Context Navigation

Back to BUSSTEPP

BUSSTEPP: exercises.nb

File exercises.nb, 228.7 KB (added by Fabio Maltoni, 7 years ago)

Line
1	(* Content-type: application/vnd.wolfram.mathematica *)
2
3	(* Wolfram Notebook File *)
4	(* http://www.wolfram.com/nb *)
5
6	(* CreatedBy='Mathematica 10.4' *)
7
8	(CacheID: 234)
9	(* Internal cache information:
10	NotebookFileLineBreakTest
11	NotebookFileLineBreakTest
12	NotebookDataPosition[ 158, 7]
13	NotebookDataLength[ 234051, 7934]
14	NotebookOptionsPosition[ 230355, 7801]
15	NotebookOutlinePosition[ 230715, 7817]
16	CellTagsIndexPosition[ 230672, 7814]
17	WindowFrame->Normal*)
18
19	(* Beginning of Notebook Content *)
20	Notebook[{
21	Cell[BoxData["\[AliasDelimiter]"], "Input",
22	CellChangeTimes->{3.693214928496599*^9}],
23
24	Cell[CellGroupData[{
25
26	Cell["BUSSTEP 2018: Exercises for QCD and collider physics", "Title",
27	CellChangeTimes->{{3.6926133610426397`^9, 3.6926133940467367`^9}, {
28	3.74436138565263^9, 3.7443613983705072`^9}}],
29
30	Cell[CellGroupData[{
31
32	Cell["Input FeynCalc (change to your installation)", "Subsection",
33	PageWidth->PaperWidth,
34	CellChangeTimes->{{3.692943673110405^9, 3.692943679690876^9}}],
35
36	Cell[BoxData[
37	RowBox[{"(*",
38	RowBox[{"<<", "HighEnergyPhysics`fc`"}], "*)"}]], "Input",
39	PageWidth->PaperWidth,
40	CellChangeTimes->{{3.692943682731015^9, 3.6929436847790537`^9}}],
41
42	Cell[BoxData[
43	RowBox[{"<<",
44	"\"\</Users/marcozaro/Physics/feyncalc-master/FeynCalc/fc.m\>\""}]], "Input"]
45	}, Open ]],
46
47	Cell[CellGroupData[{
48
49	Cell["2.2.1: 4-gluon vertex", "Chapter",
50	CellChangeTimes->{{3.692613426285159^9, 3.6926134324672422`^9}, {
51	3.6926219265062923`^9, 3.69262192669018^9}}],
52
53	Cell[CellGroupData[{
54
55	Cell["Verteces and Propagators", "Subsection",
56	PageWidth->PaperWidth],
57
58	Cell[TextData[{
59	StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \
60	outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ",
61	FontSize->14],
62	Cell[BoxData[
63	FormBox[
64	SubscriptBox["g", "s"], TraditionalForm]],
65	FontSize->14],
66	StyleBox[" ) (\[ImaginaryI] ",
67	FontSize->14],
68	Cell[BoxData[
69	FormBox[
70	SuperscriptBox["f", "abc"], TraditionalForm]],
71	FontSize->14],
72	StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ",
73	FontSize->14],
74	Cell[BoxData[
75	FormBox[
76	SubscriptBox["g", "s"], TraditionalForm]],
77	FontSize->14],
78	")",
79	StyleBox[" ",
80	FontSize->14],
81	Cell[BoxData[
82	FormBox[
83	SubscriptBox[
84	RowBox[{"(",
85	SuperscriptBox["T", "a"], ")"}], "ij"], TraditionalForm]],
86	FontSize->14],
87	StyleBox[" ",
88	FontSize->14],
89	Cell[BoxData[
90	FormBox[
91	SuperscriptBox["\[Gamma]", "\[Mu]"], TraditionalForm]],
92	FontSize->14],
93	StyleBox["\nGluon Propagator= ",
94	FontSize->14],
95	Cell[BoxData[
96	FormBox[
97	FractionBox[
98	StyleBox[
99	RowBox[{
100	RowBox[{"-", "\[ImaginaryI]"}], " ",
101	SuperscriptBox["g", "\[Mu]\[Nu]"]}],
102	FontSize->16],
103	SuperscriptBox["p", "2"]], TraditionalForm]],
104	FontSize->14],
105	"\nQuark ",
106	StyleBox["Propagator= ",
107	FontSize->14],
108	Cell[BoxData[
109	FormBox[
110	FractionBox[
111	StyleBox[
112	RowBox[{"\[ImaginaryI]", " "}],
113	FontSize->16],
114	OverscriptBox["p", "^"]], TraditionalForm]],
115	FontSize->14]
116	}], "Text",
117	PageWidth->PaperWidth],
118
119	Cell[BoxData[{
120	RowBox[{
121	RowBox[{
122	RowBox[{"GGG", "[",
123	RowBox[{
124	"p1_", ",", "p2_", ",", "p3_", ",", "m1_", ",", "m2_", ",", "m3_"}],
125	"]"}], ":=",
126	RowBox[{
127	RowBox[{
128	RowBox[{"FourVector", "[",
129	RowBox[{
130	RowBox[{"p1", "-", "p2"}], ",", "m3"}], "]"}], " ",
131	RowBox[{"MT", "[",
132	RowBox[{"m1", ",", "m2"}], "]"}]}], "+",
133	RowBox[{
134	RowBox[{"FourVector", "[",
135	RowBox[{
136	RowBox[{"p2", "-", "p3"}], ",", "m1"}], "]"}], " ",
137	RowBox[{"MT", "[",
138	RowBox[{"m2", ",", "m3"}], "]"}]}], "+",
139	RowBox[{
140	RowBox[{"FourVector", "[",
141	RowBox[{
142	RowBox[{"p3", "-", "p1"}], ",", "m2"}], "]"}], " ",
143	RowBox[{"MT", "[",
144	RowBox[{"m1", ",", "m3"}], "]"}]}]}]}], ";"}], "\[IndentingNewLine]",
145	RowBox[{
146	RowBox[{"PropQuark", "=", "I"}], ";"}], "\[IndentingNewLine]",
147	RowBox[{
148	RowBox[{
149	RowBox[{"PropGluon", "[",
150	RowBox[{"p_", ",", "mu_", ",", "nu_"}], "]"}], " ", ":=",
151	RowBox[{
152	RowBox[{"I",
153	RowBox[{
154	RowBox[{"(",
155	RowBox[{
156	RowBox[{"-",
157	RowBox[{"MT", "[",
158	RowBox[{"mu", ",", "nu"}], "]"}]}], " ", "+", " ",
159	RowBox[{
160	RowBox[{"(",
161	RowBox[{"1", "-", "\[Xi]"}], ")"}],
162	RowBox[{"FV", "[",
163	RowBox[{"p", ",", "mu"}], "]"}],
164	RowBox[{
165	RowBox[{"FV", "[",
166	RowBox[{"p", ",", "nu"}], "]"}], "/",
167	RowBox[{"SP", "[", "p", "]"}]}]}]}], ")"}], "/",
168	RowBox[{"SP", "[", "p", "]"}]}]}], "//", "ExpandScalarProduct"}]}],
169	";"}], "\[IndentingNewLine]",
170	RowBox[{
171	RowBox[{"vtx", "=",
172	RowBox[{
173	RowBox[{"-", "I"}], " ", "gs"}]}], ";"}]}], "Input",
174	PageWidth->PaperWidth,
175	CellChangeTimes->{{3.692613545298967^9, 3.692613576234826^9}, {
176	3.692615113418083^9, 3.69261518221594^9}, {3.692615289413321*^9,
177	3.6926152953216248`^9}, {3.692619057332005^9, 3.692619059218863*^9}}]
178	}, Open ]],
179
180	Cell[CellGroupData[{
181
182	Cell["Scalar products (p1 p2 > p3 p4)", "Subsection",
183	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}}],
184
185	Cell[BoxData[{
186	RowBox[{
187	RowBox[{
188	RowBox[{"ScalarProduct", "[",
189	RowBox[{"p1", ",", "p1"}], "]"}], " ", "=", " ", "0"}],
190	";"}], "\[IndentingNewLine]",
191	RowBox[{
192	RowBox[{
193	RowBox[{"ScalarProduct", "[",
194	RowBox[{"p2", ",", "p2"}], "]"}], " ", "=", " ", "0"}],
195	";"}], "\[IndentingNewLine]",
196	RowBox[{
197	RowBox[{
198	RowBox[{"ScalarProduct", "[",
199	RowBox[{"p3", ",", "p3"}], "]"}], " ", "=", " ", "0"}],
200	";"}], "\[IndentingNewLine]",
201	RowBox[{
202	RowBox[{
203	RowBox[{
204	RowBox[{"ScalarProduct", "[",
205	RowBox[{"p4", ",", "p4"}], "]"}], " ", "=", " ", "0"}], ";"}],
206	"\[IndentingNewLine]"}], "\[IndentingNewLine]",
207	RowBox[{
208	RowBox[{
209	RowBox[{"ScalarProduct", "[",
210	RowBox[{"p1", ",", "e1"}], "]"}], " ", "=", " ", "0"}],
211	";"}], "\[IndentingNewLine]",
212	RowBox[{
213	RowBox[{
214	RowBox[{"ScalarProduct", "[",
215	RowBox[{"p2", ",", "e2"}], "]"}], " ", "=", " ", "0"}],
216	";"}], "\[IndentingNewLine]",
217	RowBox[{
218	RowBox[{
219	RowBox[{"ScalarProduct", "[",
220	RowBox[{"p3", ",", "e3"}], "]"}], " ", "=", " ", "0"}],
221	";"}], "\[IndentingNewLine]",
222	RowBox[{
223	RowBox[{
224	RowBox[{
225	RowBox[{"ScalarProduct", "[",
226	RowBox[{"p4", ",", "e4"}], "]"}], " ", "=", " ", "0"}], ";"}],
227	"\[IndentingNewLine]"}], "\[IndentingNewLine]",
228	RowBox[{
229	RowBox[{
230	RowBox[{"ScalarProduct", "[",
231	RowBox[{"p1", ",", "p2"}], "]"}], " ", "=", " ",
232	RowBox[{"s", "/", "2"}]}], ";"}], "\[IndentingNewLine]",
233	RowBox[{
234	RowBox[{
235	RowBox[{"ScalarProduct", "[",
236	RowBox[{"p3", ",", "p4"}], "]"}], " ", "=", " ",
237	RowBox[{"s", "/", "2"}]}], ";"}], "\[IndentingNewLine]",
238	RowBox[{
239	RowBox[{
240	RowBox[{"ScalarProduct", "[",
241	RowBox[{"p1", ",", "p3"}], "]"}], " ", "=", " ",
242	RowBox[{
243	RowBox[{"-", "t"}], "/", "2"}]}], ";"}], "\[IndentingNewLine]",
244	RowBox[{
245	RowBox[{
246	RowBox[{"ScalarProduct", "[",
247	RowBox[{"p2", ",", "p4"}], "]"}], " ", "=", " ",
248	RowBox[{
249	RowBox[{"-", "t"}], "/", "2"}]}], ";"}], "\[IndentingNewLine]",
250	RowBox[{
251	RowBox[{
252	RowBox[{"ScalarProduct", "[",
253	RowBox[{"p1", ",", "p4"}], "]"}], " ", "=", " ",
254	RowBox[{
255	RowBox[{"-", "u"}], "/", "2"}]}], ";"}], "\[IndentingNewLine]",
256	RowBox[{
257	RowBox[{
258	RowBox[{"ScalarProduct", "[",
259	RowBox[{"p2", ",", "p3"}], "]"}], " ", "=", " ",
260	RowBox[{
261	RowBox[{"-", "u"}], "/", "2"}]}], ";"}]}], "Input",
262	CellChangeTimes->{{3.692613638907701^9, 3.692613661866912^9}, {
263	3.692616305122611^9, 3.692616330720419^9}, {3.6926163755266438`*^9,
264	3.692616402595954^9}, {3.692616457656052^9, 3.692616495830961*^9}, {
265	3.692616545674967^9, 3.692616651884303^9}, {3.6926167096078253`*^9,
266	3.692616713317981^9}, {3.692616874648031^9, 3.692616886428197*^9}, {
267	3.692616941697814^9, 3.6926169483044987`^9}, {3.692617048978189*^9,
268	3.6926170523196774`^9}, 3.692940715985712^9, {3.692941530040926*^9,
269	3.692941559445057*^9}}],
270
271	Cell[BoxData[
272	RowBox[{"(*", " ",
273	RowBox[{
274	"check", " ", "that", " ", "everything", " ", "is", " ", "consistent"}],
275	"*)"}]], "Input",
276	CellChangeTimes->{{3.692941543394758^9, 3.69294155349966^9},
277	3.69312402160058*^9}],
278
279	Cell[CellGroupData[{
280
281	Cell[BoxData[
282	RowBox[{
283	RowBox[{
284	RowBox[{"SP", "[",
285	RowBox[{"p1", "+", "p2", "-", "p3", "-", "p4"}], "]"}], "//",
286	"ExpandScalarProduct"}], "//", "Simplify"}]], "Input",
287	CellChangeTimes->{{3.692614045713563^9, 3.692614059870042^9}, {
288	3.6926146056833763`^9, 3.692614611937336^9}}],
289
290	Cell[BoxData[
291	FormBox[
292	RowBox[{"2", " ",
293	RowBox[{"(",
294	RowBox[{"s", "+", "t", "+", "u"}], ")"}]}], TraditionalForm]], "Output",
295	CellChangeTimes->{{3.692614052820998^9, 3.692614060801106^9},
296	3.692614612344434^9, 3.692615217236927^9, 3.6926153042645206`*^9,
297	3.692616414235757^9, {3.692616477002037^9, 3.692616505938065*^9},
298	3.692616661409111^9, 3.692616726099124^9, 3.69261689455098*^9,
299	3.692616957471612^9, 3.692617062422418^9, 3.6926172366788588`*^9,
300	3.692617355272015^9, 3.692617930682822^9, 3.692619069069003*^9,
301	3.6926220577810163`^9, 3.692939282165938^9, 3.6929407310064096`*^9,
302	3.692941450708425^9, 3.693124017639719^9, 3.693124176364273*^9}]
303	}, Open ]]
304	}, Open ]],
305
306	Cell[CellGroupData[{
307
308	Cell["\<\
309	Let us re-derive the case for q(p2) qx(p1) > g g. Excluding the 3-gluon \
310	vertex, there are two diagrams\
311	\>", "Subsection",
312	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
313	3.6926137866433907`^9, 3.69261380584085^9}, {3.692939027863969*^9,
314	3.692939066178437^9}, {3.692939656978591^9, 3.692939660888845*^9}}],
315
316	Cell[CellGroupData[{
317
318	Cell[BoxData[{
319	RowBox[{
320	RowBox[{"diag1", " ", "=", " ",
321	RowBox[{
322	RowBox[{
323	RowBox[{
324	RowBox[{"SpinorV", "[", "p1", "]"}], ".",
325	RowBox[{"GA", "[", "mu3", "]"}], ".",
326	RowBox[{"GS", "[",
327	RowBox[{"p1", "-", "p3"}], "]"}], ".",
328	RowBox[{"GA", "[", "mu4", "]"}], ".",
329	RowBox[{"SpinorU", "[", "p2", "]"}]}], " ",
330	RowBox[{
331	RowBox[{"SUNT", "[", "a3", "]"}], ".",
332	RowBox[{"SUNT", "[", "a4", "]"}]}],
333	RowBox[{"1", "/",
334	RowBox[{"SP", "[",
335	RowBox[{"p1", "-", "p3"}], "]"}]}]}], "//", "ExpandScalarProduct"}]}],
336	" "}], "\[IndentingNewLine]",
337	RowBox[{"diag2", " ", "=", " ",
338	RowBox[{
339	RowBox[{
340	RowBox[{
341	RowBox[{"SpinorV", "[", "p1", "]"}], ".",
342	RowBox[{"GA", "[", "mu4", "]"}], ".",
343	RowBox[{"GS", "[",
344	RowBox[{"p2", "-", "p3"}], "]"}], ".",
345	RowBox[{"GA", "[", "mu3", "]"}], ".",
346	RowBox[{"SpinorU", "[", "p2", "]"}]}], " ",
347	RowBox[{
348	RowBox[{"SUNT", "[", "a4", "]"}], ".",
349	RowBox[{"SUNT", "[", "a3", "]"}]}],
350	RowBox[{"1", "/",
351	RowBox[{"SP", "[",
352	RowBox[{"p1", "-", "p4"}], "]"}]}]}], "//",
353	"ExpandScalarProduct"}]}]}], "Input",
354	CellChangeTimes->{{3.6929390728150377`^9, 3.692939081998135^9}, {
355	3.692939120935576^9, 3.692939317921928^9}, {3.692939463980194*^9,
356	3.6929395071402483`^9}, {3.6931240307916517`^9, 3.6931240351412783`*^9}}],
357
358	Cell[BoxData[
359	FormBox[
360	FractionBox[
361	RowBox[{
362	RowBox[{
363	SuperscriptBox["T",
364	FormBox[
365	FormBox["a3",
366	TraditionalForm],
367	TraditionalForm]], ".",
368	SuperscriptBox["T",
369	FormBox[
370	FormBox["a4",
371	TraditionalForm],
372	TraditionalForm]]}], " ",
373	RowBox[{
374	RowBox[{"(",
375	RowBox[{
376	FormBox["\<\"\[CurlyPhi]\"\>",
377	TraditionalForm],
378	FormBox["\<\"(\"\>",
379	TraditionalForm],
380	FormBox[
381	OverscriptBox[
382	FormBox["p1",
383	TraditionalForm], "_"],
384	TraditionalForm],
385	FormBox["\<\")\"\>",
386	TraditionalForm]}], ")"}], ".",
387	SuperscriptBox[
388	OverscriptBox["\[Gamma]", "_"],
389	FormBox[
390	FormBox["mu3",
391	TraditionalForm],
392	TraditionalForm]], ".",
393	RowBox[{"(",
394	RowBox[{
395	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]", "(",
396	FormBox[
397	FormBox[
398	RowBox[{
399	OverscriptBox[
400	FormBox["p1",
401	TraditionalForm], "_"], "-",
402	OverscriptBox[
403	FormBox["p3",
404	TraditionalForm], "_"]}],
405	TraditionalForm],
406	TraditionalForm], ")"}], ")"}], ".",
407	SuperscriptBox[
408	OverscriptBox["\[Gamma]", "_"],
409	FormBox[
410	FormBox["mu4",
411	TraditionalForm],
412	TraditionalForm]], ".",
413	RowBox[{"(",
414	RowBox[{
415	FormBox["\<\"\[CurlyPhi]\"\>",
416	TraditionalForm],
417	FormBox["\<\"(\"\>",
418	TraditionalForm],
419	FormBox[
420	OverscriptBox[
421	FormBox["p2",
422	TraditionalForm], "_"],
423	TraditionalForm],
424	FormBox["\<\")\"\>",
425	TraditionalForm]}], ")"}]}]}], "t"], TraditionalForm]], "Output",
426	CellChangeTimes->{3.693124035741987^9, 3.6931241763941708`^9}],
427
428	Cell[BoxData[
429	FormBox[
430	FractionBox[
431	RowBox[{
432	RowBox[{
433	SuperscriptBox["T",
434	FormBox[
435	FormBox["a4",
436	TraditionalForm],
437	TraditionalForm]], ".",
438	SuperscriptBox["T",
439	FormBox[
440	FormBox["a3",
441	TraditionalForm],
442	TraditionalForm]]}], " ",
443	RowBox[{
444	RowBox[{"(",
445	RowBox[{
446	FormBox["\<\"\[CurlyPhi]\"\>",
447	TraditionalForm],
448	FormBox["\<\"(\"\>",
449	TraditionalForm],
450	FormBox[
451	OverscriptBox[
452	FormBox["p1",
453	TraditionalForm], "_"],
454	TraditionalForm],
455	FormBox["\<\")\"\>",
456	TraditionalForm]}], ")"}], ".",
457	SuperscriptBox[
458	OverscriptBox["\[Gamma]", "_"],
459	FormBox[
460	FormBox["mu4",
461	TraditionalForm],
462	TraditionalForm]], ".",
463	RowBox[{"(",
464	RowBox[{
465	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]", "(",
466	FormBox[
467	FormBox[
468	RowBox[{
469	OverscriptBox[
470	FormBox["p2",
471	TraditionalForm], "_"], "-",
472	OverscriptBox[
473	FormBox["p3",
474	TraditionalForm], "_"]}],
475	TraditionalForm],
476	TraditionalForm], ")"}], ")"}], ".",
477	SuperscriptBox[
478	OverscriptBox["\[Gamma]", "_"],
479	FormBox[
480	FormBox["mu3",
481	TraditionalForm],
482	TraditionalForm]], ".",
483	RowBox[{"(",
484	RowBox[{
485	FormBox["\<\"\[CurlyPhi]\"\>",
486	TraditionalForm],
487	FormBox["\<\"(\"\>",
488	TraditionalForm],
489	FormBox[
490	OverscriptBox[
491	FormBox["p2",
492	TraditionalForm], "_"],
493	TraditionalForm],
494	FormBox["\<\")\"\>",
495	TraditionalForm]}], ")"}]}]}], "u"], TraditionalForm]], "Output",
496	CellChangeTimes->{3.693124035741987^9, 3.693124176399372^9}]
497	}, Open ]]
498	}, Open ]],
499
500	Cell[CellGroupData[{
501
502	Cell["\<\
503	We check gauge invariance by contracting the amplitude with the gluon \
504	momentum (p3) instead of the polarisation e3\
505	\>", "Subsection",
506	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
507	3.6926137866433907`^9, 3.69261380584085^9}, {3.692939027863969*^9,
508	3.692939066178437^9}, {3.692939656978591^9, 3.692939660888845*^9}, {
509	3.692941076605266^9, 3.692941138816452^9}, {3.692941201395165*^9,
510	3.692941212146483^9}, {3.692941490672728^9, 3.692941504694332*^9}, {
511	3.692943492248446^9, 3.692943559761331^9}}],
512
513	Cell[CellGroupData[{
514
515	Cell[BoxData[
516	RowBox[{"Ampqqgg", "=",
517	RowBox[{
518	RowBox[{
519	RowBox[{
520	RowBox[{
521	RowBox[{"(",
522	RowBox[{"diag1", "+", "diag2"}], ")"}], " ",
523	RowBox[{"FV", "[",
524	RowBox[{"p3", ",", "mu3"}], "]"}], " ",
525	RowBox[{"FV", "[",
526	RowBox[{"e4", ",", "mu4"}], "]"}]}], "//", "Contract"}], "//",
527	"Simplify"}], "//", "DiracSimplify2"}]}]], "Input",
528	CellChangeTimes->{{3.6929393019278107`^9, 3.692939408564536^9},
529	3.6929394479749527`*^9}],
530
531	Cell[BoxData[
532	FormBox[
533	RowBox[{
534	RowBox[{"-",
535	RowBox[{
536	SuperscriptBox["T",
537	FormBox[
538	FormBox["a3",
539	TraditionalForm],
540	TraditionalForm]], ".",
541	SuperscriptBox["T",
542	FormBox[
543	FormBox["a4",
544	TraditionalForm],
545	TraditionalForm]], ".",
546	RowBox[{"(",
547	RowBox[{
548	FormBox["\<\"\[CurlyPhi]\"\>",
549	TraditionalForm],
550	FormBox["\<\"(\"\>",
551	TraditionalForm],
552	FormBox[
553	OverscriptBox[
554	FormBox["p1",
555	TraditionalForm], "_"],
556	TraditionalForm],
557	FormBox["\<\")\"\>",
558	TraditionalForm]}], ")"}], ".",
559	RowBox[{"(",
560	RowBox[{
561	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
562	FormBox[
563	OverscriptBox[
564	FormBox["e4",
565	TraditionalForm], "_"],
566	TraditionalForm]}], ")"}], ".",
567	RowBox[{"(",
568	RowBox[{
569	FormBox["\<\"\[CurlyPhi]\"\>",
570	TraditionalForm],
571	FormBox["\<\"(\"\>",
572	TraditionalForm],
573	FormBox[
574	OverscriptBox[
575	FormBox["p2",
576	TraditionalForm], "_"],
577	TraditionalForm],
578	FormBox["\<\")\"\>",
579	TraditionalForm]}], ")"}]}]}], "-",
580	RowBox[{
581	SuperscriptBox["T",
582	FormBox[
583	FormBox["a4",
584	TraditionalForm],
585	TraditionalForm]], ".",
586	SuperscriptBox["T",
587	FormBox[
588	FormBox["a3",
589	TraditionalForm],
590	TraditionalForm]], ".",
591	RowBox[{"(",
592	RowBox[{
593	FormBox["\<\"\[CurlyPhi]\"\>",
594	TraditionalForm],
595	FormBox["\<\"(\"\>",
596	TraditionalForm],
597	FormBox[
598	OverscriptBox[
599	FormBox["p1",
600	TraditionalForm], "_"],
601	TraditionalForm],
602	FormBox["\<\")\"\>",
603	TraditionalForm]}], ")"}], ".",
604	RowBox[{"(",
605	RowBox[{
606	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
607	FormBox[
608	OverscriptBox[
609	FormBox["e4",
610	TraditionalForm], "_"],
611	TraditionalForm]}], ")"}], ".",
612	RowBox[{"(",
613	RowBox[{
614	FormBox["\<\"\[CurlyPhi]\"\>",
615	TraditionalForm],
616	FormBox["\<\"(\"\>",
617	TraditionalForm],
618	FormBox[
619	OverscriptBox[
620	FormBox["p2",
621	TraditionalForm], "_"],
622	TraditionalForm],
623	FormBox["\<\")\"\>",
624	TraditionalForm]}], ")"}]}]}], TraditionalForm]], "Output",
