalGamma]", " ", "\[Sigma]pw", " ",
179 RowBox[{
180 InterpretationBox["\[Delta]",
181 DiracDelta,
182 Editable->False,
183 Selectable->False], "(",
184 RowBox[{"sp", "-",
185 SuperscriptBox["mw", "2"]}], ")"}]}], TraditionalForm]], "Output",
186 CellChangeTimes->{{3.42894212937996*^9, 3.4289421574093657`*^9},
187 3.428942389290112*^9, 3.4289426110242777`*^9}]
188}, Open ]],
189
190Cell["\<\
191where \[Sigma]pw is \[Sigma]p evaluated at sp = mw^2\
192\>", "Text",
193 CellChangeTimes->{{3.4289421593269777`*^9, 3.428942176694747*^9}}]
194}, Open ]],
195
196Cell[CellGroupData[{
197
198Cell["The hadronic cross - section @ Tevatron", "Section",
199 CellChangeTimes->{{3.4289421851120443`*^9, 3.4289421961269503`*^9}}],
200
201Cell["We now fold the partonic cross - section with the pdf' s", "Text",
202 CellChangeTimes->{{3.428942204656125*^9, 3.4289422175994797`*^9}}],
203
204Cell[BoxData[{
205 RowBox[{
206 RowBox[{
207 RowBox[{"u", "[", "x_", "]"}], ":=",
208 RowBox[{"6",
209 RowBox[{
210 RowBox[{"(",
211 RowBox[{"1", "-", "x"}], ")"}], "^", "2"}]}]}],
212 ";"}], "\[IndentingNewLine]",
213 RowBox[{
214 RowBox[{
215 RowBox[{"d", "[", "x_", "]"}], ":=",
216 RowBox[{"3",
217 RowBox[{
218 RowBox[{"(",
219 RowBox[{"1", "-", "x"}], ")"}], "^", "2"}]}]}], ";"}]}], "Input",
220 CellChangeTimes->{{3.4289422228675957`*^9, 3.428942241154451*^9}}],
221
222Cell[TextData[{
223 "The limits of the integration are \n\nx1: [0, 1]\nx2: ",
224 Cell[BoxData[
225 FormBox["[", TraditionalForm]]],
226 " ",
227 Cell[BoxData[
228 FormBox[
229 RowBox[{
230 RowBox[{
231 RowBox[{
232 SuperscriptBox["mw", "2"], "/", "s"}], ",", " ", "1"}], "]"}],
233 TraditionalForm]]]
234}], "Text",
235 CellChangeTimes->{{3.428942264303419*^9, 3.428942324108612*^9}}],
236
237Cell["Integration in x1:", "Text",
238 CellChangeTimes->{{3.42894232586976*^9, 3.428942341099662*^9}}],
239
240Cell[CellGroupData[{
241
242Cell[BoxData[
243 RowBox[{
244 RowBox[{
245 RowBox[{"u", "[", "x1", "]"}],
246 RowBox[{"d", "[", "x2", "]"}], "\[Sigma]pNW"}], "/.",
247 RowBox[{"{",
248 RowBox[{"sp", "\[Rule]",
249 RowBox[{"s", " ", "x1", " ", "x2"}]}], "}"}]}]], "Input",
250 CellChangeTimes->{{3.4289431681668463`*^9, 3.4289431694285183`*^9}}],
251
252Cell[BoxData[
253 FormBox[
254 RowBox[{"162", " ", "mw", " ",
255 SuperscriptBox[
256 RowBox[{"(",
257 RowBox[{"1", "-", "x1"}], ")"}], "2"], " ",
258 SuperscriptBox[
259 RowBox[{"(",
260 RowBox[{"1", "-", "x2"}], ")"}], "2"], " ", "\[CapitalGamma]", " ",
261 "\[Sigma]pw", " ",
262 RowBox[{
263 InterpretationBox["\[Delta]",
264 DiracDelta,
265 Editable->False,
266 Selectable->False], "(",
267 RowBox[{
268 SuperscriptBox["mw", "2"], "-",
269 FractionBox["s",
270 RowBox[{"x1", " ", "x2"}]]}], ")"}]}], TraditionalForm]], "Output",
271 CellChangeTimes->{3.428943155912224*^9}]
272}, Open ]],
273
274Cell[CellGroupData[{
275
276Cell[BoxData[
277 RowBox[{"\[Sigma]", "=",
278 RowBox[{"Integrate", "[",
279 RowBox[{
280 RowBox[{
281 RowBox[{
282 RowBox[{"u", "[", "x1", "]"}],
283 RowBox[{"d", "[", "x2", "]"}], "\[Sigma]pNW"}], "/.",
284 RowBox[{"{",
285 RowBox[{"sp", "\[Rule]",
286 RowBox[{"s", " ", "x1", " ", "x2"}]}], "}"}]}], ",",
287 RowBox[{"{",
288 RowBox[{"x1", ",", "0", ",", "1"}], "}"}], ",",
289 RowBox[{"Assumptions", "\[Rule]",