625	CellChangeTimes->{{3.692939364247861^9, 3.69293937171659^9},
626	3.692939409260255^9, {3.692939448343082^9, 3.692939514106633*^9},
627	3.6929407311426497`^9, 3.692941450846944^9, 3.6931241765470257`*^9}]
628	}, Open ]]
629	}, Open ]],
630
631	Cell[CellGroupData[{
632
633	Cell["\<\
634	Let us try to build the 3-glu vertex starting from this result, uncontracting \
635	some Lorentz indexes \
636	\>", "Subsection",
637	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
638	3.6926137866433907`^9, 3.69261380584085^9}, {3.692939027863969*^9,
639	3.692939066178437^9}, {3.692939656978591^9, 3.692939660888845*^9}, {
640	3.692941076605266^9, 3.692941138816452^9}, {3.692941201395165*^9,
641	3.692941212146483^9}, {3.692941490672728^9, 3.692941504694332*^9}}],
642
643	Cell[CellGroupData[{
644
645	Cell[BoxData[
646	RowBox[{"diag3", " ", "=", " ",
647	RowBox[{
648	RowBox[{
649	RowBox[{
650	RowBox[{
651	RowBox[{"SpinorV", "[", "p1", "]"}], ".",
652	RowBox[{"GA", "[", "mu", "]"}], ".",
653	RowBox[{"SpinorU", "[", "p2", "]"}]}],
654	RowBox[{"(",
655	RowBox[{
656	RowBox[{"MT", "[",
657	RowBox[{"mu", ",", "nua"}], "]"}],
658	RowBox[{"1", "/",
659	RowBox[{"SP", "[",
660	RowBox[{"p1", "+", "p2"}], "]"}]}]}], ")"}],
661	RowBox[{"FV", "[",
662	RowBox[{"e4", ",", "nub"}], "]"}], " ",
663	RowBox[{"FV", "[",
664	RowBox[{"p3", ",", "nuc"}], "]"}],
665	RowBox[{"(",
666	RowBox[{
667	RowBox[{"MT", "[",
668	RowBox[{"nub", ",", "nua"}], "]"}],
669	RowBox[{"FV", "[",
670	RowBox[{
671	RowBox[{"p4", "+",
672	RowBox[{"(",
673	RowBox[{"p3", "+", "p4"}], ")"}]}], ",", "nuc"}], "]"}]}], " ",
674	")"}]}], "//", "Contract"}], "//", "DiracSimplify"}]}]], "Input",
675	CellChangeTimes->{{3.6929396422623243`^9, 3.692939748150084^9}, {
676	3.692939799828507^9, 3.692939816926909^9}, {3.6929399094098043`*^9,
677	3.692939910766464^9}, {3.692940112132062^9, 3.692940229822454*^9}, {
678	3.6929402657779408`^9, 3.692940315577467^9}, {3.692940370143971*^9,
679	3.6929404297212353`^9}, {3.692940478023863^9, 3.692940491686718*^9}, {
680	3.692940538480098^9, 3.6929406150288963`^9}, {3.692940658281687*^9,
681	3.6929406794289303`^9}, {3.692940959878302^9, 3.692940980258583*^9}, {
682	3.692941057736762^9, 3.692941059050293^9}, {3.692941180909075*^9,
683	3.692941181860361*^9}}],
684
685	Cell[BoxData[
686	FormBox[
687	RowBox[{
688	RowBox[{"(",
689	RowBox[{
690	FormBox["\<\"\[CurlyPhi]\"\>",
691	TraditionalForm],
692	FormBox["\<\"(\"\>",
693	TraditionalForm],
694	FormBox[
695	OverscriptBox[
696	FormBox["p1",
697	TraditionalForm], "_"],
698	TraditionalForm],
699	FormBox["\<\")\"\>",
700	TraditionalForm]}], ")"}], ".",
701	RowBox[{"(",
702	RowBox[{
703	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
704	FormBox[
705	OverscriptBox[
706	FormBox["e4",
707	TraditionalForm], "_"],
708	TraditionalForm]}], ")"}], ".",
709	RowBox[{"(",
710	RowBox[{
711	FormBox["\<\"\[CurlyPhi]\"\>",
712	TraditionalForm],
713	FormBox["\<\"(\"\>",
714	TraditionalForm],
715	FormBox[
716	OverscriptBox[
717	FormBox["p2",
718	TraditionalForm], "_"],
719	TraditionalForm],
720	FormBox["\<\")\"\>",
721	TraditionalForm]}], ")"}]}], TraditionalForm]], "Output",
722	CellChangeTimes->{{3.6929398058886127`^9, 3.692939817251027^9},
723	3.692939911086081^9, {3.692940190722788^9, 3.692940199758647*^9},
724	3.69294023131234^9, 3.6929404936932573`^9, {3.6929405609022093`*^9,
725	3.692940615525085^9}, {3.6929406598057537`^9, 3.692940680276863*^9},
726	3.692940731291744^9, 3.69294098067632^9, 3.6929410656314793`*^9,
727	3.692941182193736^9, 3.692941450889511^9, 3.693124176599927*^9}]
728	}, Open ]]
729	}, Open ]],
730
731	Cell[CellGroupData[{
732
733	Cell["\<\
734	This gives the correct result (bar for the color factor). We can complete it \
735	by requiring that the momentum-space dependence of the 3-glu vertex is \
736	anti-symmetric, as the color part.\
737	\>", "Subsection",
738	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
739	3.6926137866433907`^9, 3.69261380584085^9}, {3.692939027863969*^9,
740	3.692939066178437^9}, {3.692939656978591^9, 3.692939660888845*^9}, {
741	3.692941076605266^9, 3.692941138816452^9}, {3.692941201395165*^9,
742	3.692941271647245*^9}}],
743
744	Cell[CellGroupData[{
745
746	Cell[BoxData[
747	RowBox[{"diag3full", " ", "=", " ",
748	RowBox[{
749	RowBox[{
750	RowBox[{
751	RowBox[{
752	RowBox[{"SpinorV", "[", "p1", "]"}], ".",
753	RowBox[{"GA", "[", "mu", "]"}], ".",
754	RowBox[{"SpinorU", "[", "p2", "]"}]}],
755	RowBox[{"(",
756	RowBox[{
757	RowBox[{"MT", "[",
758	RowBox[{"mu", ",", "nua"}], "]"}],
759	RowBox[{"1", "/",
760	RowBox[{"SP", "[",
761	RowBox[{"p1", "+", "p2"}], "]"}]}]}], ")"}],
762	RowBox[{"FV", "[",
763	RowBox[{"e4", ",", "nub"}], "]"}], " ",
764	RowBox[{"FV", "[",
765	RowBox[{"p3", ",", "nuc"}], "]"}],
766	RowBox[{"(",
767	RowBox[{
768	RowBox[{
769	RowBox[{"MT", "[",
770	RowBox[{"nub", ",", "nua"}], "]"}],
771	RowBox[{"FV", "[",
772	RowBox[{
773	RowBox[{"p4", "+",
774	RowBox[{"(",
775	RowBox[{"p3", "+", "p4"}], ")"}]}], ",", "nuc"}], "]"}]}], " ",
776	"+", " ",
777	RowBox[{
778	RowBox[{"MT", "[",
779	RowBox[{"nua", ",", "nuc"}], "]"}], " ",
780	RowBox[{"FV", "[",
781	RowBox[{
782	RowBox[{
783	RowBox[{"-",
784	RowBox[{"(",
785	RowBox[{"p3", "+", "p4"}], ")"}]}], "-", "p3"}], ",", "nub"}],
786	"]"}]}], "\[IndentingNewLine]", "+",
787	RowBox[{
788	RowBox[{"MT", "[",
789	RowBox[{"nuc", ",", "nub"}], "]"}],
790	RowBox[{"FV", "[",
791	RowBox[{
792	RowBox[{"p3", "-", "p4"}], ",", "nua"}], "]"}]}]}], ")"}]}], "//",
793	"Contract"}], "//", "DiracSimplify"}]}]], "Input",
794	CellChangeTimes->{{3.6929396422623243`^9, 3.692939748150084^9}, {
795	3.692939799828507^9, 3.692939816926909^9}, {3.6929399094098043`*^9,
796	3.692939910766464^9}, {3.692940112132062^9, 3.692940229822454*^9}, {
797	3.6929402657779408`^9, 3.692940315577467^9}, {3.692940370143971*^9,
798	3.6929404297212353`^9}, {3.692940478023863^9, 3.692940491686718*^9}, {
799	3.692940538480098^9, 3.6929406150288963`^9}, {3.692940658281687*^9,
800	3.6929406794289303`^9}, {3.692940959878302^9, 3.692940980258583*^9}, {
801	3.692941057736762^9, 3.692941059050293^9}, {3.692941175141336*^9,
802	3.6929411755894213`*^9}}],
803
804	Cell[BoxData[
805	FormBox[
806	RowBox[{
807	RowBox[{
808	RowBox[{"(",
809	RowBox[{
810	FormBox["\<\"\[CurlyPhi]\"\>",
811	TraditionalForm],
812	FormBox["\<\"(\"\>",
813	TraditionalForm],
814	FormBox[
815	OverscriptBox[
816	FormBox["p1",
817	TraditionalForm], "_"],
818	TraditionalForm],
819	FormBox["\<\")\"\>",
820	TraditionalForm]}], ")"}], ".",
821	RowBox[{"(",
822	RowBox[{
823	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
824	FormBox[
825	OverscriptBox[
826	FormBox["e4",
827	TraditionalForm], "_"],
828	TraditionalForm]}], ")"}], ".",
829	RowBox[{"(",
830	RowBox[{
831	FormBox["\<\"\[CurlyPhi]\"\>",
832	TraditionalForm],
833	FormBox["\<\"(\"\>",
834	TraditionalForm],
835	FormBox[
836	OverscriptBox[
837	FormBox["p2",
838	TraditionalForm], "_"],
839	TraditionalForm],
840	FormBox["\<\")\"\>",
841	TraditionalForm]}], ")"}]}], "-",
842	FractionBox[
843	RowBox[{
844	RowBox[{"(",
845	RowBox[{
846	FormBox[
847	OverscriptBox[
848	FormBox["e4",
849	TraditionalForm], "_"],
850	TraditionalForm],
851	FormBox["\<\"\[CenterDot]\"\>",
852	TraditionalForm],
853	FormBox[
854	OverscriptBox[
855	FormBox["p3",
856	TraditionalForm], "_"],
857	TraditionalForm]}], ")"}], " ",
858	RowBox[{
859	RowBox[{"(",
860	RowBox[{
861	FormBox["\<\"\[CurlyPhi]\"\>",
862	TraditionalForm],
863	FormBox["\<\"(\"\>",
864	TraditionalForm],
865	FormBox[
866	OverscriptBox[
867	FormBox["p1",
868	TraditionalForm], "_"],
869	TraditionalForm],
870	FormBox["\<\")\"\>",
871	TraditionalForm]}], ")"}], ".",
872	RowBox[{"(",
873	RowBox[{
874	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
875	FormBox[
876	OverscriptBox[
877	FormBox["p4",
878	TraditionalForm], "_"],
879	TraditionalForm]}], ")"}], ".",
880	RowBox[{"(",
881	RowBox[{
882	FormBox["\<\"\[CurlyPhi]\"\>",
883	TraditionalForm],
884	FormBox["\<\"(\"\>",
885	TraditionalForm],
886	FormBox[
887	OverscriptBox[
888	FormBox["p2",
889	TraditionalForm], "_"],
890	TraditionalForm],
891	FormBox["\<\")\"\>",
892	TraditionalForm]}], ")"}]}]}], "s"], "-",
893	FractionBox[
894	RowBox[{
895	RowBox[{"(",
896	RowBox[{
897	FormBox[
898	OverscriptBox[
899	FormBox["e4",
900	TraditionalForm], "_"],
901	TraditionalForm],
902	FormBox["\<\"\[CenterDot]\"\>",
903	TraditionalForm],
904	FormBox[
905	OverscriptBox[
906	FormBox["p3",
907	TraditionalForm], "_"],
908	TraditionalForm]}], ")"}], " ",
909	RowBox[{
910	RowBox[{"(",
911	RowBox[{
912	FormBox["\<\"\[CurlyPhi]\"\>",
913	TraditionalForm],
914	FormBox["\<\"(\"\>",
915	TraditionalForm],
916	FormBox[
917	OverscriptBox[
918	FormBox["p1",
919	TraditionalForm], "_"],
920	TraditionalForm],
921	FormBox["\<\")\"\>",
922	TraditionalForm]}], ")"}], ".",
923	RowBox[{"(",
924	RowBox[{
925	OverscriptBox["\[Gamma]", "_"], "\[CenterDot]",
926	FormBox[
927	OverscriptBox[
928	FormBox["p3",
929	TraditionalForm], "_"],
930	TraditionalForm]}], ")"}], ".",
931	RowBox[{"(",
932	RowBox[{
933	FormBox["\<\"\[CurlyPhi]\"\>",
934	TraditionalForm],
935	FormBox["\<\"(\"\>",
936	TraditionalForm],
937	FormBox[
938	OverscriptBox[
939	FormBox["p2",
940	TraditionalForm], "_"],
941	TraditionalForm],
942	FormBox["\<\")\"\>",
943	TraditionalForm]}], ")"}]}]}], "s"]}], TraditionalForm]], "Output",
944	CellChangeTimes->{3.692941175997158^9, 3.692941450997813^9,
945	3.693124176732133*^9}]
946	}, Open ]]
947	}, Open ]],
948
949	Cell["\<\
950	The two extra term cancel to zero because of the Dirac eq. (summing them one \
951	gets V[p1].GS[p3+p4].U[p2], plus momentum conservation( p3+p4=p1+p2)\
952	\>", "Subsection",
953	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
954	3.6926137866433907`^9, 3.69261380584085^9}, {3.692939027863969*^9,
955	3.692939066178437^9}, {3.692939656978591^9, 3.692939660888845*^9}, {
956	3.692941076605266^9, 3.692941138816452^9}, {3.692941201395165*^9,
957	3.692941364052196^9}, {3.6931241116170073`^9, 3.69312413253623*^9}}],
958
959	Cell[CellGroupData[{
960
961	Cell["\<\
962	Let us go back to the 4-gluon case. Excluding the 4-gluon vertex, there are 3 \
963	diagrams (s, t and u channel)\
964	\>", "Subsection",
965	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
966	3.6926137866433907`^9, 3.69261380584085^9}, {3.692941372915619*^9,
967	3.692941383306806*^9}}],
968
969	Cell[BoxData[
970	RowBox[{
971	RowBox[{"(*", " ",
972	RowBox[{
973	RowBox[{
974	RowBox[{
975	RowBox[{
976	"A", " ", "bunch", " ", "of", " ", "new", " ", "scalar", " ",
977	"products"}], "..."}], " ", "\[IndentingNewLine]", "we", " ",
978	"eliminate", " ", "p4", " ", "in", " ", "the", " ", "SP", " ", "with",
979	" ", "the", " ", "polarisations", " ", "1"}], "-", "2", "-",
980	RowBox[{"3", " ", "using", " ", "momentum", " ", "conservation"}]}],
981	"*)"}], "\[IndentingNewLine]",
982	RowBox[{
983	RowBox[{
984	RowBox[{
985	RowBox[{"ScalarProduct", "[",
986	RowBox[{"p4", ",", "e1"}], "]"}], " ", "=",
987	RowBox[{
988	RowBox[{"SP", "[",
989	RowBox[{"p1", ",", "e1"}], "]"}], "+",
990	RowBox[{"SP", "[",
991	RowBox[{"p2", ",", "e1"}], "]"}], "-",
992	RowBox[{"SP", "[",
993	RowBox[{"p3", ",", "e1"}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
994	RowBox[{
995	RowBox[{
996	RowBox[{"ScalarProduct", "[",
997	RowBox[{"p4", ",", "e2"}], "]"}], " ", "=",
998	RowBox[{
999	RowBox[{"SP", "[",
1000	RowBox[{"p1", ",", "e2"}], "]"}], "+",
1001	RowBox[{"SP", "[",
1002	RowBox[{"p2", ",", "e2"}], "]"}], "-",
1003	RowBox[{"SP", "[",
1004	RowBox[{"p3", ",", "e2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
1005	RowBox[{
1006	RowBox[{
1007	RowBox[{"ScalarProduct", "[",
1008	RowBox[{"p4", ",", "e3"}], "]"}], " ", "=",
1009	RowBox[{
1010	RowBox[{"SP", "[",
1011	RowBox[{"p1", ",", "e3"}], "]"}], "+",
1012	RowBox[{"SP", "[",
1013	RowBox[{"p2", ",", "e3"}], "]"}], "-",
1014	RowBox[{"SP", "[",
1015	RowBox[{"p3", ",", "e3"}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
1016	"\[IndentingNewLine]",
1017	RowBox[{"(*", " ",
1018	RowBox[{
1019	"we", " ", "eliminate", " ", "p3", " ", "in", " ", "the", " ", "SP", " ",
1020	"with", " ", "the", " ", "polarisations", " ", "4", " ", "using", " ",
1021	"momentum", " ", "conservation"}], "*)"}], "\[IndentingNewLine]",
1022	RowBox[{
1023	RowBox[{
1024	RowBox[{"ScalarProduct", "[",
1025	RowBox[{"p3", ",", "e4"}], "]"}], "=",
1026	RowBox[{
1027	RowBox[{"SP", "[",
1028	RowBox[{"p1", ",", "e4"}], "]"}], "+",
1029	RowBox[{"SP", "[",
1030	RowBox[{"p2", ",", "e4"}], "]"}]}]}], ";"}]}]}]], "Input",
1031	CellChangeTimes->{{3.692943587077097^9, 3.6929435966042624`^9}}],
1032
1033	Cell[BoxData[{
1034	RowBox[{
1035	RowBox[{"diagS", "=", " ",
1036	RowBox[{
1037	RowBox[{
1038	RowBox[{
1039	RowBox[{"vtx", " ", "^", "2"}], " ",
1040	RowBox[{"GGG", "[",
1041	RowBox[{
1042	RowBox[{"-", "p1"}], ",", " ",
1043	RowBox[{"p1", "+", "p2"}], ",",
1044	RowBox[{"-", "p2"}], ",", " ", "mu1", ",", " ", "mu", ",", " ",
1045	"mu2"}], "]"}],
1046	RowBox[{"GGG", "[",
1047	RowBox[{
1048	RowBox[{
1049	RowBox[{"-", "p1"}], "-", "p2"}], ",", " ", "p3", ",", " ", "p4",
1050	",", " ", "nu", ",", " ", "mu3", ",", " ", "mu4"}], "]"}],
1051	RowBox[{"PropGluon", "[",
1052	RowBox[{
1053	RowBox[{"p1", "+", "p2"}], ",", " ", "mu", ",", "nu"}], "]"}],
1054	RowBox[{"(",
1055	RowBox[{
1056	RowBox[{
1057	RowBox[{"SUNF", "[",
1058	RowBox[{"a1", ",", "a", ",", "a2"}], "]"}],
1059	RowBox[{"SUNF", "[",
1060	RowBox[{"a", ",", "a3", ",", "a4"}], "]"}]}], "//", "SUNSimplify"}],
1061	")"}]}], "//", "Contract"}], " ", "//", "Simplify"}]}],
1062	";"}], "\[IndentingNewLine]",
1063	RowBox[{
1064	RowBox[{"diagT", "=", " ",
1065	RowBox[{
1066	RowBox[{
1067	RowBox[{
1068	RowBox[{"vtx", " ", "^", "2"}],
1069	RowBox[{"GGG", "[",
1070	RowBox[{
1071	RowBox[{"-", "p1"}], ",", "p3", ",",
1072	RowBox[{"p1", "-", "p3"}], ",", " ", "mu1", ",", " ", "mu3", ",", " ",
1073	"mu"}], "]"}],
1074	RowBox[{"GGG", "[",
1075	RowBox[{
1076	RowBox[{
1077	RowBox[{"-", "p1"}], "+", "p3"}], ",", " ", "p4", ",", " ",
1078	RowBox[{"-", "p2"}], ",", " ", "nu", ",", " ", "mu4", ",", " ",
1079	"mu2"}], "]"}],
1080	RowBox[{"PropGluon", "[",
1081	RowBox[{
1082	RowBox[{"p1", "-", "p3"}], ",", " ", "mu", ",", "nu"}], "]"}], " ",
1083	RowBox[{"(",
1084	RowBox[{
1085	RowBox[{
1086	RowBox[{"SUNF", "[",
1087	RowBox[{"a1", ",", "a3", ",", "a"}], "]"}],
1088	RowBox[{"SUNF", "[",
1089	RowBox[{"a", ",", "a4", ",", "a2"}], "]"}]}], "//", "SUNSimplify"}],
1090	")"}]}], "//", "Contract"}], " ", "//", "Simplify"}]}], ";",
1091	RowBox[{"diagU", "=", " ",
1092	RowBox[{
1093	RowBox[{
1094	RowBox[{
1095	RowBox[{"vtx", " ", "^", "2"}],
1096	RowBox[{"GGG", "[",
1097	RowBox[{
1098	RowBox[{"-", "p1"}], ",", "p4", ",",
1099	RowBox[{"p1", "-", "p4"}], ",", " ", "mu1", ",", " ", "mu4", ",", " ",
1100	"mu"}], "]"}],
1101	RowBox[{"GGG", "[",
1102	RowBox[{
1103	RowBox[{
1104	RowBox[{"-", "p1"}], "+", "p4"}], ",", " ", "p3", ",", " ",
1105	RowBox[{"-", "p2"}], ",", " ", "nu", ",", " ", "mu3", ",", " ",
1106	"mu2"}], "]"}],
1107	RowBox[{"PropGluon", "[",
1108	RowBox[{
1109	RowBox[{"p1", "-", "p4"}], ",", " ", "mu", ",", "nu"}], "]"}],
1110	RowBox[{"(",
1111	RowBox[{
1112	RowBox[{
1113	RowBox[{"SUNF", "[",
1114	RowBox[{"a1", ",", "a4", ",", "a"}], "]"}],
1115	RowBox[{"SUNF", "[",
1116	RowBox[{"a", ",", "a3", ",", "a2"}], "]"}]}], "//", "SUNSimplify"}],
1117	")"}]}], "//", "Contract"}], " ", "//", "Simplify"}]}],
1118	";"}]}], "Input",
1119	CellChangeTimes->{{3.692613814880018^9, 3.692613917818449^9}, {
1120	3.692613950554772^9, 3.692613957521719^9}, {3.692614546932832*^9,
1121	3.6926145619196043`^9}, {3.6926146440022707`^9, 3.692614736605809*^9},
1122	3.692614770819673^9, {3.692614836052224^9, 3.692614927950474*^9}, {
1123	3.692615225380364^9, 3.692615282986795^9}, {3.692615727869773*^9,
1124	3.692615770909458^9}, {3.692615892953589^9, 3.6926159013934717`*^9}, {
1125	3.692615938099309^9, 3.692615943833743^9}, {3.692617190120591*^9,
1126	3.69261722729668^9}, {3.692617668883873^9, 3.6926176872260857`*^9},
1127	3.692617920994087*^9}]
1128	}, Open ]],
1129
1130	Cell[CellGroupData[{
1131
1132	Cell["\<\
1133	Now we contract mu1 with p1 and all the other momenta with the physical \
1134	polarizations\
1135	\>", "Subsection",
1136	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
1137	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
1138	3.692615820540888*^9}}],
1139
1140	Cell[CellGroupData[{
1141
1142	Cell[BoxData["diagU"], "Input",
1143	CellChangeTimes->{{3.692615926878953^9, 3.692615928354073^9}, {
1144	3.69261611622011^9, 3.692616128241695^9}}],
1145
1146	Cell[BoxData[
1147	FormBox[
1148	RowBox[{
1149	FractionBox["1",
1150	SuperscriptBox["u", "2"]],
1151	RowBox[{"\[ImaginaryI]", " ",
1152	SuperscriptBox["gs", "2"], " ",
1153	RowBox[{"(",
1154	RowBox[{
1155	RowBox[{"2", " ", "u", " ",
1156	FormBox[
1157	SuperscriptBox[
1158	FormBox[
1159	OverscriptBox[
1160	FormBox["p4",
1161	TraditionalForm], "_"],
1162	TraditionalForm],
1163	FormBox[
1164	FormBox[
1165	FormBox["mu1",
1166	TraditionalForm],
1167	TraditionalForm],
1168	TraditionalForm]],
1169	TraditionalForm], " ",
1170	FormBox[
1171	SuperscriptBox[
1172	FormBox[
1173	OverscriptBox[
1174	FormBox["p1",
1175	TraditionalForm], "_"],
1176	TraditionalForm],
1177	FormBox[
1178	FormBox[
1179	FormBox["mu2",
1180	TraditionalForm],
1181	TraditionalForm],
1182	TraditionalForm]],
1183	TraditionalForm], " ",
1184	FormBox[
1185	SuperscriptBox[
1186	OverscriptBox["g", "_"],
1187	RowBox[{
1188	FormBox[
1189	FormBox[
1190	FormBox["mu3",
1191	TraditionalForm],
1192	TraditionalForm],
1193	TraditionalForm],
1194	FormBox[
1195	FormBox[
1196	FormBox["mu4",
1197	TraditionalForm],
1198	TraditionalForm],
1199	TraditionalForm]}]],
1200	TraditionalForm]}], "+",
1201	RowBox[{"2", " ", "u", " ",
1202	FormBox[
1203	SuperscriptBox[
1204	FormBox[
1205	OverscriptBox[
1206	FormBox["p4",
1207	TraditionalForm], "_"],
1208	TraditionalForm],