290 RowBox[{
291 RowBox[{"(",
292 RowBox[{
293 RowBox[{
294 RowBox[{"mw", "^", "2"}], "/", "s"}], "<", "x2", "<", "1"}], ")"}], "&&",
295
296 RowBox[{"(",
297 RowBox[{"mw", ">", "0"}], ")"}], "&&",
298 RowBox[{"(",
299 RowBox[{"s", ">", "0"}], ")"}]}]}]}], "]"}]}]], "Input",
300 CellChangeTimes->{{3.428942342103321*^9, 3.428942375261568*^9}, {
301 3.428942406883278*^9, 3.428942462485826*^9}, {3.428942620176518*^9,
302 3.428942620422737*^9}, {3.428943124604053*^9, 3.42894321367962*^9}}],
303
304Cell[BoxData[
305 FormBox[
306 FractionBox[
307 RowBox[{"162", " ", "mw", " ",
308 SuperscriptBox[
309 RowBox[{"(",
310 RowBox[{"x2", "-", "1"}], ")"}], "2"], " ",
311 SuperscriptBox[
312 RowBox[{"(",
313 RowBox[{
314 SuperscriptBox["mw", "2"], "-",
315 RowBox[{"s", " ", "x2"}]}], ")"}], "2"], " ", "\[CapitalGamma]", " ",
316 "\[Sigma]pw"}],
317 RowBox[{
318 SuperscriptBox["s", "3"], " ",
319 SuperscriptBox["x2", "3"]}]], TraditionalForm]], "Output",
320 CellChangeTimes->{
321 3.428942400145671*^9, {3.4289424466850643`*^9, 3.428942465656705*^9},
322 3.4289425295180407`*^9, 3.428942621257402*^9, 3.428942755459882*^9,
323 3.4289429048806868`*^9, 3.428943215227071*^9}]
324}, Open ]],
325
326Cell["Integration in x2:", "Text",
327 CellChangeTimes->{{3.42894232586976*^9, 3.428942341099662*^9}, {
328 3.428942481054343*^9, 3.428942481204427*^9}}],
329
330Cell[CellGroupData[{
331
332Cell[BoxData[
333 RowBox[{"\[Sigma]", "=",
334 RowBox[{
335 RowBox[{
336 RowBox[{"Integrate", "[",
337 RowBox[{"\[Sigma]", ",",
338 RowBox[{"{",
339 RowBox[{"x2", ",",
340 RowBox[{
341 RowBox[{"mw", "^", "2"}], "/", "s"}], ",", "1"}], "}"}]}], "]"}], "//",
342 "Expand"}], "//", "Simplify"}]}]], "Input",
343 CellChangeTimes->{{3.428942483394573*^9, 3.42894252477667*^9}, {
344 3.4289427584498053`*^9, 3.4289427585792522`*^9}, {3.428942966307247*^9,
345 3.428942997247208*^9}, {3.428943223022503*^9, 3.428943273216826*^9}}],
346
347Cell[BoxData[
348 FormBox[
349 RowBox[{"-",
350 FractionBox[
351 RowBox[{
352 "162", " ", "mw", " ", "\[CapitalGamma]", " ", "\[Sigma]pw", " ",
353 RowBox[{"(",
354 RowBox[{
355 RowBox[{
356 RowBox[{"-", "3"}], " ",
357 SuperscriptBox["mw", "4"]}], "+",
358 RowBox[{"3", " ",
359 SuperscriptBox["s", "2"]}], "+",
360 RowBox[{
361 RowBox[{"(",
362 RowBox[{
363 SuperscriptBox["mw", "4"], "+",
364 RowBox[{"4", " ", "s", " ",
365 SuperscriptBox["mw", "2"]}], "+",
366 SuperscriptBox["s", "2"]}], ")"}], " ",
367 RowBox[{"log", "(",
368 FractionBox[
369 SuperscriptBox["mw", "2"], "s"], ")"}]}]}], ")"}]}],
370 SuperscriptBox["s", "3"]]}], TraditionalForm]], "Output",
371 CellChangeTimes->{{3.428942519914587*^9, 3.428942553721361*^9},
372 3.428942628541798*^9, 3.428942765391282*^9, 3.428942937050954*^9,
373 3.428942998707487*^9, {3.428943229928903*^9, 3.428943278718606*^9}}]
374}, Open ]],
375
376Cell[CellGroupData[{
377
378Cell[BoxData[
379 RowBox[{"\[Sigma]pw", "=",
380 RowBox[{"\[Sigma]p", "/.",
381 RowBox[{"sp", "\[Rule]",
382 RowBox[{"mw", "^", "2"}]}]}]}]], "Input",
383 CellChangeTimes->{{3.428943280891795*^9, 3.428943298023265*^9}}],
384
385Cell[BoxData[
386 FormBox[
387 FractionBox[
388 RowBox[{"4", " ", "\[Pi]"}],
389 RowBox[{"81", " ",
390 SuperscriptBox["mw", "2"]}]], TraditionalForm]], "Output",