1209	FormBox[
1210	FormBox[
1211	FormBox["mu1",
1212	TraditionalForm],
1213	TraditionalForm],
1214	TraditionalForm]],
1215	TraditionalForm], " ",
1216	FormBox[
1217	SuperscriptBox[
1218	FormBox[
1219	OverscriptBox[
1220	FormBox["p3",
1221	TraditionalForm], "_"],
1222	TraditionalForm],
1223	FormBox[
1224	FormBox[
1225	FormBox["mu2",
1226	TraditionalForm],
1227	TraditionalForm],
1228	TraditionalForm]],
1229	TraditionalForm], " ",
1230	FormBox[
1231	SuperscriptBox[
1232	OverscriptBox["g", "_"],
1233	RowBox[{
1234	FormBox[
1235	FormBox[
1236	FormBox["mu3",
1237	TraditionalForm],
1238	TraditionalForm],
1239	TraditionalForm],
1240	FormBox[
1241	FormBox[
1242	FormBox["mu4",
1243	TraditionalForm],
1244	TraditionalForm],
1245	TraditionalForm]}]],
1246	TraditionalForm]}], "-",
1247	RowBox[{"2", " ", "u", " ",
1248	FormBox[
1249	SuperscriptBox[
1250	FormBox[
1251	OverscriptBox[
1252	FormBox["p4",
1253	TraditionalForm], "_"],
1254	TraditionalForm],
1255	FormBox[
1256	FormBox[
1257	FormBox["mu1",
1258	TraditionalForm],
1259	TraditionalForm],
1260	TraditionalForm]],
1261	TraditionalForm], " ",
1262	FormBox[
1263	SuperscriptBox[
1264	FormBox[
1265	OverscriptBox[
1266	FormBox["p4",
1267	TraditionalForm], "_"],
1268	TraditionalForm],
1269	FormBox[
1270	FormBox[
1271	FormBox["mu2",
1272	TraditionalForm],
1273	TraditionalForm],
1274	TraditionalForm]],
1275	TraditionalForm], " ",
1276	FormBox[
1277	SuperscriptBox[
1278	OverscriptBox["g", "_"],
1279	RowBox[{
1280	FormBox[
1281	FormBox[
1282	FormBox["mu3",
1283	TraditionalForm],
1284	TraditionalForm],
1285	TraditionalForm],
1286	FormBox[
1287	FormBox[
1288	FormBox["mu4",
1289	TraditionalForm],
1290	TraditionalForm],
1291	TraditionalForm]}]],
1292	TraditionalForm]}], "-",
1293	RowBox[{"2", " ", "u", " ",
1294	FormBox[
1295	SuperscriptBox[
1296	FormBox[
1297	OverscriptBox[
1298	FormBox["p4",
1299	TraditionalForm], "_"],
1300	TraditionalForm],
1301	FormBox[
1302	FormBox[
1303	FormBox["mu1",
1304	TraditionalForm],
1305	TraditionalForm],
1306	TraditionalForm]],
1307	TraditionalForm], " ",
1308	FormBox[
1309	SuperscriptBox[
1310	OverscriptBox["g", "_"],
1311	RowBox[{
1312	FormBox[
1313	FormBox[
1314	FormBox["mu2",
1315	TraditionalForm],
1316	TraditionalForm],
1317	TraditionalForm],
1318	FormBox[
1319	FormBox[
1320	FormBox["mu4",
1321	TraditionalForm],
1322	TraditionalForm],
1323	TraditionalForm]}]],
1324	TraditionalForm], " ",
1325	FormBox[
1326	SuperscriptBox[
1327	FormBox[
1328	OverscriptBox[
1329	FormBox["p1",
1330	TraditionalForm], "_"],
1331	TraditionalForm],
1332	FormBox[
1333	FormBox[
1334	FormBox["mu3",
1335	TraditionalForm],
1336	TraditionalForm],
1337	TraditionalForm]],
1338	TraditionalForm]}], "+",
1339	RowBox[{"2", " ", "u", " ",
1340	FormBox[
1341	SuperscriptBox[
1342	FormBox[
1343	OverscriptBox[
1344	FormBox["p4",
1345	TraditionalForm], "_"],
1346	TraditionalForm],
1347	FormBox[
1348	FormBox[
1349	FormBox["mu1",
1350	TraditionalForm],
1351	TraditionalForm],
1352	TraditionalForm]],
1353	TraditionalForm], " ",
1354	FormBox[
1355	SuperscriptBox[
1356	OverscriptBox["g", "_"],
1357	RowBox[{
1358	FormBox[
1359	FormBox[
1360	FormBox["mu2",
1361	TraditionalForm],
1362	TraditionalForm],
1363	TraditionalForm],
1364	FormBox[
1365	FormBox[
1366	FormBox["mu4",
1367	TraditionalForm],
1368	TraditionalForm],
1369	TraditionalForm]}]],
1370	TraditionalForm], " ",
1371	FormBox[
1372	SuperscriptBox[
1373	FormBox[
1374	OverscriptBox[
1375	FormBox["p2",
1376	TraditionalForm], "_"],
1377	TraditionalForm],
1378	FormBox[
1379	FormBox[
1380	FormBox["mu3",
1381	TraditionalForm],
1382	TraditionalForm],
1383	TraditionalForm]],
1384	TraditionalForm]}], "+",
1385	RowBox[{"2", " ", "u", " ",
1386	FormBox[
1387	SuperscriptBox[
1388	FormBox[
1389	OverscriptBox[
1390	FormBox["p4",
1391	TraditionalForm], "_"],
1392	TraditionalForm],
1393	FormBox[
1394	FormBox[
1395	FormBox["mu1",
1396	TraditionalForm],
1397	TraditionalForm],
1398	TraditionalForm]],
1399	TraditionalForm], " ",
1400	FormBox[
1401	SuperscriptBox[
1402	OverscriptBox["g", "_"],
1403	RowBox[{
1404	FormBox[
1405	FormBox[
1406	FormBox["mu2",
1407	TraditionalForm],
1408	TraditionalForm],
1409	TraditionalForm],
1410	FormBox[
1411	FormBox[
1412	FormBox["mu4",
1413	TraditionalForm],
1414	TraditionalForm],
1415	TraditionalForm]}]],
1416	TraditionalForm], " ",
1417	FormBox[
1418	SuperscriptBox[
1419	FormBox[
1420	OverscriptBox[
1421	FormBox["p4",
1422	TraditionalForm], "_"],
1423	TraditionalForm],
1424	FormBox[
1425	FormBox[
1426	FormBox["mu3",
1427	TraditionalForm],
1428	TraditionalForm],
1429	TraditionalForm]],
1430	TraditionalForm]}], "+",
1431	RowBox[{"u", " ",
1432	FormBox[
1433	SuperscriptBox[
1434	OverscriptBox["g", "_"],
1435	RowBox[{
1436	FormBox[
1437	FormBox[
1438	FormBox["mu1",
1439	TraditionalForm],
1440	TraditionalForm],
1441	TraditionalForm],
1442	FormBox[
1443	FormBox[
1444	FormBox["mu4",
1445	TraditionalForm],
1446	TraditionalForm],
1447	TraditionalForm]}]],
1448	TraditionalForm], " ",
1449	RowBox[{"(",
1450	RowBox[{
1451	RowBox[{
1452	RowBox[{"(",
1453	RowBox[{"s", "-", "t"}], ")"}], " ",
1454	FormBox[
1455	SuperscriptBox[
1456	OverscriptBox["g", "_"],
1457	RowBox[{
1458	FormBox[
1459	FormBox[
1460	FormBox["mu2",
1461	TraditionalForm],
1462	TraditionalForm],
1463	TraditionalForm],
1464	FormBox[
1465	FormBox[
1466	FormBox["mu3",
1467	TraditionalForm],
1468	TraditionalForm],
1469	TraditionalForm]}]],
1470	TraditionalForm]}], "+",
1471	RowBox[{"2", " ",
1472	FormBox[
1473	SuperscriptBox[
1474	FormBox[
1475	OverscriptBox[
1476	FormBox["p4",
1477	TraditionalForm], "_"],
1478	TraditionalForm],
1479	FormBox[
1480	FormBox[
1481	FormBox["mu2",
1482	TraditionalForm],
1483	TraditionalForm],
1484	TraditionalForm]],
1485	TraditionalForm], " ",
1486	FormBox[
1487	SuperscriptBox[
1488	FormBox[
1489	OverscriptBox[
1490	FormBox["p1",
1491	TraditionalForm], "_"],
1492	TraditionalForm],
1493	FormBox[
1494	FormBox[
1495	FormBox["mu3",
1496	TraditionalForm],
1497	TraditionalForm],
1498	TraditionalForm]],
1499	TraditionalForm]}], "-",
1500	RowBox[{
1501	FormBox[
1502	SuperscriptBox[
1503	FormBox[
1504	OverscriptBox[
1505	FormBox["p1",
1506	TraditionalForm], "_"],
1507	TraditionalForm],
1508	FormBox[
1509	FormBox[
1510	FormBox["mu2",
1511	TraditionalForm],
1512	TraditionalForm],
1513	TraditionalForm]],
1514	TraditionalForm], " ",
1515	FormBox[
1516	SuperscriptBox[
1517	FormBox[
1518	OverscriptBox[
1519	FormBox["p2",
1520	TraditionalForm], "_"],
1521	TraditionalForm],
1522	FormBox[
1523	FormBox[
1524	FormBox["mu3",
1525	TraditionalForm],
1526	TraditionalForm],
1527	TraditionalForm]],
1528	TraditionalForm]}], "-",
1529	RowBox[{
1530	FormBox[
1531	SuperscriptBox[
1532	FormBox[
1533	OverscriptBox[
1534	FormBox["p4",
1535	TraditionalForm], "_"],
1536	TraditionalForm],
1537	FormBox[
1538	FormBox[
1539	FormBox["mu2",
1540	TraditionalForm],
1541	TraditionalForm],
1542	TraditionalForm]],
1543	TraditionalForm], " ",
1544	FormBox[
1545	SuperscriptBox[
1546	FormBox[
1547	OverscriptBox[
1548	FormBox["p2",
1549	TraditionalForm], "_"],
1550	TraditionalForm],
1551	FormBox[
1552	FormBox[
1553	FormBox["mu3",
1554	TraditionalForm],
1555	TraditionalForm],
1556	TraditionalForm]],
1557	TraditionalForm]}], "-",
1558	RowBox[{"2", " ",
1559	FormBox[
1560	SuperscriptBox[
1561	FormBox[
1562	OverscriptBox[
1563	FormBox["p1",
1564	TraditionalForm], "_"],
1565	TraditionalForm],
1566	FormBox[
1567	FormBox[
1568	FormBox["mu2",
1569	TraditionalForm],
1570	TraditionalForm],
1571	TraditionalForm]],
1572	TraditionalForm], " ",
1573	FormBox[
1574	SuperscriptBox[
1575	FormBox[
1576	OverscriptBox[
1577	FormBox["p4",
1578	TraditionalForm], "_"],
1579	TraditionalForm],
1580	FormBox[
1581	FormBox[
1582	FormBox["mu3",
1583	TraditionalForm],
1584	TraditionalForm],
1585	TraditionalForm]],
1586	TraditionalForm]}], "-",
1587	RowBox[{
1588	FormBox[
1589	SuperscriptBox[
1590	FormBox[
1591	OverscriptBox[
1592	FormBox["p3",
1593	TraditionalForm], "_"],
1594	TraditionalForm],
1595	FormBox[
1596	FormBox[
1597	FormBox["mu2",
1598	TraditionalForm],
1599	TraditionalForm],
1600	TraditionalForm]],
1601	TraditionalForm], " ",
1602	RowBox[{"(",
1603	RowBox[{
1604	FormBox[
1605	SuperscriptBox[
1606	FormBox[
1607	OverscriptBox[
1608	FormBox["p1",
1609	TraditionalForm], "_"],
1610	TraditionalForm],
1611	FormBox[
1612	FormBox[
1613	FormBox["mu3",
1614	TraditionalForm],
1615	TraditionalForm],
1616	TraditionalForm]],
1617	TraditionalForm], "+",
1618	FormBox[
1619	SuperscriptBox[
1620	FormBox[
1621	OverscriptBox[
1622	FormBox["p4",
1623	TraditionalForm], "_"],
1624	TraditionalForm],
1625	FormBox[
1626	FormBox[
1627	FormBox["mu3",
1628	TraditionalForm],
1629	TraditionalForm],
1630	TraditionalForm]],
1631	TraditionalForm]}], ")"}]}]}], ")"}]}], "-",
1632	RowBox[{"2", " ", "u", " ",
1633	FormBox[
1634	SuperscriptBox[
1635	FormBox[
1636	OverscriptBox[
1637	FormBox["p2",
1638	TraditionalForm], "_"],
1639	TraditionalForm],
1640	FormBox[
1641	FormBox[
1642	FormBox["mu1",
1643	TraditionalForm],
1644	TraditionalForm],
1645	TraditionalForm]],
1646	TraditionalForm], " ",
1647	FormBox[
1648	SuperscriptBox[
1649	OverscriptBox["g", "_"],
1650	RowBox[{
1651	FormBox[
1652	FormBox[
1653	FormBox["mu2",
1654	TraditionalForm],
1655	TraditionalForm],
1656	TraditionalForm],
1657	FormBox[
1658	FormBox[
1659	FormBox["mu3",
1660	TraditionalForm],
1661	TraditionalForm],
1662	TraditionalForm]}]],
1663	TraditionalForm], " ",
1664	FormBox[
1665	SuperscriptBox[
1666	FormBox[
1667	OverscriptBox[
1668	FormBox["p1",
1669	TraditionalForm], "_"],
1670	TraditionalForm],
1671	FormBox[
1672	FormBox[
1673	FormBox["mu4",
1674	TraditionalForm],
1675	TraditionalForm],
1676	TraditionalForm]],
1677	TraditionalForm]}], "-",
1678	RowBox[{"2", " ", "u", " ",
1679	FormBox[
1680	SuperscriptBox[
1681	FormBox[
1682	OverscriptBox[
1683	FormBox["p3",
1684	TraditionalForm], "_"],
1685	TraditionalForm],
1686	FormBox[
1687	FormBox[
1688	FormBox["mu1",
1689	TraditionalForm],
1690	TraditionalForm],
1691	TraditionalForm]],
1692	TraditionalForm], " ",
1693	FormBox[
1694	SuperscriptBox[
1695	OverscriptBox["g", "_"],
1696	RowBox[{
1697	FormBox[
1698	FormBox[
1699	FormBox["mu2",
1700	TraditionalForm],
1701	TraditionalForm],
1702	TraditionalForm],
1703	FormBox[
1704	FormBox[
1705	FormBox["mu3",
1706	TraditionalForm],
1707	TraditionalForm],
1708	TraditionalForm]}]],
1709	TraditionalForm], " ",
1710	FormBox[
1711	SuperscriptBox[
1712	FormBox[
1713	OverscriptBox[
1714	FormBox["p1",
1715	TraditionalForm], "_"],
1716	TraditionalForm],
1717	FormBox[
1718	FormBox[
1719	FormBox["mu4",
1720	TraditionalForm],
1721	TraditionalForm],
1722	TraditionalForm]],
1723	TraditionalForm]}], "+",
1724	RowBox[{"2", " ", "u", " ",
1725	FormBox[
1726	SuperscriptBox[
1727	OverscriptBox["g", "_"],
1728	RowBox[{
1729	FormBox[
1730	FormBox[
1731	FormBox["mu1",
1732	TraditionalForm],
1733	TraditionalForm],
1734	TraditionalForm],
1735	FormBox[
1736	FormBox[
1737	FormBox["mu3",
1738	TraditionalForm],
1739	TraditionalForm],
1740	TraditionalForm]}]],
1741	TraditionalForm], " ",
1742	FormBox[
1743	SuperscriptBox[
1744	FormBox[
1745	OverscriptBox[
1746	FormBox["p1",
1747	TraditionalForm], "_"],
1748	TraditionalForm],
1749	FormBox[
1750	FormBox[
1751	FormBox["mu2",
1752	TraditionalForm],
1753	TraditionalForm],
1754	TraditionalForm]],
1755	TraditionalForm], " ",
1756	FormBox[
1757	SuperscriptBox[
1758	FormBox[
1759	OverscriptBox[
1760	FormBox["p1",
1761	TraditionalForm], "_"],
1762	TraditionalForm],
1763	FormBox[
1764	FormBox[
1765	FormBox["mu4",
1766	TraditionalForm],
1767	TraditionalForm],
1768	TraditionalForm]],
1769	TraditionalForm]}], "+",
1770	RowBox[{"2", " ", "u", " ",
1771	FormBox[
1772	SuperscriptBox[
1773	OverscriptBox["g", "_"],
1774	RowBox[{
1775	FormBox[
1776	FormBox[
1777	FormBox["mu1",
1778	TraditionalForm],
1779	TraditionalForm],
1780	TraditionalForm],
1781	FormBox[
1782	FormBox[
1783	FormBox["mu3",
1784	TraditionalForm],
1785	TraditionalForm],
1786	TraditionalForm]}]],
1787	TraditionalForm], " ",
1788	FormBox[
1789	SuperscriptBox[
1790	FormBox[
1791	OverscriptBox[
1792	FormBox["p3",
1793	TraditionalForm], "_"],
1794	TraditionalForm],
1795	FormBox[
1796	FormBox[
1797	FormBox["mu2",
1798	TraditionalForm],
1799	TraditionalForm],
1800	TraditionalForm]],
1801	TraditionalForm], " ",
1802	FormBox[
1803	SuperscriptBox[
1804	FormBox[
1805	OverscriptBox[
1806	FormBox["p1",
1807	TraditionalForm], "_"],
1808	TraditionalForm],
1809	FormBox[
1810	FormBox[
1811	FormBox["mu4",
1812	TraditionalForm],
1813	TraditionalForm],
1814	TraditionalForm]],
1815	TraditionalForm]}], "-",
1816	RowBox[{"2", " ", "u", " ",
1817	FormBox[
1818	SuperscriptBox[
1819	OverscriptBox["g", "_"],
1820	RowBox[{
1821	FormBox[
1822	FormBox[
1823	FormBox["mu1",
1824	TraditionalForm],
1825	TraditionalForm],
1826	TraditionalForm],
1827	FormBox[
1828	FormBox[
1829	FormBox["mu3",
1830	TraditionalForm],
1831	TraditionalForm],
1832	TraditionalForm]}]],
1833	TraditionalForm], " ",
1834	FormBox[
1835	SuperscriptBox[
1836	FormBox[
1837	OverscriptBox[
1838	FormBox["p4",
1839	TraditionalForm], "_"],
1840	TraditionalForm],
1841	FormBox[
1842	FormBox[
1843	FormBox["mu2",
1844	TraditionalForm],
1845	TraditionalForm],
1846	TraditionalForm]],
1847	TraditionalForm], " ",
1848	FormBox[
1849	SuperscriptBox[
1850	FormBox[
1851	OverscriptBox[
1852	FormBox["p1",
1853	TraditionalForm], "_"],
1854	TraditionalForm],
1855	FormBox[
1856	FormBox[
1857	FormBox["mu4",
1858	TraditionalForm],
1859	TraditionalForm],
1860	TraditionalForm]],
1861	TraditionalForm]}], "-",
1862	RowBox[{"2", " ", "u", " ",
1863	FormBox[
1864	SuperscriptBox[
1865	OverscriptBox["g", "_"],
1866	RowBox[{
1867	FormBox[
1868	FormBox[
1869	FormBox["mu1",
1870	TraditionalForm],
1871	TraditionalForm],
1872	TraditionalForm],
1873	FormBox[
1874	FormBox[
1875	FormBox["mu2",
1876	TraditionalForm],
1877	TraditionalForm],
1878	TraditionalForm]}]],
1879	TraditionalForm], " ",
1880	FormBox[
1881	SuperscriptBox[
1882	FormBox[
1883	OverscriptBox[
1884	FormBox["p1",
1885	TraditionalForm], "_"],
1886	TraditionalForm],
1887	FormBox[
1888	FormBox[
1889	FormBox["mu3",
1890	TraditionalForm],
1891	TraditionalForm],
1892	TraditionalForm]],
1893	TraditionalForm], " ",
1894	FormBox[
1895	SuperscriptBox[
1896	FormBox[
1897	OverscriptBox[
1898	FormBox["p1",
1899	TraditionalForm], "_"],
1900	TraditionalForm],
1901	FormBox[
1902	FormBox[
1903	FormBox["mu4",
1904	TraditionalForm],
1905	TraditionalForm],
1906	TraditionalForm]],
1907	TraditionalForm]}], "+",
1908	RowBox[{"2", " ", "u", " ",
1909	FormBox[
1910	SuperscriptBox[
1911	OverscriptBox["g", "_"],
1912	RowBox[{
1913	FormBox[
1914	FormBox[
1915	FormBox["mu1",
1916	TraditionalForm],
1917	TraditionalForm],
1918	TraditionalForm],
1919	FormBox[
1920	FormBox[
1921	FormBox["mu2",
1922	TraditionalForm],
1923	TraditionalForm],
1924	TraditionalForm]}]],
1925	TraditionalForm], " ",
1926	FormBox[
1927	SuperscriptBox[
1928	FormBox[
1929	OverscriptBox[
1930	FormBox["p2",
1931	TraditionalForm], "_"],
1932	TraditionalForm],
1933	FormBox[
1934	FormBox[
1935	FormBox["mu3",
1936	TraditionalForm],
1937	TraditionalForm],
1938	TraditionalForm]],
1939	TraditionalForm], " ",
1940	FormBox[
1941	SuperscriptBox[
1942	FormBox[
1943	OverscriptBox[
1944	FormBox["p1",
1945	TraditionalForm], "_"],
1946	TraditionalForm],
1947	FormBox[
1948	FormBox[
1949	FormBox["mu4",
1950	TraditionalForm],
1951	TraditionalForm],
1952	TraditionalForm]],
1953	TraditionalForm]}], "+",
1954	RowBox[{"2", " ", "u", " ",
1955	FormBox[
1956	SuperscriptBox[
1957	OverscriptBox["g", "_"],
1958	RowBox[{
1959	FormBox[
1960	FormBox[
1961	FormBox["mu1",
1962	TraditionalForm],
1963	TraditionalForm],
1964	TraditionalForm],
1965	FormBox[
1966	FormBox[
1967	FormBox["mu2",
1968	TraditionalForm],
1969	TraditionalForm],
1970	TraditionalForm]}]],
1971	TraditionalForm], " ",
1972	FormBox[
1973	SuperscriptBox[
1974	FormBox[
1975	OverscriptBox[
1976	FormBox["p4",
1977	TraditionalForm], "_"],
1978	TraditionalForm],
1979	FormBox[
1980	FormBox[
1981	FormBox["mu3",
1982	TraditionalForm],
1983	TraditionalForm],
1984	TraditionalForm]],
1985	TraditionalForm], " ",
1986	FormBox[
1987	SuperscriptBox[
1988	FormBox[
1989	OverscriptBox[
1990	FormBox["p1",
1991	TraditionalForm], "_"],
1992	TraditionalForm],
1993	FormBox[
1994	FormBox[
1995	FormBox["mu4",
1996	TraditionalForm],
1997	TraditionalForm],
1998	TraditionalForm]],
1999	TraditionalForm]}], "-",
2000	RowBox[{"2", " ", "u", " ",
2001	FormBox[
2002	SuperscriptBox[
2003	FormBox[
2004	OverscriptBox[
2005	FormBox["p4",
2006	TraditionalForm], "_"],
2007	TraditionalForm],
2008	FormBox[
2009	FormBox[
2010	FormBox["mu1",
2011	TraditionalForm],
2012	TraditionalForm],
2013	TraditionalForm]],
2014	TraditionalForm], " ",
2015	FormBox[
2016	SuperscriptBox[
2017	OverscriptBox["g", "_"],
2018	RowBox[{
2019	FormBox[
2020	FormBox[
2021	FormBox["mu2",
2022	TraditionalForm],
2023	TraditionalForm],
2024	TraditionalForm],
2025	FormBox[
2026	FormBox[
2027	FormBox["mu3",
2028	TraditionalForm],
2029	TraditionalForm],
2030	TraditionalForm]}]],
2031	TraditionalForm], " ",
2032	FormBox[
2033	SuperscriptBox[
2034	FormBox[
2035	OverscriptBox[
2036	FormBox["p2",
2037	TraditionalForm], "_"],
2038	TraditionalForm],
2039	FormBox[
2040	FormBox[
2041	FormBox["mu4",
2042	TraditionalForm],
2043	TraditionalForm],
2044	TraditionalForm]],
2045	TraditionalForm]}], "-",
2046	RowBox[{"2", " ", "u", " ",
2047	FormBox[
2048	SuperscriptBox[
2049	FormBox[
2050	OverscriptBox[
2051	FormBox["p4",
2052	TraditionalForm], "_"],
2053	TraditionalForm],
2054	FormBox[
2055	FormBox[
2056	FormBox["mu1",
2057	TraditionalForm],
2058	TraditionalForm],
2059	TraditionalForm]],
2060	TraditionalForm], " ",
2061	FormBox[
2062	SuperscriptBox[
2063	OverscriptBox["g", "_"],
2064	RowBox[{
2065	FormBox[
2066	FormBox[
2067	FormBox["mu2",
2068	TraditionalForm],
2069	TraditionalForm],
2070	TraditionalForm],
2071	FormBox[
2072	FormBox[
2073	FormBox["mu3",
2074	TraditionalForm],