391 CellChangeTimes->{{3.428943298212599*^9, 3.4289433089075317`*^9}}]
392}, Open ]],
393
394Cell[CellGroupData[{
395
396Cell[BoxData[
397 RowBox[{
398 RowBox[{"\[Sigma]", "//", "Expand"}], "//", "Simplify"}]], "Input",
399 CellChangeTimes->{{3.42894331467972*^9, 3.428943330039212*^9}}],
400
401Cell[BoxData[
402 FormBox[
403 RowBox[{"-",
404 FractionBox[
405 RowBox[{"8", " ", "\[Pi]", " ", "\[CapitalGamma]", " ",
406 RowBox[{"(",
407 RowBox[{
408 RowBox[{
409 RowBox[{"-", "3"}], " ",
410 SuperscriptBox["mw", "4"]}], "+",
411 RowBox[{"3", " ",
412 SuperscriptBox["s", "2"]}], "+",
413 RowBox[{
414 RowBox[{"(",
415 RowBox[{
416 SuperscriptBox["mw", "4"], "+",
417 RowBox[{"4", " ", "s", " ",
418 SuperscriptBox["mw", "2"]}], "+",
419 SuperscriptBox["s", "2"]}], ")"}], " ",
420 RowBox[{"log", "(",
421 FractionBox[
422 SuperscriptBox["mw", "2"], "s"], ")"}]}]}], ")"}]}],
423 RowBox[{"mw", " ",
424 SuperscriptBox["s", "3"]}]]}], TraditionalForm]], "Output",
425 CellChangeTimes->{{3.428943318260807*^9, 3.42894333060226*^9}}]
426}, Open ]]
427}, Open ]]
428}, Open ]]
429},
430WindowSize->{1280, 683},
431WindowMargins->{{0, Automatic}, {0, Automatic}},
432ShowSelection->True,
433Magnification->1.5,
434FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)",
435StyleDefinitions->"Default.nb"
436]
437(* End of Notebook Content *)
438
439(* Internal cache information *)
440(*CellTagsOutline
441CellTagsIndex->{}
442*)
443(*CellTagsIndex
444CellTagsIndex->{}
445*)
446(*NotebookFileOutline
447Notebook[{
448Cell[CellGroupData[{
449Cell[590, 23, 251, 5, 185, "Title"],
450Cell[CellGroupData[{
451Cell[866, 32, 36, 0, 51, "Subsection"],
452Cell[905, 34, 149, 3, 40, "Input"]
453}, Open ]],
454Cell[CellGroupData[{
455Cell[1091, 42, 113, 1, 100, "Section"],
456Cell[1207, 45, 531, 12, 39, "Text"],
457Cell[CellGroupData[{
458Cell[1763, 61, 574, 15, 40, "Input"],
459Cell[2340, 78, 573, 17, 75, "Output"]
460}, Open ]],
461Cell[2928, 98, 469, 15, 287, "Text"],
462Cell[CellGroupData[{
463Cell[3422, 117, 444, 10, 40, "Input"],
464Cell[3869, 129, 511, 15, 79, "Output"]
465}, Open ]],
466Cell[4395, 147, 462, 14, 144, "Text"],
467Cell[CellGroupData[{
468Cell[4882, 165, 403, 8, 49, "Input"],
469Cell[5288, 175, 427, 11, 49, "Output"]
470}, Open ]],
471Cell[5730, 189, 144, 3, 39, "Text"]
472}, Open ]],
473Cell[CellGroupData[{
474Cell[5911, 197, 128, 1, 100, "Section"],
475Cell[6042, 200, 140, 1, 39, "Text"],
476Cell[6185, 203, 464, 16, 66, "Input"],
477Cell[6652, 221, 365, 13, 115, "Text"],
478Cell[7020, 236, 99, 1, 39, "Text"],
479Cell[CellGroupData[{
480Cell[7144, 241, 304, 8, 40, "Input"],
481Cell[7451, 251, 584, 19, 59, "Output"]
482}, Open ]],
483Cell[CellGroupData[{
484Cell[8072, 275, 948, 26, 64, "Input"],
485Cell[9023, 303, 682, 19, 75, "Output"]
486}, Open ]],
487Cell[9720, 325, 148, 2, 39, "Text"],
488Cell[CellGroupData[{
489Cell[9893, 331, 532, 13, 40, "Input"],
490Cell[10428, 346, 954, 26, 83, "Output"]
491}, Open ]],
492Cell[CellGroupData[{
493Cell[11419, 377, 213, 5, 40, "Input"],
494Cell[11635, 384, 225, 6, 67, "Output"]
495}, Open ]],
496Cell[CellGroupData[{
497Cell[11897, 395, 158, 3, 40, "Input"],
498Cell[12058, 400, 815, 24, 83, "Output"]
499}, Open ]]
500}, Open ]]
501}, Open ]]
502}
503]
504*)
505
506(* End of internal cache information *)