2075	TraditionalForm],
2076	TraditionalForm]}]],
2077	TraditionalForm], " ",
2078	FormBox[
2079	SuperscriptBox[
2080	FormBox[
2081	OverscriptBox[
2082	FormBox["p3",
2083	TraditionalForm], "_"],
2084	TraditionalForm],
2085	FormBox[
2086	FormBox[
2087	FormBox["mu4",
2088	TraditionalForm],
2089	TraditionalForm],
2090	TraditionalForm]],
2091	TraditionalForm]}], "+",
2092	RowBox[{
2093	FormBox[
2094	SuperscriptBox[
2095	FormBox[
2096	OverscriptBox[
2097	FormBox["p1",
2098	TraditionalForm], "_"],
2099	TraditionalForm],
2100	FormBox[
2101	FormBox[
2102	FormBox["mu1",
2103	TraditionalForm],
2104	TraditionalForm],
2105	TraditionalForm]],
2106	TraditionalForm], " ",
2107	RowBox[{"(",
2108	RowBox[{
2109	RowBox[{"u", " ",
2110	FormBox[
2111	SuperscriptBox[
2112	FormBox[
2113	OverscriptBox[
2114	FormBox["p4",
2115	TraditionalForm], "_"],
2116	TraditionalForm],
2117	FormBox[
2118	FormBox[
2119	FormBox["mu2",
2120	TraditionalForm],
2121	TraditionalForm],
2122	TraditionalForm]],
2123	TraditionalForm], " ",
2124	FormBox[
2125	SuperscriptBox[
2126	OverscriptBox["g", "_"],
2127	RowBox[{
2128	FormBox[
2129	FormBox[
2130	FormBox["mu3",
2131	TraditionalForm],
2132	TraditionalForm],
2133	TraditionalForm],
2134	FormBox[
2135	FormBox[
2136	FormBox["mu4",
2137	TraditionalForm],
2138	TraditionalForm],
2139	TraditionalForm]}]],
2140	TraditionalForm]}], "+",
2141	RowBox[{"u", " ",
2142	FormBox[
2143	SuperscriptBox[
2144	OverscriptBox["g", "_"],
2145	RowBox[{
2146	FormBox[
2147	FormBox[
2148	FormBox["mu2",
2149	TraditionalForm],
2150	TraditionalForm],
2151	TraditionalForm],
2152	FormBox[
2153	FormBox[
2154	FormBox["mu4",
2155	TraditionalForm],
2156	TraditionalForm],
2157	TraditionalForm]}]],
2158	TraditionalForm], " ",
2159	FormBox[
2160	SuperscriptBox[
2161	FormBox[
2162	OverscriptBox[
2163	FormBox["p1",
2164	TraditionalForm], "_"],
2165	TraditionalForm],
2166	FormBox[
2167	FormBox[
2168	FormBox["mu3",
2169	TraditionalForm],
2170	TraditionalForm],
2171	TraditionalForm]],
2172	TraditionalForm]}], "-",
2173	RowBox[{"u", " ",
2174	FormBox[
2175	SuperscriptBox[
2176	OverscriptBox["g", "_"],
2177	RowBox[{
2178	FormBox[
2179	FormBox[
2180	FormBox["mu2",
2181	TraditionalForm],
2182	TraditionalForm],
2183	TraditionalForm],
2184	FormBox[
2185	FormBox[
2186	FormBox["mu4",
2187	TraditionalForm],
2188	TraditionalForm],
2189	TraditionalForm]}]],
2190	TraditionalForm], " ",
2191	FormBox[
2192	SuperscriptBox[
2193	FormBox[
2194	OverscriptBox[
2195	FormBox["p2",
2196	TraditionalForm], "_"],
2197	TraditionalForm],
2198	FormBox[
2199	FormBox[
2200	FormBox["mu3",
2201	TraditionalForm],
2202	TraditionalForm],
2203	TraditionalForm]],
2204	TraditionalForm]}], "-",
2205	RowBox[{"u", " ",
2206	FormBox[
2207	SuperscriptBox[
2208	OverscriptBox["g", "_"],
2209	RowBox[{
2210	FormBox[
2211	FormBox[
2212	FormBox["mu2",
2213	TraditionalForm],
2214	TraditionalForm],
2215	TraditionalForm],
2216	FormBox[
2217	FormBox[
2218	FormBox["mu4",
2219	TraditionalForm],
2220	TraditionalForm],
2221	TraditionalForm]}]],
2222	TraditionalForm], " ",
2223	FormBox[
2224	SuperscriptBox[
2225	FormBox[
2226	OverscriptBox[
2227	FormBox["p4",
2228	TraditionalForm], "_"],
2229	TraditionalForm],
2230	FormBox[
2231	FormBox[
2232	FormBox["mu3",
2233	TraditionalForm],
2234	TraditionalForm],
2235	TraditionalForm]],
2236	TraditionalForm]}], "-",
2237	RowBox[{"\[Xi]", " ",
2238	FormBox[
2239	SuperscriptBox[
2240	FormBox[
2241	OverscriptBox[
2242	FormBox["p4",
2243	TraditionalForm], "_"],
2244	TraditionalForm],
2245	FormBox[
2246	FormBox[
2247	FormBox["mu2",
2248	TraditionalForm],
2249	TraditionalForm],
2250	TraditionalForm]],
2251	TraditionalForm], " ",
2252	FormBox[
2253	SuperscriptBox[
2254	FormBox[
2255	OverscriptBox[
2256	FormBox["p2",
2257	TraditionalForm], "_"],
2258	TraditionalForm],
2259	FormBox[
2260	FormBox[
2261	FormBox["mu3",
2262	TraditionalForm],
2263	TraditionalForm],
2264	TraditionalForm]],
2265	TraditionalForm], " ",
2266	FormBox[
2267	SuperscriptBox[
2268	FormBox[
2269	OverscriptBox[
2270	FormBox["p1",
2271	TraditionalForm], "_"],
2272	TraditionalForm],
2273	FormBox[
2274	FormBox[
2275	FormBox["mu4",
2276	TraditionalForm],
2277	TraditionalForm],
2278	TraditionalForm]],
2279	TraditionalForm]}], "+",
2280	RowBox[{
2281	FormBox[
2282	SuperscriptBox[
2283	FormBox[
2284	OverscriptBox[
2285	FormBox["p4",
2286	TraditionalForm], "_"],
2287	TraditionalForm],
2288	FormBox[
2289	FormBox[
2290	FormBox["mu2",
2291	TraditionalForm],
2292	TraditionalForm],
2293	TraditionalForm]],
2294	TraditionalForm], " ",
2295	FormBox[
2296	SuperscriptBox[
2297	FormBox[
2298	OverscriptBox[
2299	FormBox["p2",
2300	TraditionalForm], "_"],
2301	TraditionalForm],
2302	FormBox[
2303	FormBox[
2304	FormBox["mu3",
2305	TraditionalForm],
2306	TraditionalForm],
2307	TraditionalForm]],
2308	TraditionalForm], " ",
2309	FormBox[
2310	SuperscriptBox[
2311	FormBox[
2312	OverscriptBox[
2313	FormBox["p1",
2314	TraditionalForm], "_"],
2315	TraditionalForm],
2316	FormBox[
2317	FormBox[
2318	FormBox["mu4",
2319	TraditionalForm],
2320	TraditionalForm],
2321	TraditionalForm]],
2322	TraditionalForm]}], "-",
2323	RowBox[{
2324	FormBox[
2325	SuperscriptBox[
2326	FormBox[
2327	OverscriptBox[
2328	FormBox["p1",
2329	TraditionalForm], "_"],
2330	TraditionalForm],
2331	FormBox[
2332	FormBox[
2333	FormBox["mu2",
2334	TraditionalForm],
2335	TraditionalForm],
2336	TraditionalForm]],
2337	TraditionalForm], " ",
2338	RowBox[{"(",
2339	RowBox[{
2340	RowBox[{"u", " ",
2341	FormBox[
2342	SuperscriptBox[
2343	OverscriptBox["g", "_"],
2344	RowBox[{
2345	FormBox[
2346	FormBox[
2347	FormBox["mu3",
2348	TraditionalForm],
2349	TraditionalForm],
2350	TraditionalForm],
2351	FormBox[
2352	FormBox[
2353	FormBox["mu4",
2354	TraditionalForm],
2355	TraditionalForm],
2356	TraditionalForm]}]],
2357	TraditionalForm]}], "-",
2358	RowBox[{
2359	RowBox[{"(",
2360	RowBox[{"\[Xi]", "-", "1"}], ")"}], " ",
2361	FormBox[
2362	SuperscriptBox[
2363	FormBox[
2364	OverscriptBox[
2365	FormBox["p2",
2366	TraditionalForm], "_"],
2367	TraditionalForm],
2368	FormBox[
2369	FormBox[
2370	FormBox["mu3",
2371	TraditionalForm],
2372	TraditionalForm],
2373	TraditionalForm]],
2374	TraditionalForm], " ",
2375	FormBox[
2376	SuperscriptBox[
2377	FormBox[
2378	OverscriptBox[
2379	FormBox["p1",
2380	TraditionalForm], "_"],
2381	TraditionalForm],
2382	FormBox[
2383	FormBox[
2384	FormBox["mu4",
2385	TraditionalForm],
2386	TraditionalForm],
2387	TraditionalForm]],
2388	TraditionalForm]}]}], ")"}]}], "-",
2389	RowBox[{
2390	FormBox[
2391	SuperscriptBox[
2392	FormBox[
2393	OverscriptBox[
2394	FormBox["p3",
2395	TraditionalForm], "_"],
2396	TraditionalForm],
2397	FormBox[
2398	FormBox[
2399	FormBox["mu2",
2400	TraditionalForm],
2401	TraditionalForm],
2402	TraditionalForm]],
2403	TraditionalForm], " ",
2404	RowBox[{"(",
2405	RowBox[{
2406	RowBox[{"u", " ",
2407	FormBox[
2408	SuperscriptBox[
2409	OverscriptBox["g", "_"],
2410	RowBox[{
2411	FormBox[
2412	FormBox[
2413	FormBox["mu3",
2414	TraditionalForm],
2415	TraditionalForm],
2416	TraditionalForm],
2417	FormBox[
2418	FormBox[
2419	FormBox["mu4",
2420	TraditionalForm],
2421	TraditionalForm],
2422	TraditionalForm]}]],
2423	TraditionalForm]}], "-",
2424	RowBox[{
2425	RowBox[{"(",
2426	RowBox[{"\[Xi]", "-", "1"}], ")"}], " ",
2427	RowBox[{"(",
2428	RowBox[{
2429	FormBox[
2430	SuperscriptBox[
2431	FormBox[
2432	OverscriptBox[
2433	FormBox["p1",
2434	TraditionalForm], "_"],
2435	TraditionalForm],
2436	FormBox[
2437	FormBox[
2438	FormBox["mu3",
2439	TraditionalForm],
2440	TraditionalForm],
2441	TraditionalForm]],
2442	TraditionalForm], "-",
2443	FormBox[
2444	SuperscriptBox[
2445	FormBox[
2446	OverscriptBox[
2447	FormBox["p4",
2448	TraditionalForm], "_"],
2449	TraditionalForm],
2450	FormBox[
2451	FormBox[
2452	FormBox["mu3",
2453	TraditionalForm],
2454	TraditionalForm],
2455	TraditionalForm]],
2456	TraditionalForm]}], ")"}], " ",
2457	FormBox[
2458	SuperscriptBox[
2459	FormBox[
2460	OverscriptBox[
2461	FormBox["p1",
2462	TraditionalForm], "_"],
2463	TraditionalForm],
2464	FormBox[
2465	FormBox[
2466	FormBox["mu4",
2467	TraditionalForm],
2468	TraditionalForm],
2469	TraditionalForm]],
2470	TraditionalForm]}]}], ")"}]}], "+",
2471	RowBox[{"u", " ",
2472	FormBox[
2473	SuperscriptBox[
2474	OverscriptBox["g", "_"],
2475	RowBox[{
2476	FormBox[
2477	FormBox[
2478	FormBox["mu2",
2479	TraditionalForm],
2480	TraditionalForm],
2481	TraditionalForm],
2482	FormBox[
2483	FormBox[
2484	FormBox["mu3",
2485	TraditionalForm],
2486	TraditionalForm],
2487	TraditionalForm]}]],
2488	TraditionalForm], " ",
2489	FormBox[
2490	SuperscriptBox[
2491	FormBox[
2492	OverscriptBox[
2493	FormBox["p2",
2494	TraditionalForm], "_"],
2495	TraditionalForm],
2496	FormBox[
2497	FormBox[
2498	FormBox["mu4",
2499	TraditionalForm],
2500	TraditionalForm],
2501	TraditionalForm]],
2502	TraditionalForm]}], "+",
2503	RowBox[{"u", " ",
2504	FormBox[
2505	SuperscriptBox[
2506	OverscriptBox["g", "_"],
2507	RowBox[{
2508	FormBox[
2509	FormBox[
2510	FormBox["mu2",
2511	TraditionalForm],
2512	TraditionalForm],
2513	TraditionalForm],
2514	FormBox[
2515	FormBox[
2516	FormBox["mu3",
2517	TraditionalForm],
2518	TraditionalForm],
2519	TraditionalForm]}]],
2520	TraditionalForm], " ",
2521	FormBox[
2522	SuperscriptBox[
2523	FormBox[
2524	OverscriptBox[
2525	FormBox["p3",
2526	TraditionalForm], "_"],
2527	TraditionalForm],
2528	FormBox[
2529	FormBox[
2530	FormBox["mu4",
2531	TraditionalForm],
2532	TraditionalForm],
2533	TraditionalForm]],
2534	TraditionalForm]}]}], ")"}]}], "+",
2535	RowBox[{"u", " ",
2536	FormBox[
2537	SuperscriptBox[
2538	FormBox[
2539	OverscriptBox[
2540	FormBox["p2",
2541	TraditionalForm], "_"],
2542	TraditionalForm],
2543	FormBox[
2544	FormBox[
2545	FormBox["mu1",
2546	TraditionalForm],
2547	TraditionalForm],
2548	TraditionalForm]],
2549	TraditionalForm], " ",
2550	FormBox[
2551	SuperscriptBox[
2552	OverscriptBox["g", "_"],
2553	RowBox[{
2554	FormBox[
2555	FormBox[
2556	FormBox["mu2",
2557	TraditionalForm],
2558	TraditionalForm],
2559	TraditionalForm],
2560	FormBox[
2561	FormBox[
2562	FormBox["mu3",
2563	TraditionalForm],
2564	TraditionalForm],
2565	TraditionalForm]}]],
2566	TraditionalForm], " ",
2567	FormBox[
2568	SuperscriptBox[
2569	FormBox[
2570	OverscriptBox[
2571	FormBox["p4",
2572	TraditionalForm], "_"],
2573	TraditionalForm],
2574	FormBox[
2575	FormBox[
2576	FormBox["mu4",
2577	TraditionalForm],
2578	TraditionalForm],
2579	TraditionalForm]],
2580	TraditionalForm]}], "+",
2581	RowBox[{"u", " ",
2582	FormBox[
2583	SuperscriptBox[
2584	FormBox[
2585	OverscriptBox[
2586	FormBox["p3",
2587	TraditionalForm], "_"],
2588	TraditionalForm],
2589	FormBox[
2590	FormBox[
2591	FormBox["mu1",
2592	TraditionalForm],
2593	TraditionalForm],
2594	TraditionalForm]],
2595	TraditionalForm], " ",
2596	FormBox[
2597	SuperscriptBox[
2598	OverscriptBox["g", "_"],
2599	RowBox[{
2600	FormBox[
2601	FormBox[
2602	FormBox["mu2",
2603	TraditionalForm],
2604	TraditionalForm],
2605	TraditionalForm],
2606	FormBox[
2607	FormBox[
2608	FormBox["mu3",
2609	TraditionalForm],
2610	TraditionalForm],
2611	TraditionalForm]}]],
2612	TraditionalForm], " ",
2613	FormBox[
2614	SuperscriptBox[
2615	FormBox[
2616	OverscriptBox[
2617	FormBox["p4",
2618	TraditionalForm], "_"],
2619	TraditionalForm],
2620	FormBox[
2621	FormBox[
2622	FormBox["mu4",
2623	TraditionalForm],
2624	TraditionalForm],
2625	TraditionalForm]],
2626	TraditionalForm]}], "-",
2627	RowBox[{"u", " ",
2628	FormBox[
2629	SuperscriptBox[
2630	OverscriptBox["g", "_"],
2631	RowBox[{
2632	FormBox[
2633	FormBox[
2634	FormBox["mu1",
2635	TraditionalForm],
2636	TraditionalForm],
2637	TraditionalForm],
2638	FormBox[
2639	FormBox[
2640	FormBox["mu3",
2641	TraditionalForm],
2642	TraditionalForm],
2643	TraditionalForm]}]],
2644	TraditionalForm], " ",
2645	FormBox[
2646	SuperscriptBox[
2647	FormBox[
2648	OverscriptBox[
2649	FormBox["p1",
2650	TraditionalForm], "_"],
2651	TraditionalForm],
2652	FormBox[
2653	FormBox[
2654	FormBox["mu2",
2655	TraditionalForm],
2656	TraditionalForm],
2657	TraditionalForm]],
2658	TraditionalForm], " ",
2659	FormBox[
2660	SuperscriptBox[
2661	FormBox[
2662	OverscriptBox[
2663	FormBox["p4",
2664	TraditionalForm], "_"],
2665	TraditionalForm],
2666	FormBox[
2667	FormBox[
2668	FormBox["mu4",
2669	TraditionalForm],
2670	TraditionalForm],
2671	TraditionalForm]],
2672	TraditionalForm]}], "-",
2673	RowBox[{"u", " ",
2674	FormBox[
2675	SuperscriptBox[
2676	OverscriptBox["g", "_"],
2677	RowBox[{
2678	FormBox[
2679	FormBox[
2680	FormBox["mu1",
2681	TraditionalForm],
2682	TraditionalForm],
2683	TraditionalForm],
2684	FormBox[
2685	FormBox[
2686	FormBox["mu3",
2687	TraditionalForm],
2688	TraditionalForm],
2689	TraditionalForm]}]],
2690	TraditionalForm], " ",
2691	FormBox[
2692	SuperscriptBox[
2693	FormBox[
2694	OverscriptBox[
2695	FormBox["p3",
2696	TraditionalForm], "_"],
2697	TraditionalForm],
2698	FormBox[
2699	FormBox[
2700	FormBox["mu2",
2701	TraditionalForm],
2702	TraditionalForm],
2703	TraditionalForm]],
2704	TraditionalForm], " ",
2705	FormBox[
2706	SuperscriptBox[
2707	FormBox[
2708	OverscriptBox[
2709	FormBox["p4",
2710	TraditionalForm], "_"],
2711	TraditionalForm],
2712	FormBox[
2713	FormBox[
2714	FormBox["mu4",
2715	TraditionalForm],
2716	TraditionalForm],
2717	TraditionalForm]],
2718	TraditionalForm]}], "+",
2719	RowBox[{"u", " ",
2720	FormBox[
2721	SuperscriptBox[
2722	OverscriptBox["g", "_"],
2723	RowBox[{
2724	FormBox[
2725	FormBox[
2726	FormBox["mu1",
2727	TraditionalForm],
2728	TraditionalForm],
2729	TraditionalForm],
2730	FormBox[
2731	FormBox[
2732	FormBox["mu3",
2733	TraditionalForm],
2734	TraditionalForm],
2735	TraditionalForm]}]],
2736	TraditionalForm], " ",
2737	FormBox[
2738	SuperscriptBox[
2739	FormBox[
2740	OverscriptBox[
2741	FormBox["p4",
2742	TraditionalForm], "_"],
2743	TraditionalForm],
2744	FormBox[
2745	FormBox[
2746	FormBox["mu2",
2747	TraditionalForm],
2748	TraditionalForm],
2749	TraditionalForm]],
2750	TraditionalForm], " ",
2751	FormBox[
2752	SuperscriptBox[
2753	FormBox[
2754	OverscriptBox[
2755	FormBox["p4",
2756	TraditionalForm], "_"],
2757	TraditionalForm],
2758	FormBox[
2759	FormBox[
2760	FormBox["mu4",
2761	TraditionalForm],
2762	TraditionalForm],
2763	TraditionalForm]],
2764	TraditionalForm]}], "+",
2765	RowBox[{"u", " ",
2766	FormBox[
2767	SuperscriptBox[
2768	OverscriptBox["g", "_"],
2769	RowBox[{
2770	FormBox[
2771	FormBox[
2772	FormBox["mu1",
2773	TraditionalForm],
2774	TraditionalForm],
2775	TraditionalForm],
2776	FormBox[
2777	FormBox[
2778	FormBox["mu2",
2779	TraditionalForm],
2780	TraditionalForm],
2781	TraditionalForm]}]],
2782	TraditionalForm], " ",
2783	FormBox[
2784	SuperscriptBox[
2785	FormBox[
2786	OverscriptBox[
2787	FormBox["p1",
2788	TraditionalForm], "_"],
2789	TraditionalForm],
2790	FormBox[
2791	FormBox[
2792	FormBox["mu3",
2793	TraditionalForm],
2794	TraditionalForm],
2795	TraditionalForm]],
2796	TraditionalForm], " ",
2797	FormBox[
2798	SuperscriptBox[
2799	FormBox[
2800	OverscriptBox[
2801	FormBox["p4",
2802	TraditionalForm], "_"],
2803	TraditionalForm],
2804	FormBox[
2805	FormBox[
2806	FormBox["mu4",
2807	TraditionalForm],
2808	TraditionalForm],
2809	TraditionalForm]],
2810	TraditionalForm]}], "-",
2811	RowBox[{"\[Xi]", " ",
2812	FormBox[
2813	SuperscriptBox[
2814	FormBox[
2815	OverscriptBox[
2816	FormBox["p4",
2817	TraditionalForm], "_"],
2818	TraditionalForm],
2819	FormBox[
2820	FormBox[
2821	FormBox["mu1",
2822	TraditionalForm],
2823	TraditionalForm],
2824	TraditionalForm]],
2825	TraditionalForm], " ",
2826	FormBox[
2827	SuperscriptBox[
2828	FormBox[
2829	OverscriptBox[
2830	FormBox["p3",
2831	TraditionalForm], "_"],
2832	TraditionalForm],
2833	FormBox[
2834	FormBox[
2835	FormBox["mu2",
2836	TraditionalForm],
2837	TraditionalForm],
2838	TraditionalForm]],
2839	TraditionalForm], " ",
2840	FormBox[
2841	SuperscriptBox[
2842	FormBox[
2843	OverscriptBox[
2844	FormBox["p1",
2845	TraditionalForm], "_"],
2846	TraditionalForm],
2847	FormBox[
2848	FormBox[
2849	FormBox["mu3",
2850	TraditionalForm],
2851	TraditionalForm],
2852	TraditionalForm]],
2853	TraditionalForm], " ",
2854	FormBox[
2855	SuperscriptBox[
2856	FormBox[
2857	OverscriptBox[
2858	FormBox["p4",
2859	TraditionalForm], "_"],
2860	TraditionalForm],
2861	FormBox[
2862	FormBox[
2863	FormBox["mu4",
2864	TraditionalForm],
2865	TraditionalForm],
2866	TraditionalForm]],
2867	TraditionalForm]}], "+",
2868	RowBox[{
2869	FormBox[
2870	SuperscriptBox[
2871	FormBox[
2872	OverscriptBox[
2873	FormBox["p4",
2874	TraditionalForm], "_"],
2875	TraditionalForm],
2876	FormBox[
2877	FormBox[
2878	FormBox["mu1",
2879	TraditionalForm],
2880	TraditionalForm],
2881	TraditionalForm]],
2882	TraditionalForm], " ",
2883	FormBox[
2884	SuperscriptBox[
2885	FormBox[
2886	OverscriptBox[
2887	FormBox["p3",
2888	TraditionalForm], "_"],
2889	TraditionalForm],
2890	FormBox[
2891	FormBox[
2892	FormBox["mu2",
2893	TraditionalForm],
2894	TraditionalForm],
2895	TraditionalForm]],
2896	TraditionalForm], " ",
2897	FormBox[
2898	SuperscriptBox[
2899	FormBox[
2900	OverscriptBox[
2901	FormBox["p1",
2902	TraditionalForm], "_"],
2903	TraditionalForm],
2904	FormBox[
2905	FormBox[
2906	FormBox["mu3",
2907	TraditionalForm],
2908	TraditionalForm],
2909	TraditionalForm]],
2910	TraditionalForm], " ",
2911	FormBox[
2912	SuperscriptBox[
2913	FormBox[
2914	OverscriptBox[
2915	FormBox["p4",
2916	TraditionalForm], "_"],
2917	TraditionalForm],
2918	FormBox[
2919	FormBox[
2920	FormBox["mu4",
2921	TraditionalForm],
2922	TraditionalForm],
2923	TraditionalForm]],
2924	TraditionalForm]}], "-",
2925	RowBox[{"u", " ",
2926	FormBox[
2927	SuperscriptBox[
2928	OverscriptBox["g", "_"],
2929	RowBox[{
2930	FormBox[
2931	FormBox[
2932	FormBox["mu1",
2933	TraditionalForm],
2934	TraditionalForm],
2935	TraditionalForm],
2936	FormBox[
2937	FormBox[
2938	FormBox["mu2",
2939	TraditionalForm],
2940	TraditionalForm],
2941	TraditionalForm]}]],
2942	TraditionalForm], " ",
2943	FormBox[
2944	SuperscriptBox[
2945	FormBox[
2946	OverscriptBox[
2947	FormBox["p2",
2948	TraditionalForm], "_"],
2949	TraditionalForm],
2950	FormBox[
2951	FormBox[
2952	FormBox["mu3",
2953	TraditionalForm],
2954	TraditionalForm],
2955	TraditionalForm]],
2956	TraditionalForm], " ",
2957	FormBox[
2958	SuperscriptBox[
2959	FormBox[
2960	OverscriptBox[
2961	FormBox["p4",
2962	TraditionalForm], "_"],
2963	TraditionalForm],
2964	FormBox[
2965	FormBox[
2966	FormBox["mu4",
2967	TraditionalForm],
2968	TraditionalForm],
2969	TraditionalForm]],
2970	TraditionalForm]}], "-",
2971	RowBox[{"\[Xi]", " ",
2972	FormBox[
2973	SuperscriptBox[
2974	FormBox[
2975	OverscriptBox[
2976	FormBox["p4",
2977	TraditionalForm], "_"],
2978	TraditionalForm],
2979	FormBox[
2980	FormBox[
2981	FormBox["mu1",
2982	TraditionalForm],
2983	TraditionalForm],
2984	TraditionalForm]],
2985	TraditionalForm], " ",
2986	FormBox[
2987	SuperscriptBox[
2988	FormBox[
2989	OverscriptBox[
2990	FormBox["p1",
2991	TraditionalForm], "_"],
2992	TraditionalForm],
2993	FormBox[
2994	FormBox[
2995	FormBox["mu2",
2996	TraditionalForm],
2997	TraditionalForm],
2998	TraditionalForm]],
2999	TraditionalForm], " ",
3000	FormBox[
3001	SuperscriptBox[
3002	FormBox[
3003	OverscriptBox[
3004	FormBox["p2",
3005	TraditionalForm], "_"],
3006	TraditionalForm],
3007	FormBox[
3008	FormBox[
3009	FormBox["mu3",
3010	TraditionalForm],
3011	TraditionalForm],
3012	TraditionalForm]],
3013	TraditionalForm], " ",
3014	FormBox[
3015	SuperscriptBox[
3016	FormBox[
3017	OverscriptBox[
3018	FormBox["p4",
3019	TraditionalForm], "_"],
3020	TraditionalForm],
3021	FormBox[
3022	FormBox[
3023	FormBox["mu4",
3024	TraditionalForm],
3025	TraditionalForm],
3026	TraditionalForm]],
3027	TraditionalForm]}], "+",
3028	RowBox[{
3029	FormBox[
3030	SuperscriptBox[
3031	FormBox[
3032	OverscriptBox[
3033	FormBox["p4",
3034	TraditionalForm], "_"],
3035	TraditionalForm],
3036	FormBox[
3037	FormBox[
3038	FormBox["mu1",
3039	TraditionalForm],
3040	TraditionalForm],
3041	TraditionalForm]],
3042	TraditionalForm], " ",
3043	FormBox[
3044	SuperscriptBox[
3045	FormBox[
3046	OverscriptBox[
3047	FormBox["p1",
3048	TraditionalForm], "_"],
3049	TraditionalForm],
3050	FormBox[
3051	FormBox[
3052	FormBox["mu2",
3053	TraditionalForm],
3054	TraditionalForm],
3055	TraditionalForm]],
3056	TraditionalForm], " ",
3057	FormBox[
3058	SuperscriptBox[
3059	FormBox[
3060	OverscriptBox[
3061	FormBox["p2",
3062	TraditionalForm], "_"],
3063	TraditionalForm],
3064	FormBox[
3065	FormBox[
3066	FormBox["mu3",
3067	TraditionalForm],
3068	TraditionalForm],
3069	TraditionalForm]],
3070	TraditionalForm], " ",
3071	FormBox[
3072	SuperscriptBox[
3073	FormBox[
3074	OverscriptBox[
3075	FormBox["p4",
3076	TraditionalForm], "_"],
3077	TraditionalForm],
3078	FormBox[
3079	FormBox[
3080	FormBox["mu4",
3081	TraditionalForm],
3082	TraditionalForm],
3083	TraditionalForm]],
3084	TraditionalForm]}], "+",
3085	RowBox[{"\[Xi]", " ",
3086	FormBox[
3087	SuperscriptBox[
3088	FormBox[
3089	OverscriptBox[
3090	FormBox["p4",
3091	TraditionalForm], "_"],
3092	TraditionalForm],
3093	FormBox[
3094	FormBox[
3095	FormBox["mu1",
3096	TraditionalForm],
3097	TraditionalForm],
3098	TraditionalForm]],
3099	TraditionalForm], " ",
3100	FormBox[
3101	SuperscriptBox[
3102	FormBox[
3103	OverscriptBox[
3104	FormBox["p4",
3105	TraditionalForm], "_"],
3106	TraditionalForm],
3107	FormBox[
3108	FormBox[
3109	FormBox["mu2",
3110	TraditionalForm],
3111	TraditionalForm],
3112	TraditionalForm]],
3113	TraditionalForm], " ",
3114	FormBox[
3115	SuperscriptBox[
3116	FormBox[
3117	OverscriptBox[
3118	FormBox["p2",
3119	TraditionalForm], "_"],
3120	TraditionalForm],
3121	FormBox[
3122	FormBox[
3123	FormBox["mu3",
3124	TraditionalForm],
3125	TraditionalForm],
3126	TraditionalForm]],
3127	TraditionalForm], " ",
3128	FormBox[
3129	SuperscriptBox[
3130	FormBox[
3131	OverscriptBox[
3132	FormBox["p4",
3133	TraditionalForm], "_"],
3134	TraditionalForm],
3135	FormBox[
3136	FormBox[
3137	FormBox["mu4",
3138	TraditionalForm],
3139	TraditionalForm],
3140	TraditionalForm]],
3141	TraditionalForm]}], "-",
3142	RowBox[{
3143	FormBox[
3144	SuperscriptBox[
3145	FormBox[
3146	OverscriptBox[
3147	FormBox["p4",
3148	TraditionalForm], "_"],
3149	TraditionalForm],
3150	FormBox[
3151	FormBox[
3152	FormBox["mu1",
3153	TraditionalForm],
3154	TraditionalForm],
3155	TraditionalForm]],
3156	TraditionalForm], " ",
3157	FormBox[
3158	SuperscriptBox[
3159	FormBox[
3160	OverscriptBox[
3161	FormBox["p4",
3162	TraditionalForm], "_"],
3163	TraditionalForm],
3164	FormBox[
3165	FormBox[
3166	FormBox["mu2",
3167	TraditionalForm],
3168	TraditionalForm],
3169	TraditionalForm]],
3170	TraditionalForm], " ",
3171	FormBox[
3172	SuperscriptBox[
3173	FormBox[
3174	OverscriptBox[
3175	FormBox["p2",
3176	TraditionalForm], "_"],
3177	TraditionalForm],
3178	FormBox[
3179	FormBox[
3180	FormBox["mu3",
3181	TraditionalForm],
3182	TraditionalForm],
3183	TraditionalForm]],
3184	TraditionalForm], " ",
3185	FormBox[
3186	SuperscriptBox[
3187	FormBox[
3188	OverscriptBox[
3189	FormBox["p4",
3190	TraditionalForm], "_"],
3191	TraditionalForm],
3192	FormBox[
3193	FormBox[
3194	FormBox["mu4",
3195	TraditionalForm],
3196	TraditionalForm],
3197	TraditionalForm]],
3198	TraditionalForm]}], "-",
3199	RowBox[{"u", " ",
3200	FormBox[
3201	SuperscriptBox[
3202	OverscriptBox["g", "_"],
3203	RowBox[{
3204	FormBox[
3205	FormBox[
3206	FormBox["mu1",
3207	TraditionalForm],
3208	TraditionalForm],
3209	TraditionalForm],
3210	FormBox[
3211	FormBox[
3212	FormBox["mu2",
3213	TraditionalForm],
3214	TraditionalForm],
3215	TraditionalForm]}]],
3216	TraditionalForm], " ",
3217	FormBox[
3218	SuperscriptBox[
3219	FormBox[
3220	OverscriptBox[
3221	FormBox["p4",
3222	TraditionalForm], "_"],
3223	TraditionalForm],
3224	FormBox[
3225	FormBox[
3226	FormBox["mu3",
3227	TraditionalForm],
3228	TraditionalForm],
3229	TraditionalForm]],
3230	TraditionalForm], " ",
3231	FormBox[
3232	SuperscriptBox[
3233	FormBox[
3234	OverscriptBox[
3235	FormBox["p4",
3236	TraditionalForm], "_"],
3237	TraditionalForm],
3238	FormBox[
3239	FormBox[
3240	FormBox["mu4",
3241	TraditionalForm],
3242	TraditionalForm],
3243	TraditionalForm]],
3244	TraditionalForm]}], "+",
3245	RowBox[{"\[Xi]", " ",
3246	FormBox[
3247	SuperscriptBox[
3248	FormBox[
3249	OverscriptBox[
3250	FormBox["p4",
3251	TraditionalForm], "_"],
3252	TraditionalForm],
3253	FormBox[
3254	FormBox[
3255	FormBox["mu1",
3256	TraditionalForm],
3257	TraditionalForm],
3258	TraditionalForm]],
3259	TraditionalForm], " ",
3260	FormBox[
3261	SuperscriptBox[
3262	FormBox[
3263	OverscriptBox[
3264	FormBox["p3",
3265	TraditionalForm], "_"],
3266	TraditionalForm],
3267	FormBox[
3268	FormBox[
3269	FormBox["mu2",
3270	TraditionalForm],
3271	TraditionalForm],
3272	TraditionalForm]],
3273	TraditionalForm], " ",
3274	FormBox[
3275	SuperscriptBox[
3276	FormBox[
3277	OverscriptBox[
3278	FormBox["p4",
3279	TraditionalForm], "_"],
3280	TraditionalForm],
3281	FormBox[
3282	FormBox[
3283	FormBox["mu3",
3284	TraditionalForm],
3285	TraditionalForm],
3286	TraditionalForm]],
3287	TraditionalForm], " ",
3288	FormBox[
3289	SuperscriptBox[
3290	FormBox[
3291	OverscriptBox[
3292	FormBox["p4",
3293	TraditionalForm], "_"],
3294	TraditionalForm],
3295	FormBox[
3296	FormBox[
3297	FormBox["mu4",
3298	TraditionalForm],
3299	TraditionalForm],
3300	TraditionalForm]],
3301	TraditionalForm]}], "-",
3302	RowBox[{
3303	FormBox[
3304	SuperscriptBox[
3305	FormBox[
3306	OverscriptBox[
3307	FormBox["p4",
3308	TraditionalForm], "_"],
3309	TraditionalForm],
3310	FormBox[
3311	FormBox[
3312	FormBox["mu1",
3313	TraditionalForm],
3314	TraditionalForm],
3315	TraditionalForm]],
3316	TraditionalForm], " ",
3317	FormBox[
3318	SuperscriptBox[
3319	FormBox[
3320	OverscriptBox[
3321	FormBox["p3",
3322	TraditionalForm], "_"],
3323	TraditionalForm],
3324	FormBox[
3325	FormBox[
3326	FormBox["mu2",
3327	TraditionalForm],
3328	TraditionalForm],
3329	TraditionalForm]],
3330	TraditionalForm], " ",
3331	FormBox[
3332	SuperscriptBox[
3333	FormBox[
3334	OverscriptBox[
3335	FormBox["p4",
3336	TraditionalForm], "_"],
3337	TraditionalForm],
3338	FormBox[
3339	FormBox[
3340	FormBox["mu3",
3341	TraditionalForm],
3342	TraditionalForm],
3343	TraditionalForm]],
3344	TraditionalForm], " ",
3345	FormBox[
3346	SuperscriptBox[
3347	FormBox[
3348	OverscriptBox[
3349	FormBox["p4",
3350	TraditionalForm], "_"],
3351	TraditionalForm],
3352	FormBox[
3353	FormBox[
3354	FormBox["mu4",
3355	TraditionalForm],
3356	TraditionalForm],
3357	TraditionalForm]],
3358	TraditionalForm]}]}], ")"}], " ",
3359	SuperscriptBox["f",
3360	RowBox[{
3361	FormBox[
3362	FormBox["a",
3363	TraditionalForm],
3364	TraditionalForm],
3365	FormBox[
3366	FormBox["a1",
3367	TraditionalForm],
3368	TraditionalForm],
3369	FormBox[
3370	FormBox["a4",
3371	TraditionalForm],
3372	TraditionalForm]}]], " ",
3373	SuperscriptBox["f",
3374	RowBox[{
3375	FormBox[
3376	FormBox["a",
3377	TraditionalForm],
3378	TraditionalForm],
3379	FormBox[
3380	FormBox["a2",
3381	TraditionalForm],
3382	TraditionalForm],
3383	FormBox[
3384	FormBox["a3",
3385	TraditionalForm],
3386	TraditionalForm]}]]}]}], TraditionalForm]], "Output",
3387	CellChangeTimes->{{3.692615928720977^9, 3.692615951914709^9}, {
3388	3.692616117201549^9, 3.692616128680936^9}, 3.692616419301153*^9, {
3389	3.692616482168438^9, 3.692616510932992^9}, 3.6926166662361393`*^9,
3390	3.6926167310253363`^9, 3.6926168994114122`^9, 3.6926169631011066`*^9,
3391	3.6926170672860193`^9, 3.6926172417327423`^9, 3.692617360263195*^9,
3392	3.692617935490901^9, 3.69261907267662^9, 3.692622060882275*^9,
3393	3.692940734448763^9, 3.692941454330248^9, {3.69312417316147*^9,
3394	3.693124180346133*^9}}]
3395	}, Open ]],
3396
3397	Cell[CellGroupData[{
3398
3399	Cell[BoxData[
3400	RowBox[{"Amp4g", "=",
3401	RowBox[{
3402	RowBox[{"Contract", "[",
3403	RowBox[{
3404	RowBox[{"FV", "[",
3405	RowBox[{"p1", ",", "mu1"}], "]"}], " ",
3406	RowBox[{"FV", "[",
3407	RowBox[{"e2", ",", "mu2"}], "]"}], " ",
3408	RowBox[{"FV", "[",
3409	RowBox[{"e3", ",", "mu3"}], "]"}], " ",
3410	RowBox[{"FV", "[",
3411	RowBox[{"e4", ",", "mu4"}], "]"}],
3412	RowBox[{"(",
3413	RowBox[{"diagS", "+", "diagT", "+", "diagU"}], ")"}]}], "]"}], "//",
3414	"Simplify"}]}]], "Input",
3415	CellChangeTimes->{{3.692615828534959^9, 3.692615877130622^9}, {
3416	3.69261597173477^9, 3.692616003979336^9}, {3.692616043275064*^9,
3417	3.692616079909423^9}, {3.6926161937429934`^9, 3.692616196644211*^9}, {
3418	3.692617056334735^9, 3.6926170565665483`^9}, {3.692617315076572*^9,
3419	3.692617319555043^9}, {3.6926173862851553`^9, 3.692617436293618*^9}, {
3420	3.6926174700247602`^9, 3.692617524395886^9}, {3.69262202405676*^9,
3421	3.692622024494514*^9}}],
3422
3423	Cell[BoxData[
3424	FormBox[
3425	RowBox[{"-",
3426	RowBox[{
3427	FractionBox["1",
3428	RowBox[{"s", " ", "t", " ", "u"}]],
3429	RowBox[{"\[ImaginaryI]", " ",
3430	SuperscriptBox["gs", "2"], " ",
3431	RowBox[{"(",
3432	RowBox[{
3433	RowBox[{"s", " ",
3434	RowBox[{"(",
3435	RowBox[{
3436	RowBox[{
3437	RowBox[{"-", "t"}], " ", "u", " ",
3438	RowBox[{"(",
3439	RowBox[{
3440	FormBox[
3441	OverscriptBox[
3442	FormBox["e2",
3443	TraditionalForm], "_"],
3444	TraditionalForm],
3445	FormBox["\<\"\[CenterDot]\"\>",
3446	TraditionalForm],
3447	FormBox[
3448	OverscriptBox[
3449	FormBox["e3",
3450	TraditionalForm], "_"],
3451	TraditionalForm]}], ")"}], " ",
3452	RowBox[{"(",
3453	RowBox[{
3454	FormBox[
3455	OverscriptBox[
3456	FormBox["e4",
3457	TraditionalForm], "_"],
3458	TraditionalForm],
3459	FormBox["\<\"\[CenterDot]\"\>",
3460	TraditionalForm],
3461	FormBox[
3462	OverscriptBox[
3463	FormBox["p2",
3464	TraditionalForm], "_"],
3465	TraditionalForm]}], ")"}], " ",
3466	SuperscriptBox["f",
3467	RowBox[{
3468	FormBox[
3469	FormBox["a",
3470	TraditionalForm],
3471	TraditionalForm],
3472	FormBox[
3473	FormBox["a1",
3474	TraditionalForm],
3475	TraditionalForm],
3476	FormBox[
3477	FormBox["a4",
3478	TraditionalForm],
3479	TraditionalForm]}]], " ",
3480	SuperscriptBox["f",
3481	RowBox[{
3482	FormBox[
3483	FormBox["a",
3484	TraditionalForm],
3485	TraditionalForm],
3486	FormBox[
3487	FormBox["a2",
3488	TraditionalForm],
3489	TraditionalForm],
3490	FormBox[
3491	FormBox["a3",
3492	TraditionalForm],
3493	TraditionalForm]}]]}], "-",
3494	RowBox[{"t", " ", "u", " ",
3495	RowBox[{"(",
3496	RowBox[{
3497	FormBox[
3498	OverscriptBox[
3499	FormBox["e3",
3500	TraditionalForm], "_"],
3501	TraditionalForm],
3502	FormBox["\<\"\[CenterDot]\"\>",
3503	TraditionalForm],
3504	FormBox[
3505	OverscriptBox[
3506	FormBox["e4",
3507	TraditionalForm], "_"],
3508	TraditionalForm]}], ")"}], " ",
3509	RowBox[{"(",
3510	FormBox[
3511	RowBox[{
3512	FormBox[
3513	OverscriptBox[
3514	FormBox["e2",
3515	TraditionalForm], "_"],
3516	TraditionalForm],
3517	FormBox["\<\"\[CenterDot]\"\>",
3518	TraditionalForm],
3519	FormBox[
3520	OverscriptBox[
3521	FormBox["p1",
3522	TraditionalForm], "_"],
3523	TraditionalForm]}],
3524	TraditionalForm], ")"}], " ",
3525	SuperscriptBox["f",
3526	RowBox[{
3527	FormBox[
3528	FormBox["a",
3529	TraditionalForm],
3530	TraditionalForm],
3531	FormBox[
3532	FormBox["a1",
3533	TraditionalForm],
3534	TraditionalForm],
3535	FormBox[
3536	FormBox["a4",
3537	TraditionalForm],
3538	TraditionalForm]}]], " ",
3539	SuperscriptBox["f",
3540	RowBox[{
3541	FormBox[
3542	FormBox["a",
3543	TraditionalForm],
3544	TraditionalForm],
3545	FormBox[
3546	FormBox["a2",
3547	TraditionalForm],
3548	TraditionalForm],
3549	FormBox[
3550	FormBox["a3",
3551	TraditionalForm],
3552	TraditionalForm]}]]}], "+",
3553	RowBox[{"t", " ",
3554	RowBox[{"(",
3555	RowBox[{
3556	FormBox[
3557	OverscriptBox[
3558	FormBox["e3",
3559	TraditionalForm], "_"],
3560	TraditionalForm],
3561	FormBox["\<\"\[CenterDot]\"\>",
3562	TraditionalForm],
3563	FormBox[
3564	OverscriptBox[
3565	FormBox["p2",
3566	TraditionalForm], "_"],
3567	TraditionalForm]}], ")"}], " ",
3568	RowBox[{"(",
3569	RowBox[{
3570	FormBox[
3571	OverscriptBox[
3572	FormBox["e4",
3573	TraditionalForm], "_"],
3574	TraditionalForm],
3575	FormBox["\<\"\[CenterDot]\"\>",
3576	TraditionalForm],
3577	FormBox[
3578	OverscriptBox[
3579	FormBox["p1",
3580	TraditionalForm], "_"],
3581	TraditionalForm]}], ")"}], " ",
3582	RowBox[{"(",
3583	FormBox[
3584	RowBox[{
3585	FormBox[
3586	OverscriptBox[
3587	FormBox["e2",
3588	TraditionalForm], "_"],
3589	TraditionalForm],
3590	FormBox["\<\"\[CenterDot]\"\>",
3591	TraditionalForm],
3592	FormBox[
3593	OverscriptBox[
3594	FormBox["p1",
3595	TraditionalForm], "_"],
3596	TraditionalForm]}],
3597	TraditionalForm], ")"}], " ",
3598	SuperscriptBox["f",
3599	RowBox[{
3600	FormBox[
3601	FormBox["a",
3602	TraditionalForm],
3603	TraditionalForm],
3604	FormBox[
3605	FormBox["a1",
3606	TraditionalForm],
3607	TraditionalForm],
3608	FormBox[
3609	FormBox["a4",
3610	TraditionalForm],
3611	TraditionalForm]}]], " ",
3612	SuperscriptBox["f",
3613	RowBox[{
3614	FormBox[
3615	FormBox["a",
3616	TraditionalForm],
3617	TraditionalForm],
3618	FormBox[
3619	FormBox["a2",
3620	TraditionalForm],
3621	TraditionalForm],
3622	FormBox[
3623	FormBox["a3",
3624	TraditionalForm],
3625	TraditionalForm]}]]}], "+",
3626	RowBox[{"t", " ", "u", " ",
3627	RowBox[{"(",
3628	RowBox[{
3629	FormBox[
3630	OverscriptBox[
3631	FormBox["e3",
3632	TraditionalForm], "_"],
3633	TraditionalForm],
3634	FormBox["\<\"\[CenterDot]\"\>",
3635	TraditionalForm],
3636	FormBox[
3637	OverscriptBox[
3638	FormBox["e4",
3639	TraditionalForm], "_"],
3640	TraditionalForm]}], ")"}], " ",
3641	RowBox[{"(",
3642	FormBox[
3643	RowBox[{
3644	FormBox[
3645	OverscriptBox[
3646	FormBox["e2",
3647	TraditionalForm], "_"],
3648	TraditionalForm],
3649	FormBox["\<\"\[CenterDot]\"\>",
3650	TraditionalForm],
3651	FormBox[
3652	OverscriptBox[
3653	FormBox["p3",
3654	TraditionalForm], "_"],
3655	TraditionalForm]}],
3656	TraditionalForm], ")"}], " ",
3657	SuperscriptBox["f",
3658	RowBox[{
3659	FormBox[
3660	FormBox["a",
3661	TraditionalForm],
3662	TraditionalForm],
3663	FormBox[
3664	FormBox["a1",
3665	TraditionalForm],
3666	TraditionalForm],
3667	FormBox[
3668	FormBox["a4",
3669	TraditionalForm],
3670	TraditionalForm]}]], " ",
3671	SuperscriptBox["f",
3672	RowBox[{
3673	FormBox[
3674	FormBox["a",
3675	TraditionalForm],
3676	TraditionalForm],
3677	FormBox[
3678	FormBox["a2",
3679	TraditionalForm],
3680	TraditionalForm],
3681	FormBox[
3682	FormBox["a3",
3683	TraditionalForm],
3684	TraditionalForm]}]]}], "-",
3685	RowBox[{"t", " ",
3686	RowBox[{"(",
3687	RowBox[{
3688	FormBox[
3689	OverscriptBox[
3690	FormBox["e3",
3691	TraditionalForm], "_"],
3692	TraditionalForm],
3693	FormBox["\<\"\[CenterDot]\"\>",
3694	TraditionalForm],
3695	FormBox[
3696	OverscriptBox[
3697	FormBox["p2",
3698	TraditionalForm], "_"],
3699	TraditionalForm]}], ")"}], " ",
3700	RowBox[{"(",
3701	RowBox[{
3702	FormBox[
3703	OverscriptBox[
3704	FormBox["e4",
3705	TraditionalForm], "_"],
3706	TraditionalForm],
3707	FormBox["\<\"\[CenterDot]\"\>",
3708	TraditionalForm],
3709	FormBox[
3710	OverscriptBox[
3711	FormBox["p1",
3712	TraditionalForm], "_"],
3713	TraditionalForm]}], ")"}], " ",
3714	RowBox[{"(",
3715	FormBox[
3716	RowBox[{
3717	FormBox[
3718	OverscriptBox[
3719	FormBox["e2",
3720	TraditionalForm], "_"],
3721	TraditionalForm],
3722	FormBox["\<\"\[CenterDot]\"\>",
3723	TraditionalForm],
3724	FormBox[
3725	OverscriptBox[
3726	FormBox["p3",
3727	TraditionalForm], "_"],
3728	TraditionalForm]}],
3729	TraditionalForm], ")"}], " ",
3730	SuperscriptBox["f",
3731	RowBox[{
3732	FormBox[
3733	FormBox["a",
3734	TraditionalForm],
3735	TraditionalForm],
3736	FormBox[
3737	FormBox["a1",
3738	TraditionalForm],
3739	TraditionalForm],
3740	FormBox[
3741	FormBox["a4",
3742	TraditionalForm],
3743	TraditionalForm]}]], " ",
3744	SuperscriptBox["f",
3745	RowBox[{
3746	FormBox[
3747	FormBox["a",
3748	TraditionalForm],
3749	TraditionalForm],
3750	FormBox[
3751	FormBox["a2",
3752	TraditionalForm],
3753	TraditionalForm],
3754	FormBox[
3755	FormBox["a3",
3756	TraditionalForm],
3757	TraditionalForm]}]]}], "-",
3758	RowBox[{"t", " ", "u", " ",
3759	RowBox[{"(",
3760	RowBox[{
3761	FormBox[
3762	OverscriptBox[
3763	FormBox["e2",
3764	TraditionalForm], "_"],
3765	TraditionalForm],
3766	FormBox["\<\"\[CenterDot]\"\>",
3767	TraditionalForm],
3768	FormBox[
3769	OverscriptBox[
3770	FormBox["e3",
3771	TraditionalForm], "_"],
3772	TraditionalForm]}], ")"}], " ",
3773	RowBox[{"(",
3774	FormBox[
3775	RowBox[{
3776	FormBox[
3777	OverscriptBox[
3778	FormBox["e4",
3779	TraditionalForm], "_"],
3780	TraditionalForm],
3781	FormBox["\<\"\[CenterDot]\"\>",
3782	TraditionalForm],
3783	FormBox[
3784	OverscriptBox[
3785	FormBox["p1",
3786	TraditionalForm], "_"],
3787	TraditionalForm]}],
3788	TraditionalForm], ")"}], " ",
3789	SuperscriptBox["f",
3790	RowBox[{
3791	FormBox[
3792	FormBox["a",
3793	TraditionalForm],
3794	TraditionalForm],
3795	FormBox[
3796	FormBox["a1",
3797	TraditionalForm],
3798	TraditionalForm],
3799	FormBox[
3800	FormBox["a4",
3801	TraditionalForm],
3802	TraditionalForm]}]], " ",
3803	SuperscriptBox["f",
3804	RowBox[{
3805	FormBox[
3806	FormBox["a",
3807	TraditionalForm],
3808	TraditionalForm],
3809	FormBox[
3810	FormBox["a2",
3811	TraditionalForm],
3812	TraditionalForm],
3813	FormBox[
3814	FormBox["a3",
3815	TraditionalForm],
3816	TraditionalForm]}]]}], "-",
3817	RowBox[{"t", " ", "u", " ",
3818	RowBox[{"(",
3819	RowBox[{
3820	FormBox[
3821	OverscriptBox[
3822	FormBox["e2",
3823	TraditionalForm], "_"],
3824	TraditionalForm],
3825	FormBox["\<\"\[CenterDot]\"\>",
3826	TraditionalForm],
3827	FormBox[
3828	OverscriptBox[
3829	FormBox["e3",
3830	TraditionalForm], "_"],
3831	TraditionalForm]}], ")"}], " ",
3832	RowBox[{"(",
3833	FormBox[
3834	RowBox[{
3835	FormBox[
3836	OverscriptBox[
3837	FormBox["e4",
3838	TraditionalForm], "_"],
3839	TraditionalForm],
3840	FormBox["\<\"\[CenterDot]\"\>",
3841	TraditionalForm],
3842	FormBox[
3843	OverscriptBox[
3844	FormBox["p2",
3845	TraditionalForm], "_"],
3846	TraditionalForm]}],
3847	TraditionalForm], ")"}], " ",
3848	SuperscriptBox["f",
3849	RowBox[{
3850	FormBox[
3851	FormBox["a",
3852	TraditionalForm],
3853	TraditionalForm],
3854	FormBox[
3855	FormBox["a1",
3856	TraditionalForm],
3857	TraditionalForm],
3858	FormBox[
3859	FormBox["a4",
3860	TraditionalForm],
3861	TraditionalForm]}]], " ",
3862	SuperscriptBox["f",
3863	RowBox[{
3864	FormBox[
3865	FormBox["a",
3866	TraditionalForm],
3867	TraditionalForm],
3868	FormBox[
3869	FormBox["a2",
3870	TraditionalForm],
3871	TraditionalForm],
3872	FormBox[
3873	FormBox["a3",
3874	TraditionalForm],
3875	TraditionalForm]}]]}], "-",
3876	RowBox[{"t", " ", "u", " ",
3877	RowBox[{"(",
3878	RowBox[{
3879	FormBox[
3880	OverscriptBox[
3881	FormBox["e2",
3882	TraditionalForm], "_"],
3883	TraditionalForm],
3884	FormBox["\<\"\[CenterDot]\"\>",
3885	TraditionalForm],
3886	FormBox[
3887	OverscriptBox[
3888	FormBox["e3",
3889	TraditionalForm], "_"],
3890	TraditionalForm]}], ")"}], " ",
3891	RowBox[{"(",
3892	RowBox[{
3893	FormBox[
3894	OverscriptBox[
3895	FormBox["e4",
3896	TraditionalForm], "_"],
3897	TraditionalForm],
3898	FormBox["\<\"\[CenterDot]\"\>",
3899	TraditionalForm],
3900	FormBox[
3901	OverscriptBox[
3902	FormBox["p1",
3903	TraditionalForm], "_"],
3904	TraditionalForm]}], ")"}], " ",
3905	SuperscriptBox["f",
3906	RowBox[{
3907	FormBox[
3908	FormBox["a",
3909	TraditionalForm],
3910	TraditionalForm],
3911	FormBox[
3912	FormBox["a1",
3913	TraditionalForm],
3914	TraditionalForm],
3915	FormBox[
3916	FormBox["a3",
3917	TraditionalForm],
3918	TraditionalForm]}]], " ",
3919	SuperscriptBox["f",
3920	RowBox[{
3921	FormBox[
3922	FormBox["a",
3923	TraditionalForm],
3924	TraditionalForm],
3925	FormBox[
3926	FormBox["a2",
3927	TraditionalForm],
3928	TraditionalForm],
3929	FormBox[
3930	FormBox["a4",
3931	TraditionalForm],
3932	TraditionalForm]}]]}], "+",
3933	RowBox[{"t", " ", "u", " ",
3934	RowBox[{"(",
3935	RowBox[{
3936	FormBox[
3937	OverscriptBox[
3938	FormBox["e2",
3939	TraditionalForm], "_"],
3940	TraditionalForm],
3941	FormBox["\<\"\[CenterDot]\"\>",
3942	TraditionalForm],
3943	FormBox[
3944	OverscriptBox[
3945	FormBox["e3",
3946	TraditionalForm], "_"],
3947	TraditionalForm]}], ")"}], " ",
3948	RowBox[{"(",
3949	RowBox[{
3950	FormBox[
3951	OverscriptBox[
3952	FormBox["e4",
3953	TraditionalForm], "_"],
3954	TraditionalForm],
3955	FormBox["\<\"\[CenterDot]\"\>",
3956	TraditionalForm],
3957	FormBox[
3958	OverscriptBox[
3959	FormBox["p2",
3960	TraditionalForm], "_"],
3961	TraditionalForm]}], ")"}], " ",
3962	SuperscriptBox["f",
3963	RowBox[{
3964	FormBox[
3965	FormBox["a",
3966	TraditionalForm],
3967	TraditionalForm],
3968	FormBox[
3969	FormBox["a1",
3970	TraditionalForm],
3971	TraditionalForm],
3972	FormBox[
3973	FormBox["a3",
3974	TraditionalForm],
3975	TraditionalForm]}]], " ",
3976	SuperscriptBox["f",
3977	RowBox[{
3978	FormBox[
3979	FormBox["a",
3980	TraditionalForm],
3981	TraditionalForm],
3982	FormBox[
3983	FormBox["a2",
3984	TraditionalForm],
3985	TraditionalForm],
3986	FormBox[
3987	FormBox["a4",
3988	TraditionalForm],
3989	TraditionalForm]}]]}], "+",
3990	RowBox[{"t", " ", "u", " ",
3991	RowBox[{"(",
3992	RowBox[{
3993	FormBox[
3994	OverscriptBox[
3995	FormBox["e3",
3996	TraditionalForm], "_"],
3997	TraditionalForm],
3998	FormBox["\<\"\[CenterDot]\"\>",
3999	TraditionalForm],
4000	FormBox[
4001	OverscriptBox[
4002	FormBox["e4",
4003	TraditionalForm], "_"],
4004	TraditionalForm]}], ")"}], " ",
4005	RowBox[{"(",
4006	FormBox[
4007	RowBox[{
4008	FormBox[
4009	OverscriptBox[
4010	FormBox["e2",
4011	TraditionalForm], "_"],
4012	TraditionalForm],
4013	FormBox["\<\"\[CenterDot]\"\>",
4014	TraditionalForm],
4015	FormBox[
4016	OverscriptBox[
4017	FormBox["p1",
4018	TraditionalForm], "_"],
4019	TraditionalForm]}],
4020	TraditionalForm], ")"}], " ",
4021	SuperscriptBox["f",
4022	RowBox[{
4023	FormBox[
4024	FormBox["a",
4025	TraditionalForm],
4026	TraditionalForm],
4027	FormBox[
4028	FormBox["a1",
4029	TraditionalForm],
4030	TraditionalForm],
4031	FormBox[
4032	FormBox["a3",
4033	TraditionalForm],
4034	TraditionalForm]}]], " ",
4035	SuperscriptBox["f",
4036	RowBox[{
4037	FormBox[
4038	FormBox["a",
4039	TraditionalForm],
4040	TraditionalForm],
4041	FormBox[
4042	FormBox["a2",
4043	TraditionalForm],
4044	TraditionalForm],
4045	FormBox[
4046	FormBox["a4",
4047	TraditionalForm],
4048	TraditionalForm]}]]}], "-",
4049	RowBox[{"u", " ",
4050	RowBox[{"(",
4051	RowBox[{
4052	FormBox[
4053	OverscriptBox[
4054	FormBox["e3",
4055	TraditionalForm], "_"],
4056	TraditionalForm],
4057	FormBox["\<\"\[CenterDot]\"\>",
4058	TraditionalForm],
4059	FormBox[
4060	OverscriptBox[
4061	FormBox["p1",
4062	TraditionalForm], "_"],
4063	TraditionalForm]}], ")"}], " ",
4064	RowBox[{"(",
4065	RowBox[{
4066	FormBox[
4067	OverscriptBox[
4068	FormBox["e4",
4069	TraditionalForm], "_"],
4070	TraditionalForm],
4071	FormBox["\<\"\[CenterDot]\"\>",
4072	TraditionalForm],
4073	FormBox[
4074	OverscriptBox[
4075	FormBox["p1",
4076	TraditionalForm], "_"],
4077	TraditionalForm]}], ")"}], " ",
4078	RowBox[{"(",
4079	FormBox[
4080	RowBox[{
4081	FormBox[
4082	OverscriptBox[
4083	FormBox["e2",
4084	TraditionalForm], "_"],
4085	TraditionalForm],
4086	FormBox["\<\"\[CenterDot]\"\>",
4087	TraditionalForm],
4088	FormBox[
4089	OverscriptBox[
4090	FormBox["p1",
4091	TraditionalForm], "_"],
4092	TraditionalForm]}],
4093	TraditionalForm], ")"}], " ",
4094	SuperscriptBox["f",
4095	RowBox[{
4096	FormBox[
4097	FormBox["a",
4098	TraditionalForm],
4099	TraditionalForm],
4100	FormBox[
4101	FormBox["a1",
4102	TraditionalForm],
4103	TraditionalForm],
4104	FormBox[
4105	FormBox["a3",
4106	TraditionalForm],
4107	TraditionalForm]}]], " ",
4108	SuperscriptBox["f",
4109	RowBox[{
4110	FormBox[
4111	FormBox["a",
4112	TraditionalForm],
4113	TraditionalForm],
4114	FormBox[
4115	FormBox["a2",
4116	TraditionalForm],
4117	TraditionalForm],
4118	FormBox[
4119	FormBox["a4",
4120	TraditionalForm],
4121	TraditionalForm]}]]}], "-",
4122	RowBox[{"t", " ", "u", " ",
4123	RowBox[{"(",
4124	RowBox[{
4125	FormBox[
4126	OverscriptBox[
4127	FormBox["e3",
4128	TraditionalForm], "_"],
4129	TraditionalForm],
4130	FormBox["\<\"\[CenterDot]\"\>",
4131	TraditionalForm],
4132	FormBox[
4133	OverscriptBox[
4134	FormBox["e4",
4135	TraditionalForm], "_"],
4136	TraditionalForm]}], ")"}], " ",
4137	RowBox[{"(",
4138	FormBox[
4139	RowBox[{
4140	FormBox[
4141	OverscriptBox[
4142	FormBox["e2",
4143	TraditionalForm], "_"],
4144	TraditionalForm],
4145	FormBox["\<\"\[CenterDot]\"\>",
4146	TraditionalForm],
4147	FormBox[
4148	OverscriptBox[
4149	FormBox["p3",
4150	TraditionalForm], "_"],
4151	TraditionalForm]}],
4152	TraditionalForm], ")"}], " ",
4153	SuperscriptBox["f",
4154	RowBox[{
4155	FormBox[
4156	FormBox["a",
4157	TraditionalForm],
4158	TraditionalForm],
4159	FormBox[
4160	FormBox["a1",
4161	TraditionalForm],
4162	TraditionalForm],
4163	FormBox[
4164	FormBox["a3",
4165	TraditionalForm],
4166	TraditionalForm]}]], " ",
4167	SuperscriptBox["f",
4168	RowBox[{
4169	FormBox[
4170	FormBox["a",
4171	TraditionalForm],
4172	TraditionalForm],
4173	FormBox[
4174	FormBox["a2",
4175	TraditionalForm],
4176	TraditionalForm],
4177	FormBox[
4178	FormBox["a4",
4179	TraditionalForm],
4180	TraditionalForm]}]]}], "+",
4181	RowBox[{"u", " ",
4182	RowBox[{"(",
4183	RowBox[{
4184	FormBox[
4185	OverscriptBox[
4186	FormBox["e3",
4187	TraditionalForm], "_"],
4188	TraditionalForm],
4189	FormBox["\<\"\[CenterDot]\"\>",
4190	TraditionalForm],
4191	FormBox[
4192	OverscriptBox[
4193	FormBox["p1",
4194	TraditionalForm], "_"],
4195	TraditionalForm]}], ")"}], " ",
4196	RowBox[{"(",
4197	RowBox[{
4198	FormBox[
4199	OverscriptBox[
4200	FormBox["e4",
4201	TraditionalForm], "_"],
4202	TraditionalForm],
4203	FormBox["\<\"\[CenterDot]\"\>",
4204	TraditionalForm],
4205	FormBox[
4206	OverscriptBox[
4207	FormBox["p1",
4208	TraditionalForm], "_"],
4209	TraditionalForm]}], ")"}], " ",
4210	RowBox[{"(",
4211	FormBox[
4212	RowBox[{
4213	FormBox[
4214	OverscriptBox[
4215	FormBox["e2",
4216	TraditionalForm], "_"],
4217	TraditionalForm],
4218	FormBox["\<\"\[CenterDot]\"\>",
4219	TraditionalForm],
4220	FormBox[
4221	OverscriptBox[
4222	FormBox["p3",
4223	TraditionalForm], "_"],
4224	TraditionalForm]}],
4225	TraditionalForm], ")"}], " ",
4226	SuperscriptBox["f",
4227	RowBox[{
4228	FormBox[
4229	FormBox["a",
4230	TraditionalForm],
4231	TraditionalForm],
4232	FormBox[
4233	FormBox["a1",
4234	TraditionalForm],
4235	TraditionalForm],
4236	FormBox[
4237	FormBox["a3",
4238	TraditionalForm],
4239	TraditionalForm]}]], " ",
4240	SuperscriptBox["f",
4241	RowBox[{
4242	FormBox[
4243	FormBox["a",
4244	TraditionalForm],
4245	TraditionalForm],
4246	FormBox[
4247	FormBox["a2",
4248	TraditionalForm],
4249	TraditionalForm],
4250	FormBox[
4251	FormBox["a4",
4252	TraditionalForm],
4253	TraditionalForm]}]]}], "+",
4254	RowBox[{"t", " ", "u", " ",
4255	RowBox[{"(",
4256	RowBox[{
4257	FormBox[
4258	OverscriptBox[
4259	FormBox["e2",
4260	TraditionalForm], "_"],
4261	TraditionalForm],
4262	FormBox["\<\"\[CenterDot]\"\>",
4263	TraditionalForm],
4264	FormBox[
4265	OverscriptBox[
4266	FormBox["e3",
4267	TraditionalForm], "_"],
4268	TraditionalForm]}], ")"}], " ",
4269	RowBox[{"(",
4270	FormBox[
4271	RowBox[{
4272	FormBox[
4273	OverscriptBox[
4274	FormBox["e4",
4275	TraditionalForm], "_"],
4276	TraditionalForm],
4277	FormBox["\<\"\[CenterDot]\"\>",
4278	TraditionalForm],
4279	FormBox[
4280	OverscriptBox[
4281	FormBox["p1",
4282	TraditionalForm], "_"],
4283	TraditionalForm]}],
4284	TraditionalForm], ")"}], " ",
4285	SuperscriptBox["f",
4286	RowBox[{
4287	FormBox[
4288	FormBox["a",
4289	TraditionalForm],
4290	TraditionalForm],
4291	FormBox[
4292	FormBox["a1",
4293	TraditionalForm],
4294	TraditionalForm],
4295	FormBox[
4296	FormBox["a3",
4297	TraditionalForm],
4298	TraditionalForm]}]], " ",
4299	SuperscriptBox["f",
4300	RowBox[{
4301	FormBox[
4302	FormBox["a",
4303	TraditionalForm],
4304	TraditionalForm],
4305	FormBox[
4306	FormBox["a2",
4307	TraditionalForm],
4308	TraditionalForm],
4309	FormBox[
4310	FormBox["a4",
4311	TraditionalForm],
4312	TraditionalForm]}]]}], "+",
4313	RowBox[{"u", " ",
4314	RowBox[{"(",
4315	RowBox[{
4316	FormBox[
4317	OverscriptBox[
4318	FormBox["e3",
4319	TraditionalForm], "_"],
4320	TraditionalForm],
4321	FormBox["\<\"\[CenterDot]\"\>",
4322	TraditionalForm],
4323	FormBox[
4324	OverscriptBox[
4325	FormBox["p1",
4326	TraditionalForm], "_"],
4327	TraditionalForm]}], ")"}], " ",
4328	RowBox[{"(",
4329	FormBox[
4330	RowBox[{
4331	FormBox[
4332	OverscriptBox[
4333	FormBox["e2",
4334	TraditionalForm], "_"],
4335	TraditionalForm],
4336	FormBox["\<\"\[CenterDot]\"\>",
4337	TraditionalForm],
4338	FormBox[
4339	OverscriptBox[
4340	FormBox["p1",
4341	TraditionalForm], "_"],
4342	TraditionalForm]}],
4343	TraditionalForm], ")"}], " ",
4344	RowBox[{"(",
4345	FormBox[
4346	RowBox[{
4347	FormBox[
4348	OverscriptBox[
4349	FormBox["e4",
4350	TraditionalForm], "_"],
4351	TraditionalForm],
4352	FormBox["\<\"\[CenterDot]\"\>",
4353	TraditionalForm],
4354	FormBox[
4355	OverscriptBox[
4356	FormBox["p1",
4357	TraditionalForm], "_"],
4358	TraditionalForm]}],
4359	TraditionalForm], ")"}], " ",
4360	SuperscriptBox["f",
4361	RowBox[{
4362	FormBox[
4363	FormBox["a",
4364	TraditionalForm],
4365	TraditionalForm],
4366	FormBox[
4367	FormBox["a1",
4368	TraditionalForm],
4369	TraditionalForm],
4370	FormBox[
4371	FormBox["a3",
4372	TraditionalForm],
4373	TraditionalForm]}]], " ",
4374	SuperscriptBox["f",
4375	RowBox[{
4376	FormBox[
4377	FormBox["a",
4378	TraditionalForm],
4379	TraditionalForm],
4380	FormBox[
4381	FormBox["a2",
4382	TraditionalForm],
4383	TraditionalForm],
4384	FormBox[
4385	FormBox["a4",
4386	TraditionalForm],
4387	TraditionalForm]}]]}], "-",
4388	RowBox[{"u", " ",
4389	RowBox[{"(",
4390	RowBox[{
4391	FormBox[
4392	OverscriptBox[
4393	FormBox["e3",
4394	TraditionalForm], "_"],
4395	TraditionalForm],
4396	FormBox["\<\"\[CenterDot]\"\>",
4397	TraditionalForm],
4398	FormBox[
4399	OverscriptBox[
4400	FormBox["p1",
4401	TraditionalForm], "_"],
4402	TraditionalForm]}], ")"}], " ",
4403	RowBox[{"(",
4404	FormBox[
4405	RowBox[{
4406	FormBox[
4407	OverscriptBox[
4408	FormBox["e2",
4409	TraditionalForm], "_"],
4410	TraditionalForm],
4411	FormBox["\<\"\[CenterDot]\"\>",
4412	TraditionalForm],
4413	FormBox[
4414	OverscriptBox[
4415	FormBox["p3",
4416	TraditionalForm], "_"],
4417	TraditionalForm]}],
4418	TraditionalForm], ")"}], " ",
4419	RowBox[{"(",
4420	FormBox[
4421	RowBox[{
4422	FormBox[
4423	OverscriptBox[
4424	FormBox["e4",
4425	TraditionalForm], "_"],
4426	TraditionalForm],
4427	FormBox["\<\"\[CenterDot]\"\>",
4428	TraditionalForm],
4429	FormBox[
4430	OverscriptBox[
4431	FormBox["p1",
4432	TraditionalForm], "_"],
4433	TraditionalForm]}],
4434	TraditionalForm], ")"}], " ",
4435	SuperscriptBox["f",
4436	RowBox[{
4437	FormBox[
4438	FormBox["a",
4439	TraditionalForm],
4440	TraditionalForm],
4441	FormBox[
4442	FormBox["a1",
4443	TraditionalForm],
4444	TraditionalForm],
4445	FormBox[
4446	FormBox["a3",
4447	TraditionalForm],
4448	TraditionalForm]}]], " ",
4449	SuperscriptBox["f",
4450	RowBox[{
4451	FormBox[
4452	FormBox["a",
4453	TraditionalForm],
4454	TraditionalForm],
4455	FormBox[
4456	FormBox["a2",
4457	TraditionalForm],
4458	TraditionalForm],
4459	FormBox[
4460	FormBox["a4",
4461	TraditionalForm],
4462	TraditionalForm]}]]}], "+",
4463	RowBox[{"t", " ", "u", " ",
4464	RowBox[{"(",
4465	RowBox[{
4466	FormBox[
4467	OverscriptBox[
4468	FormBox["e2",
4469	TraditionalForm], "_"],
4470	TraditionalForm],
4471	FormBox["\<\"\[CenterDot]\"\>",
4472	TraditionalForm],
4473	FormBox[
4474	OverscriptBox[
4475	FormBox["e3",
4476	TraditionalForm], "_"],
4477	TraditionalForm]}], ")"}], " ",
4478	RowBox[{"(",
4479	FormBox[
4480	RowBox[{
4481	FormBox[
4482	OverscriptBox[
4483	FormBox["e4",
4484	TraditionalForm], "_"],
4485	TraditionalForm],
4486	FormBox["\<\"\[CenterDot]\"\>",
4487	TraditionalForm],
4488	FormBox[
4489	OverscriptBox[
4490	FormBox["p2",
4491	TraditionalForm], "_"],
4492	TraditionalForm]}],
4493	TraditionalForm], ")"}], " ",
4494	SuperscriptBox["f",
4495	RowBox[{
4496	FormBox[
4497	FormBox["a",
4498	TraditionalForm],
4499	TraditionalForm],
4500	FormBox[
4501	FormBox["a1",
4502	TraditionalForm],
4503	TraditionalForm],
4504	FormBox[
4505	FormBox["a3",
4506	TraditionalForm],
4507	TraditionalForm]}]], " ",
4508	SuperscriptBox["f",
4509	RowBox[{
4510	FormBox[
4511	FormBox["a",
4512	TraditionalForm],
4513	TraditionalForm],
4514	FormBox[
4515	FormBox["a2",
4516	TraditionalForm],
4517	TraditionalForm],
4518	FormBox[
4519	FormBox["a4",
4520	TraditionalForm],
4521	TraditionalForm]}]]}], "+",
4522	RowBox[{"u", " ",
4523	RowBox[{"(",
4524	RowBox[{
4525	FormBox[
4526	OverscriptBox[
4527	FormBox["e3",
4528	TraditionalForm], "_"],
4529	TraditionalForm],
4530	FormBox["\<\"\[CenterDot]\"\>",
4531	TraditionalForm],
4532	FormBox[
4533	OverscriptBox[
4534	FormBox["p1",
4535	TraditionalForm], "_"],
4536	TraditionalForm]}], ")"}], " ",
4537	RowBox[{"(",
4538	FormBox[
4539	RowBox[{
4540	FormBox[
4541	OverscriptBox[
4542	FormBox["e2",
4543	TraditionalForm], "_"],
4544	TraditionalForm],
4545	FormBox["\<\"\[CenterDot]\"\>",
4546	TraditionalForm],
4547	FormBox[
4548	OverscriptBox[
4549	FormBox["p1",
4550	TraditionalForm], "_"],
4551	TraditionalForm]}],
4552	TraditionalForm], ")"}], " ",
4553	RowBox[{"(",
4554	FormBox[
4555	RowBox[{
4556	FormBox[
4557	OverscriptBox[
4558	FormBox["e4",
4559	TraditionalForm], "_"],
4560	TraditionalForm],
4561	FormBox["\<\"\[CenterDot]\"\>",
4562	TraditionalForm],
4563	FormBox[
4564	OverscriptBox[
4565	FormBox["p2",
4566	TraditionalForm], "_"],
4567	TraditionalForm]}],
4568	TraditionalForm], ")"}], " ",
4569	SuperscriptBox["f",
4570	RowBox[{
4571	FormBox[
4572	FormBox["a",
4573	TraditionalForm],
4574	TraditionalForm],
4575	FormBox[
4576	FormBox["a1",
4577	TraditionalForm],
4578	TraditionalForm],
4579	FormBox[
4580	FormBox["a3",
4581	TraditionalForm],
4582	TraditionalForm]}]], " ",
4583	SuperscriptBox["f",
4584	RowBox[{
4585	FormBox[
4586	FormBox["a",
4587	TraditionalForm],
4588	TraditionalForm],
4589	FormBox[
4590	FormBox["a2",
4591	TraditionalForm],
4592	TraditionalForm],
4593	FormBox[
4594	FormBox["a4",
4595	TraditionalForm],
4596	TraditionalForm]}]]}], "-",
4597	RowBox[{"u", " ",
4598	RowBox[{"(",
4599	RowBox[{
4600	FormBox[
4601	OverscriptBox[
4602	FormBox["e3",
4603	TraditionalForm], "_"],
4604	TraditionalForm],
4605	FormBox["\<\"\[CenterDot]\"\>",
4606	TraditionalForm],
4607	FormBox[
4608	OverscriptBox[
4609	FormBox["p1",
4610	TraditionalForm], "_"],
4611	TraditionalForm]}], ")"}], " ",
4612	RowBox[{"(",
4613	FormBox[
4614	RowBox[{
4615	FormBox[
4616	OverscriptBox[
4617	FormBox["e2",
4618	TraditionalForm], "_"],
4619	TraditionalForm],
4620	FormBox["\<\"\[CenterDot]\"\>",
4621	TraditionalForm],
4622	FormBox[
4623	OverscriptBox[
4624	FormBox["p3",
4625	TraditionalForm], "_"],
4626	TraditionalForm]}],
4627	TraditionalForm], ")"}], " ",
4628	RowBox[{"(",
4629	FormBox[
4630	RowBox[{
4631	FormBox[
4632	OverscriptBox[
4633	FormBox["e4",
4634	TraditionalForm], "_"],
4635	TraditionalForm],
4636	FormBox["\<\"\[CenterDot]\"\>",
4637	TraditionalForm],
4638	FormBox[
4639	OverscriptBox[
4640	FormBox["p2",
4641	TraditionalForm], "_"],
4642	TraditionalForm]}],
4643	TraditionalForm], ")"}], " ",
4644	SuperscriptBox["f",
4645	RowBox[{
4646	FormBox[
4647	FormBox["a",
4648	TraditionalForm],
4649	TraditionalForm],
4650	FormBox[
4651	FormBox["a1",
4652	TraditionalForm],
4653	TraditionalForm],
4654	FormBox[
4655	FormBox["a3",
4656	TraditionalForm],
4657	TraditionalForm]}]], " ",
4658	SuperscriptBox["f",
4659	RowBox[{
4660	FormBox[
4661	FormBox["a",
4662	TraditionalForm],
4663	TraditionalForm],
4664	FormBox[
4665	FormBox["a2",
4666	TraditionalForm],
4667	TraditionalForm],
4668	FormBox[
4669	FormBox["a4",
4670	TraditionalForm],
4671	TraditionalForm]}]]}], "-",
4672	RowBox[{"t", " ", "u", " ",
4673	RowBox[{"(",
4674	RowBox[{
4675	FormBox[
4676	OverscriptBox[
4677	FormBox["e2",
4678	TraditionalForm], "_"],
4679	TraditionalForm],
4680	FormBox["\<\"\[CenterDot]\"\>",
4681	TraditionalForm],
4682	FormBox[
4683	OverscriptBox[
4684	FormBox["e3",
4685	TraditionalForm], "_"],
4686	TraditionalForm]}], ")"}], " ",
4687	RowBox[{"(",
4688	RowBox[{
4689	FormBox[
4690	OverscriptBox[
4691	FormBox["e4",
4692	TraditionalForm], "_"],
4693	TraditionalForm],
4694	FormBox["\<\"\[CenterDot]\"\>",
4695	TraditionalForm],
4696	FormBox[
4697	OverscriptBox[
4698	FormBox["p1",
4699	TraditionalForm], "_"],
4700	TraditionalForm]}], ")"}], " ",
4701	SuperscriptBox["f",
4702	RowBox[{
4703	FormBox[
4704	FormBox["a",
4705	TraditionalForm],
4706	TraditionalForm],
4707	FormBox[
4708	FormBox["a1",
4709	TraditionalForm],
4710	TraditionalForm],
4711	FormBox[
4712	FormBox["a2",
4713	TraditionalForm],
4714	TraditionalForm]}]], " ",
4715	SuperscriptBox["f",
4716	RowBox[{
4717	FormBox[
4718	FormBox["a",
4719	TraditionalForm],
4720	TraditionalForm],
4721	FormBox[
4722	FormBox["a3",
4723	TraditionalForm],
4724	TraditionalForm],
4725	FormBox[
4726	FormBox["a4",
4727	TraditionalForm],
4728	TraditionalForm]}]]}], "-",
4729	RowBox[{"t", " ", "u", " ",
4730	RowBox[{"(",
4731	RowBox[{
4732	FormBox[
4733	OverscriptBox[
4734	FormBox["e2",
4735	TraditionalForm], "_"],
4736	TraditionalForm],
4737	FormBox["\<\"\[CenterDot]\"\>",
4738	TraditionalForm],
4739	FormBox[
4740	OverscriptBox[
4741	FormBox["e3",
4742	TraditionalForm], "_"],
4743	TraditionalForm]}], ")"}], " ",
4744	RowBox[{"(",
4745	RowBox[{
4746	FormBox[
4747	OverscriptBox[
4748	FormBox["e4",
4749	TraditionalForm], "_"],
4750	TraditionalForm],
4751	FormBox["\<\"\[CenterDot]\"\>",
4752	TraditionalForm],
4753	FormBox[
4754	OverscriptBox[
4755	FormBox["p2",
4756	TraditionalForm], "_"],
4757	TraditionalForm]}], ")"}], " ",
4758	SuperscriptBox["f",
4759	RowBox[{
4760	FormBox[
4761	FormBox["a",
4762	TraditionalForm],
4763	TraditionalForm],
4764	FormBox[
4765	FormBox["a1",
4766	TraditionalForm],
4767	TraditionalForm],
4768	FormBox[
4769	FormBox["a2",
4770	TraditionalForm],
4771	TraditionalForm]}]], " ",
4772	SuperscriptBox["f",
4773	RowBox[{
4774	FormBox[
4775	FormBox["a",
4776	TraditionalForm],
4777	TraditionalForm],
4778	FormBox[
4779	FormBox["a3",
4780	TraditionalForm],
4781	TraditionalForm],
4782	FormBox[
4783	FormBox["a4",
4784	TraditionalForm],
4785	TraditionalForm]}]]}], "-",
4786	RowBox[{"t", " ", "u", " ",
4787	RowBox[{"(",
4788	RowBox[{
4789	FormBox[
4790	OverscriptBox[
4791	FormBox["e3",
4792	TraditionalForm], "_"],
4793	TraditionalForm],
4794	FormBox["\<\"\[CenterDot]\"\>",
4795	TraditionalForm],
4796	FormBox[
4797	OverscriptBox[
4798	FormBox["e4",
4799	TraditionalForm], "_"],
4800	TraditionalForm]}], ")"}], " ",
4801	RowBox[{"(",
4802	FormBox[
4803	RowBox[{
4804	FormBox[
4805	OverscriptBox[
4806	FormBox["e2",
4807	TraditionalForm], "_"],
4808	TraditionalForm],
4809	FormBox["\<\"\[CenterDot]\"\>",
4810	TraditionalForm],
4811	FormBox[
4812	OverscriptBox[
4813	FormBox["p1",
4814	TraditionalForm], "_"],
4815	TraditionalForm]}],
4816	TraditionalForm], ")"}], " ",
4817	SuperscriptBox["f",
4818	RowBox[{
4819	FormBox[
4820	FormBox["a",
4821	TraditionalForm],
4822	TraditionalForm],
4823	FormBox[
4824	FormBox["a1",
4825	TraditionalForm],
4826	TraditionalForm],
4827	FormBox[
4828	FormBox["a2",
4829	TraditionalForm],
4830	TraditionalForm]}]], " ",
4831	SuperscriptBox["f",
4832	RowBox[{
4833	FormBox[
4834	FormBox["a",
4835	TraditionalForm],
4836	TraditionalForm],
4837	FormBox[
4838	FormBox["a3",
4839	TraditionalForm],
4840	TraditionalForm],
4841	FormBox[
4842	FormBox["a4",
4843	TraditionalForm],
4844	TraditionalForm]}]]}], "+",
4845	RowBox[{"t", " ", "u", " ",
4846	RowBox[{"(",
4847	RowBox[{
4848	FormBox[
4849	OverscriptBox[
4850	FormBox["e3",
4851	TraditionalForm], "_"],
4852	TraditionalForm],
4853	FormBox["\<\"\[CenterDot]\"\>",
4854	TraditionalForm],
4855	FormBox[
4856	OverscriptBox[
4857	FormBox["e4",
4858	TraditionalForm], "_"],
4859	TraditionalForm]}], ")"}], " ",
4860	RowBox[{"(",
4861	FormBox[
4862	RowBox[{
4863	FormBox[
4864	OverscriptBox[
4865	FormBox["e2",
4866	TraditionalForm], "_"],
4867	TraditionalForm],
4868	FormBox["\<\"\[CenterDot]\"\>",
4869	TraditionalForm],
4870	FormBox[
4871	OverscriptBox[
4872	FormBox["p3",
4873	TraditionalForm], "_"],
4874	TraditionalForm]}],
4875	TraditionalForm], ")"}], " ",
4876	SuperscriptBox["f",
4877	RowBox[{
4878	FormBox[
4879	FormBox["a",
4880	TraditionalForm],
4881	TraditionalForm],
4882	FormBox[
4883	FormBox["a1",
4884	TraditionalForm],
4885	TraditionalForm],
4886	FormBox[
4887	FormBox["a2",
4888	TraditionalForm],
4889	TraditionalForm]}]], " ",
4890	SuperscriptBox["f",
4891	RowBox[{
4892	FormBox[
4893	FormBox["a",
4894	TraditionalForm],
4895	TraditionalForm],
4896	FormBox[
4897	FormBox["a3",
4898	TraditionalForm],
4899	TraditionalForm],
4900	FormBox[
4901	FormBox["a4",
4902	TraditionalForm],
4903	TraditionalForm]}]]}], "-",
4904	RowBox[{"t", " ", "u", " ",
4905	RowBox[{"(",
4906	RowBox[{
4907	FormBox[
4908	OverscriptBox[
4909	FormBox["e2",
4910	TraditionalForm], "_"],
4911	TraditionalForm],
4912	FormBox["\<\"\[CenterDot]\"\>",
4913	TraditionalForm],
4914	FormBox[
4915	OverscriptBox[
4916	FormBox["e3",
4917	TraditionalForm], "_"],
4918	TraditionalForm]}], ")"}], " ",
4919	RowBox[{"(",
4920	FormBox[
4921	RowBox[{
4922	FormBox[
4923	OverscriptBox[
4924	FormBox["e4",
4925	TraditionalForm], "_"],
4926	TraditionalForm],
4927	FormBox["\<\"\[CenterDot]\"\>",
4928	TraditionalForm],
4929	FormBox[
4930	OverscriptBox[
4931	FormBox["p1",
4932	TraditionalForm], "_"],
4933	TraditionalForm]}],
4934	TraditionalForm], ")"}], " ",
4935	SuperscriptBox["f",
4936	RowBox[{
4937	FormBox[
4938	FormBox["a",
4939	TraditionalForm],
4940	TraditionalForm],
4941	FormBox[
4942	FormBox["a1",
4943	TraditionalForm],
4944	TraditionalForm],
4945	FormBox[
4946	FormBox["a2",
4947	TraditionalForm],
4948	TraditionalForm]}]], " ",
4949	SuperscriptBox["f",
4950	RowBox[{
4951	FormBox[
4952	FormBox["a",
4953	TraditionalForm],
4954	TraditionalForm],
4955	FormBox[
4956	FormBox["a3",
4957	TraditionalForm],
4958	TraditionalForm],
4959	FormBox[
4960	FormBox["a4",
4961	TraditionalForm],
4962	TraditionalForm]}]]}], "-",
4963	RowBox[{"t", " ", "u", " ",
4964	RowBox[{"(",
4965	RowBox[{
4966	FormBox[
4967	OverscriptBox[
4968	FormBox["e2",
4969	TraditionalForm], "_"],
4970	TraditionalForm],
4971	FormBox["\<\"\[CenterDot]\"\>",
4972	TraditionalForm],
4973	FormBox[
4974	OverscriptBox[
4975	FormBox["e3",
4976	TraditionalForm], "_"],
4977	TraditionalForm]}], ")"}], " ",
4978	RowBox[{"(",
4979	FormBox[
4980	RowBox[{
4981	FormBox[
4982	OverscriptBox[
4983	FormBox["e4",
4984	TraditionalForm], "_"],
4985	TraditionalForm],
4986	FormBox["\<\"\[CenterDot]\"\>",
4987	TraditionalForm],
4988	FormBox[
4989	OverscriptBox[
4990	FormBox["p2",
4991	TraditionalForm], "_"],
4992	TraditionalForm]}],
4993	TraditionalForm], ")"}], " ",
4994	SuperscriptBox["f",
4995	RowBox[{
4996	FormBox[
4997	FormBox["a",
4998	TraditionalForm],
4999	TraditionalForm],
5000	FormBox[
5001	FormBox["a1",
5002	TraditionalForm],
5003	TraditionalForm],
5004	FormBox[
5005	FormBox["a2",
5006	TraditionalForm],
5007	TraditionalForm]}]], " ",
5008	SuperscriptBox["f",
5009	RowBox[{
5010	FormBox[
5011	FormBox["a",
5012	TraditionalForm],
5013	TraditionalForm],
5014	FormBox[
5015	FormBox["a3",
5016	TraditionalForm],
5017	TraditionalForm],
5018	FormBox[
5019	FormBox["a4",
5020	TraditionalForm],
5021	TraditionalForm]}]]}], "+",
5022	RowBox[{
5023	RowBox[{"(",
5024	RowBox[{
5025	FormBox[
5026	OverscriptBox[
5027	FormBox["e2",
5028	TraditionalForm], "_"],
5029	TraditionalForm],
5030	FormBox["\<\"\[CenterDot]\"\>",
5031	TraditionalForm],
5032	FormBox[
5033	OverscriptBox[
5034	FormBox["p3",
5035	TraditionalForm], "_"],
5036	TraditionalForm]}], ")"}], " ",
5037	RowBox[{"(",
5038	RowBox[{
5039	RowBox[{"t", " ",
5040	RowBox[{"(",
5041	RowBox[{
5042	FormBox[
5043	OverscriptBox[
5044	FormBox["e4",
5045	TraditionalForm], "_"],
5046	TraditionalForm],
5047	FormBox["\<\"\[CenterDot]\"\>",
5048	TraditionalForm],
5049	FormBox[
5050	OverscriptBox[
5051	FormBox["p1",
5052	TraditionalForm], "_"],
5053	TraditionalForm]}], ")"}], " ",
5054	RowBox[{"(",
5055	RowBox[{
5056	FormBox[
5057	RowBox[{
5058	FormBox[
5059	OverscriptBox[
5060	FormBox["e3",
5061	TraditionalForm], "_"],
5062	TraditionalForm],
5063	FormBox["\<\"\[CenterDot]\"\>",
5064	TraditionalForm],
5065	FormBox[
5066	OverscriptBox[
5067	FormBox["p1",
5068	TraditionalForm], "_"],
5069	TraditionalForm]}],
5070	TraditionalForm], "+",
5071	FormBox[
5072	RowBox[{
5073	FormBox[
5074	OverscriptBox[
5075	FormBox["e3",
5076	TraditionalForm], "_"],
5077	TraditionalForm],
5078	FormBox["\<\"\[CenterDot]\"\>",
5079	TraditionalForm],
5080	FormBox[
5081	OverscriptBox[
5082	FormBox["p2",
5083	TraditionalForm], "_"],
5084	TraditionalForm]}],
5085	TraditionalForm]}], ")"}], " ",
5086	SuperscriptBox["f",
5087	RowBox[{
5088	FormBox[
5089	FormBox["a",
5090	TraditionalForm],
5091	TraditionalForm],
5092	FormBox[
5093	FormBox["a1",
5094	TraditionalForm],
5095	TraditionalForm],
5096	FormBox[
5097	FormBox["a4",
5098	TraditionalForm],
5099	TraditionalForm]}]], " ",
5100	SuperscriptBox["f",
5101	RowBox[{
5102	FormBox[
5103	FormBox["a",
5104	TraditionalForm],
5105	TraditionalForm],
5106	FormBox[
5107	FormBox["a2",
5108	TraditionalForm],
5109	TraditionalForm],
5110	FormBox[
5111	FormBox["a3",
5112	TraditionalForm],
5113	TraditionalForm]}]]}], "+",
5114	RowBox[{
5115	RowBox[{"(",
5116	RowBox[{
5117	FormBox[
5118	OverscriptBox[
5119	FormBox["e3",
5120	TraditionalForm], "_"],
5121	TraditionalForm],
5122	FormBox["\<\"\[CenterDot]\"\>",
5123	TraditionalForm],
5124	FormBox[
5125	OverscriptBox[
5126	FormBox["p1",
5127	TraditionalForm], "_"],
5128	TraditionalForm]}], ")"}], " ",
5129	RowBox[{"(",
5130	RowBox[{
5131	RowBox[{"u", " ",
5132	RowBox[{"(",
5133	RowBox[{
5134	FormBox[
5135	OverscriptBox[
5136	FormBox["e4",
5137	TraditionalForm], "_"],
5138	TraditionalForm],
5139	FormBox["\<\"\[CenterDot]\"\>",
5140	TraditionalForm],
5141	FormBox[
5142	OverscriptBox[
5143	FormBox["p2",
5144	TraditionalForm], "_"],
5145	TraditionalForm]}], ")"}], " ",
5146	SuperscriptBox["f",
5147	RowBox[{
5148	FormBox[
5149	FormBox["a",
5150	TraditionalForm],
5151	TraditionalForm],
5152	FormBox[
5153	FormBox["a1",
5154	TraditionalForm],
5155	TraditionalForm],
5156	FormBox[
5157	FormBox["a3",
5158	TraditionalForm],
5159	TraditionalForm]}]], " ",
5160	SuperscriptBox["f",
5161	RowBox[{
5162	FormBox[
5163	FormBox["a",
5164	TraditionalForm],
5165	TraditionalForm],
5166	FormBox[
5167	FormBox["a2",
5168	TraditionalForm],
5169	TraditionalForm],
5170	FormBox[
5171	FormBox["a4",
5172	TraditionalForm],
5173	TraditionalForm]}]]}], "-",
5174	RowBox[{"t", " ",
5175	RowBox[{"(",
5176	RowBox[{
5177	FormBox[
5178	OverscriptBox[
5179	FormBox["e4",
5180	TraditionalForm], "_"],
5181	TraditionalForm],
5182	FormBox["\<\"\[CenterDot]\"\>",
5183	TraditionalForm],
5184	FormBox[
5185	OverscriptBox[
5186	FormBox["p1",
5187	TraditionalForm], "_"],
5188	TraditionalForm]}], ")"}], " ",
5189	SuperscriptBox["f",
5190	RowBox[{
5191	FormBox[
5192	FormBox["a",
5193	TraditionalForm],
5194	TraditionalForm],
5195	FormBox[
5196	FormBox["a1",
5197	TraditionalForm],
5198	TraditionalForm],
5199	FormBox[
5200	FormBox["a4",
5201	TraditionalForm],
5202	TraditionalForm]}]], " ",
5203	SuperscriptBox["f",
5204	RowBox[{
5205	FormBox[
5206	FormBox["a",
5207	TraditionalForm],
5208	TraditionalForm],
5209	FormBox[
5210	FormBox["a2",
5211	TraditionalForm],
5212	TraditionalForm],
5213	FormBox[
5214	FormBox["a3",
5215	TraditionalForm],
5216	TraditionalForm]}]]}]}], ")"}]}], "+",
5217	RowBox[{"t", " ", "u", " ",
5218	RowBox[{"(",
5219	RowBox[{
5220	FormBox[
5221	OverscriptBox[
5222	FormBox["e3",
5223	TraditionalForm], "_"],
5224	TraditionalForm],
5225	FormBox["\<\"\[CenterDot]\"\>",
5226	TraditionalForm],
5227	FormBox[
5228	OverscriptBox[
5229	FormBox["e4",
5230	TraditionalForm], "_"],
5231	TraditionalForm]}], ")"}], " ",
5232	RowBox[{"(",
5233	RowBox[{
5234	RowBox[{
5235	SuperscriptBox["f",
5236	RowBox[{
5237	FormBox[
5238	FormBox["a",
5239	TraditionalForm],
5240	TraditionalForm],
5241	FormBox[
5242	FormBox["a1",
5243	TraditionalForm],
5244	TraditionalForm],
5245	FormBox[
5246	FormBox["a4",
5247	TraditionalForm],
5248	TraditionalForm]}]], " ",
5249	SuperscriptBox["f",
5250	RowBox[{
5251	FormBox[
5252	FormBox["a",
5253	TraditionalForm],
5254	TraditionalForm],
5255	FormBox[
5256	FormBox["a2",
5257	TraditionalForm],
5258	TraditionalForm],
5259	FormBox[
5260	FormBox["a3",
5261	TraditionalForm],
5262	TraditionalForm]}]]}], "-",
5263	RowBox[{
5264	SuperscriptBox["f",
5265	RowBox[{
5266	FormBox[
5267	FormBox["a",
5268	TraditionalForm],
5269	TraditionalForm],
5270	FormBox[
5271	FormBox["a1",
5272	TraditionalForm],
5273	TraditionalForm],
5274	FormBox[
5275	FormBox["a3",
5276	TraditionalForm],
5277	TraditionalForm]}]], " ",
5278	SuperscriptBox["f",
5279	RowBox[{
5280	FormBox[
5281	FormBox["a",
5282	TraditionalForm],
5283	TraditionalForm],
5284	FormBox[
5285	FormBox["a2",
5286	TraditionalForm],
5287	TraditionalForm],
5288	FormBox[
5289	FormBox["a4",
5290	TraditionalForm],
5291	TraditionalForm]}]]}], "+",
5292	RowBox[{
5293	SuperscriptBox["f",
5294	RowBox[{
5295	FormBox[
5296	FormBox["a",
5297	TraditionalForm],
5298	TraditionalForm],
5299	FormBox[
5300	FormBox["a1",
5301	TraditionalForm],
5302	TraditionalForm],
5303	FormBox[
5304	FormBox["a2",
5305	TraditionalForm],
5306	TraditionalForm]}]], " ",
5307	SuperscriptBox["f",
5308	RowBox[{
5309	FormBox[
5310	FormBox["a",
5311	TraditionalForm],
5312	TraditionalForm],
5313	FormBox[
5314	FormBox["a3",
5315	TraditionalForm],
5316	TraditionalForm],
5317	FormBox[
5318	FormBox["a4",
5319	TraditionalForm],
5320	TraditionalForm]}]]}]}], ")"}]}]}], ")"}]}], "+",
5321	RowBox[{"t", " ", "u", " ",
5322	RowBox[{"(",
5323	RowBox[{
5324	FormBox[
5325	OverscriptBox[
5326	FormBox["e2",
5327	TraditionalForm], "_"],
5328	TraditionalForm],
5329	FormBox["\<\"\[CenterDot]\"\>",
5330	TraditionalForm],
5331	FormBox[
5332	OverscriptBox[
5333	FormBox["e4",
5334	TraditionalForm], "_"],
5335	TraditionalForm]}], ")"}], " ",
5336	RowBox[{"(",
5337	RowBox[{
5338	RowBox[{
5339	RowBox[{"(",
5340	RowBox[{
5341	FormBox[
5342	OverscriptBox[
5343	FormBox["e3",
5344	TraditionalForm], "_"],
5345	TraditionalForm],
5346	FormBox["\<\"\[CenterDot]\"\>",
5347	TraditionalForm],
5348	FormBox[
5349	OverscriptBox[
5350	FormBox["p1",
5351	TraditionalForm], "_"],
5352	TraditionalForm]}], ")"}], " ",
5353	RowBox[{"(",
5354	RowBox[{
5355	RowBox[{
5356	SuperscriptBox["f",
5357	RowBox[{
5358	FormBox[
5359	FormBox["a",
5360	TraditionalForm],
5361	TraditionalForm],
5362	FormBox[
5363	FormBox["a1",
5364	TraditionalForm],
5365	TraditionalForm],
5366	FormBox[
5367	FormBox["a2",
5368	TraditionalForm],
5369	TraditionalForm]}]], " ",
5370	SuperscriptBox["f",
5371	RowBox[{
5372	FormBox[
5373	FormBox["a",
5374	TraditionalForm],
5375	TraditionalForm],
5376	FormBox[
5377	FormBox["a3",
5378	TraditionalForm],
5379	TraditionalForm],
5380	FormBox[
5381	FormBox["a4",
5382	TraditionalForm],
5383	TraditionalForm]}]]}], "-",
5384	RowBox[{
5385	SuperscriptBox["f",
5386	RowBox[{
5387	FormBox[
5388	FormBox["a",
5389	TraditionalForm],
5390	TraditionalForm],
5391	FormBox[
5392	FormBox["a1",
5393	TraditionalForm],
5394	TraditionalForm],
5395	FormBox[
5396	FormBox["a4",
5397	TraditionalForm],
5398	TraditionalForm]}]], " ",
5399	SuperscriptBox["f",
5400	RowBox[{
5401	FormBox[
5402	FormBox["a",
5403	TraditionalForm],
5404	TraditionalForm],
5405	FormBox[
5406	FormBox["a2",
5407	TraditionalForm],
5408	TraditionalForm],
5409	FormBox[
5410	FormBox["a3",
5411	TraditionalForm],
5412	TraditionalForm]}]]}]}], ")"}]}], "+",
5413	RowBox[{
5414	RowBox[{"(",
5415	RowBox[{
5416	RowBox[{
5417	FormBox[
5418	OverscriptBox[
5419	FormBox["e3",
5420	TraditionalForm], "_"],
5421	TraditionalForm],
5422	FormBox["\<\"\[CenterDot]\"\>",
5423	TraditionalForm],
5424	FormBox[
5425	OverscriptBox[
5426	FormBox["p2",
5427	TraditionalForm], "_"],
5428	TraditionalForm]}], "+",
5429	FormBox[
5430	RowBox[{
5431	FormBox[
5432	OverscriptBox[
5433	FormBox["e3",
5434	TraditionalForm], "_"],
5435	TraditionalForm],
5436	FormBox["\<\"\[CenterDot]\"\>",
5437	TraditionalForm],
5438	FormBox[
5439	OverscriptBox[
5440	FormBox["p1",
5441	TraditionalForm], "_"],
5442	TraditionalForm]}],
5443	TraditionalForm], "+",
5444	FormBox[
5445	RowBox[{
5446	FormBox[
5447	OverscriptBox[
5448	FormBox["e3",
5449	TraditionalForm], "_"],
5450	TraditionalForm],
5451	FormBox["\<\"\[CenterDot]\"\>",
5452	TraditionalForm],
5453	FormBox[
5454	OverscriptBox[
5455	FormBox["p2",
5456	TraditionalForm], "_"],
5457	TraditionalForm]}],
5458	TraditionalForm]}], ")"}], " ",
5459	RowBox[{"(",
5460	RowBox[{
5461	RowBox[{
5462	SuperscriptBox["f",
5463	RowBox[{
5464	FormBox[
5465	FormBox["a",
5466	TraditionalForm],
5467	TraditionalForm],
5468	FormBox[
5469	FormBox["a1",
5470	TraditionalForm],
5471	TraditionalForm],
5472	FormBox[
5473	FormBox["a4",
5474	TraditionalForm],
5475	TraditionalForm]}]], " ",
5476	SuperscriptBox["f",
5477	RowBox[{
5478	FormBox[
5479	FormBox["a",
5480	TraditionalForm],
5481	TraditionalForm],
5482	FormBox[
5483	FormBox["a2",
5484	TraditionalForm],
5485	TraditionalForm],
5486	FormBox[
5487	FormBox["a3",
5488	TraditionalForm],
5489	TraditionalForm]}]]}], "-",
5490	RowBox[{
5491	SuperscriptBox["f",
5492	RowBox[{
5493	FormBox[
5494	FormBox["a",
5495	TraditionalForm],
5496	TraditionalForm],
5497	FormBox[
5498	FormBox["a1",
5499	TraditionalForm],
5500	TraditionalForm],
5501	FormBox[
5502	FormBox["a3",
5503	TraditionalForm],
5504	TraditionalForm]}]], " ",
5505	SuperscriptBox["f",
5506	RowBox[{
5507	FormBox[
5508	FormBox["a",
5509	TraditionalForm],
5510	TraditionalForm],
5511	FormBox[
5512	FormBox["a2",
5513	TraditionalForm],
5514	TraditionalForm],
5515	FormBox[
5516	FormBox["a4",
5517	TraditionalForm],
5518	TraditionalForm]}]]}], "+",
5519	RowBox[{
5520	SuperscriptBox["f",
5521	RowBox[{
5522	FormBox[
5523	FormBox["a",
5524	TraditionalForm],
5525	TraditionalForm],
5526	FormBox[
5527	FormBox["a1",
5528	TraditionalForm],
5529	TraditionalForm],
5530	FormBox[
5531	FormBox["a2",
5532	TraditionalForm],
5533	TraditionalForm]}]], " ",
5534	SuperscriptBox["f",
5535	RowBox[{
5536	FormBox[
5537	FormBox["a",
5538	TraditionalForm],
5539	TraditionalForm],
5540	FormBox[
5541	FormBox["a3",
5542	TraditionalForm],
5543	TraditionalForm],
5544	FormBox[
5545	FormBox["a4",
5546	TraditionalForm],
5547	TraditionalForm]}]]}]}], ")"}]}]}], ")"}]}]}], ")"}]}],
5548	"+",
5549	RowBox[{
5550	RowBox[{"(",
5551	RowBox[{
5552	FormBox[
5553	OverscriptBox[
5554	FormBox["e2",
5555	TraditionalForm], "_"],
5556	TraditionalForm],
5557	FormBox["\<\"\[CenterDot]\"\>",
5558	TraditionalForm],
5559	FormBox[
5560	OverscriptBox[
5561	FormBox["p1",
5562	TraditionalForm], "_"],
5563	TraditionalForm]}], ")"}], " ",
5564	RowBox[{"(",
5565	RowBox[{
5566	RowBox[{"s", " ", "t", " ", "u", " ",
5567	RowBox[{"(",
5568	RowBox[{
5569	FormBox[
5570	OverscriptBox[
5571	FormBox["e3",
5572	TraditionalForm], "_"],
5573	TraditionalForm],
5574	FormBox["\<\"\[CenterDot]\"\>",
5575	TraditionalForm],
5576	FormBox[
5577	OverscriptBox[
5578	FormBox["e4",
5579	TraditionalForm], "_"],
5580	TraditionalForm]}], ")"}], " ",
5581	RowBox[{"(",
5582	RowBox[{
5583	RowBox[{
5584	SuperscriptBox["f",
5585	RowBox[{
5586	FormBox[
5587	FormBox["a",
5588	TraditionalForm],
5589	TraditionalForm],
5590	FormBox[
5591	FormBox["a1",
5592	TraditionalForm],
5593	TraditionalForm],
5594	FormBox[
5595	FormBox["a4",
5596	TraditionalForm],
5597	TraditionalForm]}]], " ",
5598	SuperscriptBox["f",
5599	RowBox[{
5600	FormBox[
5601	FormBox["a",
5602	TraditionalForm],
5603	TraditionalForm],
5604	FormBox[
5605	FormBox["a2",
5606	TraditionalForm],
5607	TraditionalForm],
5608	FormBox[
5609	FormBox["a3",
5610	TraditionalForm],
5611	TraditionalForm]}]]}], "+",
5612	RowBox[{
5613	SuperscriptBox["f",
5614	RowBox[{
5615	FormBox[
5616	FormBox["a",
5617	TraditionalForm],
5618	TraditionalForm],
5619	FormBox[
5620	FormBox["a1",
5621	TraditionalForm],
5622	TraditionalForm],
5623	FormBox[
5624	FormBox["a3",
5625	TraditionalForm],
5626	TraditionalForm]}]], " ",
5627	SuperscriptBox["f",
5628	RowBox[{
5629	FormBox[
5630	FormBox["a",
5631	TraditionalForm],
5632	TraditionalForm],
5633	FormBox[
5634	FormBox["a2",
5635	TraditionalForm],
5636	TraditionalForm],
5637	FormBox[
5638	FormBox["a4",
5639	TraditionalForm],
5640	TraditionalForm]}]]}]}], ")"}]}], "+",
5641	RowBox[{"t", " ",
5642	RowBox[{"(",
5643	RowBox[{
5644	FormBox[
5645	OverscriptBox[
5646	FormBox["e3",
5647	TraditionalForm], "_"],
5648	TraditionalForm],
5649	FormBox["\<\"\[CenterDot]\"\>",
5650	TraditionalForm],
5651	FormBox[
5652	OverscriptBox[
5653	FormBox["p2",
5654	TraditionalForm], "_"],
5655	TraditionalForm]}], ")"}], " ",
5656	RowBox[{"(",
5657	RowBox[{
5658	RowBox[{"u", " ",
5659	RowBox[{"(",
5660	RowBox[{
5661	FormBox[
5662	RowBox[{
5663	FormBox[
5664	OverscriptBox[
5665	FormBox["e4",
5666	TraditionalForm], "_"],
5667	TraditionalForm],
5668	FormBox["\<\"\[CenterDot]\"\>",
5669	TraditionalForm],
5670	FormBox[
5671	OverscriptBox[
5672	FormBox["p1",
5673	TraditionalForm], "_"],
5674	TraditionalForm]}],
5675	TraditionalForm], "+",
5676	FormBox[
5677	RowBox[{
5678	FormBox[
5679	OverscriptBox[
5680	FormBox["e4",
5681	TraditionalForm], "_"],
5682	TraditionalForm],
5683	FormBox["\<\"\[CenterDot]\"\>",
5684	TraditionalForm],
5685	FormBox[
5686	OverscriptBox[
5687	FormBox["p2",
5688	TraditionalForm], "_"],
5689	TraditionalForm]}],
5690	TraditionalForm]}], ")"}], " ",
5691	SuperscriptBox["f",
5692	RowBox[{
5693	FormBox[
5694	FormBox["a",
5695	TraditionalForm],
5696	TraditionalForm],
5697	FormBox[
5698	FormBox["a1",
5699	TraditionalForm],
5700	TraditionalForm],
5701	FormBox[
5702	FormBox["a2",
5703	TraditionalForm],
5704	TraditionalForm]}]], " ",
5705	SuperscriptBox["f",
5706	RowBox[{
5707	FormBox[
5708	FormBox["a",
5709	TraditionalForm],
5710	TraditionalForm],
5711	FormBox[
5712	FormBox["a3",
5713	TraditionalForm],
5714	TraditionalForm],
5715	FormBox[
5716	FormBox["a4",
5717	TraditionalForm],
5718	TraditionalForm]}]]}], "-",
5719	RowBox[{"s", " ",
5720	RowBox[{"(",
5721	RowBox[{
5722	FormBox[
5723	OverscriptBox[
5724	FormBox["e4",
5725	TraditionalForm], "_"],
5726	TraditionalForm],
5727	FormBox["\<\"\[CenterDot]\"\>",
5728	TraditionalForm],
5729	FormBox[
5730	OverscriptBox[
5731	FormBox["p1",
5732	TraditionalForm], "_"],
5733	TraditionalForm]}], ")"}], " ",
5734	SuperscriptBox["f",
5735	RowBox[{
5736	FormBox[
5737	FormBox["a",
5738	TraditionalForm],
5739	TraditionalForm],
5740	FormBox[
5741	FormBox["a1",
5742	TraditionalForm],
5743	TraditionalForm],
5744	FormBox[
5745	FormBox["a4",
5746	TraditionalForm],
5747	TraditionalForm]}]], " ",
5748	SuperscriptBox["f",
5749	RowBox[{
5750	FormBox[
5751	FormBox["a",
5752	TraditionalForm],
5753	TraditionalForm],
5754	FormBox[
5755	FormBox["a2",
5756	TraditionalForm],
5757	TraditionalForm],
5758	FormBox[
5759	FormBox["a3",
5760	TraditionalForm],
5761	TraditionalForm]}]]}]}], ")"}]}], "-",
5762	RowBox[{"u", " ",
5763	RowBox[{"(",
5764	RowBox[{
5765	RowBox[{"t", " ",
5766	RowBox[{"(",
5767	RowBox[{
5768	RowBox[{
5769	FormBox[
5770	OverscriptBox[
5771	FormBox["e4",
5772	TraditionalForm], "_"],
5773	TraditionalForm],
5774	FormBox["\<\"\[CenterDot]\"\>",
5775	TraditionalForm],
5776	FormBox[
5777	OverscriptBox[
5778	FormBox["p1",
5779	TraditionalForm], "_"],
5780	TraditionalForm]}], "+",
5781	RowBox[{
5782	FormBox[
5783	OverscriptBox[
5784	FormBox["e4",
5785	TraditionalForm], "_"],
5786	TraditionalForm],
5787	FormBox["\<\"\[CenterDot]\"\>",
5788	TraditionalForm],
5789	FormBox[
5790	OverscriptBox[
5791	FormBox["p2",
5792	TraditionalForm], "_"],
5793	TraditionalForm]}]}], ")"}], " ",
5794	RowBox[{"(",
5795	RowBox[{
5796	FormBox[
5797	RowBox[{
5798	FormBox[
5799	OverscriptBox[
5800	FormBox["e3",
5801	TraditionalForm], "_"],
5802	TraditionalForm],
5803	FormBox["\<\"\[CenterDot]\"\>",
5804	TraditionalForm],
5805	FormBox[
5806	OverscriptBox[
5807	FormBox["p1",
5808	TraditionalForm], "_"],
5809	TraditionalForm]}],
5810	TraditionalForm], "+",
5811	FormBox[
5812	RowBox[{
5813	FormBox[
5814	OverscriptBox[
5815	FormBox["e3",
5816	TraditionalForm], "_"],
5817	TraditionalForm],
5818	FormBox["\<\"\[CenterDot]\"\>",
5819	TraditionalForm],
5820	FormBox[
5821	OverscriptBox[
5822	FormBox["p2",
5823	TraditionalForm], "_"],
5824	TraditionalForm]}],
5825	TraditionalForm]}], ")"}], " ",
5826	SuperscriptBox["f",
5827	RowBox[{
5828	FormBox[
5829	FormBox["a",
5830	TraditionalForm],
5831	TraditionalForm],
5832	FormBox[
5833	FormBox["a1",
5834	TraditionalForm],
5835	TraditionalForm],
5836	FormBox[
5837	FormBox["a2",
5838	TraditionalForm],
5839	TraditionalForm]}]], " ",
5840	SuperscriptBox["f",
5841	RowBox[{
5842	FormBox[
5843	FormBox["a",
5844	TraditionalForm],
5845	TraditionalForm],
5846	FormBox[
5847	FormBox["a3",
5848	TraditionalForm],
5849	TraditionalForm],
5850	FormBox[
5851	FormBox["a4",
5852	TraditionalForm],
5853	TraditionalForm]}]]}], "+",
5854	RowBox[{
5855	RowBox[{"(",
5856	RowBox[{
5857	FormBox[
5858	OverscriptBox[
5859	FormBox["e3",
5860	TraditionalForm], "_"],
5861	TraditionalForm],
5862	FormBox["\<\"\[CenterDot]\"\>",
5863	TraditionalForm],
5864	FormBox[
5865	OverscriptBox[
5866	FormBox["p1",
5867	TraditionalForm], "_"],
5868	TraditionalForm]}], ")"}], " ",
5869	RowBox[{"(",
5870	RowBox[{
5871	RowBox[{"s", " ",
5872	RowBox[{"(",
5873	RowBox[{
5874	FormBox[
5875	OverscriptBox[
5876	FormBox["e4",
5877	TraditionalForm], "_"],
5878	TraditionalForm],
5879	FormBox["\<\"\[CenterDot]\"\>",
5880	TraditionalForm],
5881	FormBox[
5882	OverscriptBox[
5883	FormBox["p2",
5884	TraditionalForm], "_"],
5885	TraditionalForm]}], ")"}], " ",
5886	SuperscriptBox["f",
5887	RowBox[{
5888	FormBox[
5889	FormBox["a",
5890	TraditionalForm],
5891	TraditionalForm],
5892	FormBox[
5893	FormBox["a1",
5894	TraditionalForm],
5895	TraditionalForm],
5896	FormBox[
5897	FormBox["a3",
5898	TraditionalForm],
5899	TraditionalForm]}]], " ",
5900	SuperscriptBox["f",
5901	RowBox[{
5902	FormBox[
5903	FormBox["a",
5904	TraditionalForm],
5905	TraditionalForm],
5906	FormBox[
5907	FormBox["a2",
5908	TraditionalForm],
5909	TraditionalForm],
5910	FormBox[
5911	FormBox["a4",
5912	TraditionalForm],
5913	TraditionalForm]}]]}], "-",
5914	RowBox[{"t", " ",
5915	RowBox[{"(",
5916	RowBox[{
5917	FormBox[
5918	RowBox[{
5919	FormBox[
5920	OverscriptBox[
5921	FormBox["e4",
5922	TraditionalForm], "_"],
5923	TraditionalForm],
5924	FormBox["\<\"\[CenterDot]\"\>",
5925	TraditionalForm],
5926	FormBox[
5927	OverscriptBox[
5928	FormBox["p1",
5929	TraditionalForm], "_"],
5930	TraditionalForm]}],
5931	TraditionalForm], "+",
5932	FormBox[
5933	RowBox[{
5934	FormBox[
5935	OverscriptBox[
5936	FormBox["e4",
5937	TraditionalForm], "_"],
5938	TraditionalForm],
5939	FormBox["\<\"\[CenterDot]\"\>",
5940	TraditionalForm],
5941	FormBox[
5942	OverscriptBox[
5943	FormBox["p2",
5944	TraditionalForm], "_"],
5945	TraditionalForm]}],
5946	TraditionalForm]}], ")"}], " ",
5947	SuperscriptBox["f",
5948	RowBox[{
5949	FormBox[
5950	FormBox["a",
5951	TraditionalForm],
5952	TraditionalForm],
5953	FormBox[
5954	FormBox["a1",
5955	TraditionalForm],
5956	TraditionalForm],
5957	FormBox[
5958	FormBox["a2",
5959	TraditionalForm],
5960	TraditionalForm]}]], " ",
5961	SuperscriptBox["f",
5962	RowBox[{
5963	FormBox[
5964	FormBox["a",
5965	TraditionalForm],
5966	TraditionalForm],
5967	FormBox[
5968	FormBox["a3",
5969	TraditionalForm],
5970	TraditionalForm],
5971	FormBox[
5972	FormBox["a4",
5973	TraditionalForm],
5974	TraditionalForm]}]]}]}], ")"}]}]}], ")"}]}]}], ")"}]}]}],
5975	")"}]}]}]}], TraditionalForm]], "Output",
5976	CellChangeTimes->{
5977	3.6926179368868713`^9, 3.692619074074085^9, 3.692622062482294*^9,
5978	3.692940735779708^9, 3.692941455688466^9, {3.693124173308194*^9,
5979	3.693124181896801*^9}}]
5980	}, Open ]],
5981
5982	Cell[CellGroupData[{
5983
5984	Cell[BoxData[
5985	RowBox[{"AmpSimp4g", "=",
5986	RowBox[{
5987	RowBox[{"SUNSimplify", "[",
5988	RowBox[{"Amp4g", ",",
5989	RowBox[{"SUNFJacobi", "\[Rule]", "True"}]}], "]"}], "//",
5990	"Simplify"}]}]], "Input",
5991	CellChangeTimes->{{3.692617536061657^9, 3.692617563954812^9}, {
5992	3.692617734907619^9, 3.692617736602909^9}, {3.6926195449341183`*^9,
5993	3.69261955489699^9}, {3.692622039662298^9, 3.6926220463087053`*^9}}],
5994
5995	Cell[BoxData[
5996	FormBox[
5997	RowBox[{
5998	RowBox[{"-", "\[ImaginaryI]"}], " ",
5999	SuperscriptBox["gs", "2"], " ",
6000	RowBox[{"(",
6001	RowBox[{
6002	RowBox[{
6003	RowBox[{"(",
6004	RowBox[{
6005	FormBox[
6006	OverscriptBox[
6007	FormBox["e2",
6008	TraditionalForm], "_"],
6009	TraditionalForm],
6010	FormBox["\<\"\[CenterDot]\"\>",
6011	TraditionalForm],
6012	FormBox[
6013	OverscriptBox[
6014	FormBox["e4",
6015	TraditionalForm], "_"],
6016	TraditionalForm]}], ")"}], " ",
6017	RowBox[{"(",
6018	RowBox[{
6019	RowBox[{
6020	RowBox[{"(",
6021	RowBox[{
6022	FormBox[
6023	OverscriptBox[
6024	FormBox["e3",
6025	TraditionalForm], "_"],
6026	TraditionalForm],
6027	FormBox["\<\"\[CenterDot]\"\>",
6028	TraditionalForm],
6029	FormBox[
6030	OverscriptBox[
6031	FormBox["p1",
6032	TraditionalForm], "_"],
6033	TraditionalForm]}], ")"}], " ",
6034	RowBox[{"(",
6035	RowBox[{
6036	RowBox[{"2", " ",
6037	SuperscriptBox["f",
6038	RowBox[{
6039	FormBox[
6040	FormBox["a",
6041	TraditionalForm],
6042	TraditionalForm],
6043	FormBox[
6044	FormBox["a1",
6045	TraditionalForm],
6046	TraditionalForm],
6047	FormBox[
6048	FormBox["a2",
6049	TraditionalForm],
6050	TraditionalForm]}]], " ",
6051	SuperscriptBox["f",
6052	RowBox[{
6053	FormBox[
6054	FormBox["a",
6055	TraditionalForm],
6056	TraditionalForm],
6057	FormBox[
6058	FormBox["a3",
6059	TraditionalForm],
6060	TraditionalForm],
6061	FormBox[
6062	FormBox["a4",
6063	TraditionalForm],
6064	TraditionalForm]}]]}], "-",
6065	RowBox[{
6066	SuperscriptBox["f",
6067	RowBox[{
6068	FormBox[
6069	FormBox["a",
6070	TraditionalForm],
6071	TraditionalForm],
6072	FormBox[
6073	FormBox["a1",
6074	TraditionalForm],
6075	TraditionalForm],
6076	FormBox[
6077	FormBox["a3",
6078	TraditionalForm],
6079	TraditionalForm]}]], " ",
6080	SuperscriptBox["f",
6081	RowBox[{
6082	FormBox[
6083	FormBox["a",
6084	TraditionalForm],
6085	TraditionalForm],
6086	FormBox[
6087	FormBox["a2",
6088	TraditionalForm],
6089	TraditionalForm],
6090	FormBox[
6091	FormBox["a4",
6092	TraditionalForm],
6093	TraditionalForm]}]]}]}], ")"}]}], "+",
6094	RowBox[{"2", " ",
6095	RowBox[{"(",
6096	RowBox[{
6097	FormBox[
6098	OverscriptBox[
6099	FormBox["e3",
6100	TraditionalForm], "_"],
6101	TraditionalForm],
6102	FormBox["\<\"\[CenterDot]\"\>",
6103	TraditionalForm],
6104	FormBox[
6105	OverscriptBox[
6106	FormBox["p2",
6107	TraditionalForm], "_"],
6108	TraditionalForm]}], ")"}], " ",
6109	RowBox[{"(",
6110	RowBox[{
6111	RowBox[{
6112	SuperscriptBox["f",
6113	RowBox[{
6114	FormBox[
6115	FormBox["a",
6116	TraditionalForm],
6117	TraditionalForm],
6118	FormBox[
6119	FormBox["a1",
6120	TraditionalForm],
6121	TraditionalForm],
6122	FormBox[
6123	FormBox["a4",
6124	TraditionalForm],
6125	TraditionalForm]}]], " ",
6126	SuperscriptBox["f",
6127	RowBox[{
6128	FormBox[
6129	FormBox["a",
6130	TraditionalForm],
6131	TraditionalForm],
6132	FormBox[
6133	FormBox["a2",
6134	TraditionalForm],
6135	TraditionalForm],
6136	FormBox[
6137	FormBox["a3",
6138	TraditionalForm],
6139	TraditionalForm]}]]}], "-",
6140	RowBox[{
6141	SuperscriptBox["f",
6142	RowBox[{
6143	FormBox[
6144	FormBox["a",
6145	TraditionalForm],
6146	TraditionalForm],
6147	FormBox[
6148	FormBox["a1",
6149	TraditionalForm],
6150	TraditionalForm],
6151	FormBox[
6152	FormBox["a3",
6153	TraditionalForm],
6154	TraditionalForm]}]], " ",
6155	SuperscriptBox["f",
6156	RowBox[{
6157	FormBox[
6158	FormBox["a",
6159	TraditionalForm],
6160	TraditionalForm],
6161	FormBox[
6162	FormBox["a2",
6163	TraditionalForm],
6164	TraditionalForm],
6165	FormBox[
6166	FormBox["a4",
6167	TraditionalForm],
6168	TraditionalForm]}]]}], "+",
6169	RowBox[{
6170	SuperscriptBox["f",
6171	RowBox[{
6172	FormBox[
6173	FormBox["a",
6174	TraditionalForm],
6175	TraditionalForm],
6176	FormBox[
6177	FormBox["a1",
6178	TraditionalForm],
6179	TraditionalForm],
6180	FormBox[
6181	FormBox["a2",
6182	TraditionalForm],
6183	TraditionalForm]}]], " ",
6184	SuperscriptBox["f",
6185	RowBox[{
6186	FormBox[
6187	FormBox["a",
6188	TraditionalForm],
6189	TraditionalForm],
6190	FormBox[
6191	FormBox["a3",
6192	TraditionalForm],
6193	TraditionalForm],
6194	FormBox[
6195	FormBox["a4",
6196	TraditionalForm],
6197	TraditionalForm]}]]}]}], ")"}]}]}], ")"}]}], "-",
6198	RowBox[{
6199	RowBox[{"(",
6200	RowBox[{
6201	FormBox[
6202	OverscriptBox[
6203	FormBox["e2",
6204	TraditionalForm], "_"],
6205	TraditionalForm],
6206	FormBox["\<\"\[CenterDot]\"\>",
6207	TraditionalForm],
6208	FormBox[
6209	OverscriptBox[
6210	FormBox["e3",
6211	TraditionalForm], "_"],
6212	TraditionalForm]}], ")"}], " ",
6213	RowBox[{"(",
6214	RowBox[{
6215	FormBox[
6216	OverscriptBox[
6217	FormBox["e4",
6218	TraditionalForm], "_"],
6219	TraditionalForm],
6220	FormBox["\<\"\[CenterDot]\"\>",
6221	TraditionalForm],
6222	FormBox[
6223	OverscriptBox[
6224	FormBox["p1",
6225	TraditionalForm], "_"],
6226	TraditionalForm]}], ")"}], " ",
6227	SuperscriptBox["f",
6228	RowBox[{
6229	FormBox[
6230	FormBox["a",
6231	TraditionalForm],
6232	TraditionalForm],
6233	FormBox[
6234	FormBox["a1",
6235	TraditionalForm],
6236	TraditionalForm],
6237	FormBox[
6238	FormBox["a4",
6239	TraditionalForm],
6240	TraditionalForm]}]], " ",
6241	SuperscriptBox["f",
6242	RowBox[{
6243	FormBox[
6244	FormBox["a",
6245	TraditionalForm],
6246	TraditionalForm],
6247	FormBox[
6248	FormBox["a2",
6249	TraditionalForm],
6250	TraditionalForm],
6251	FormBox[
6252	FormBox["a3",
6253	TraditionalForm],
6254	TraditionalForm]}]]}], "+",
6255	RowBox[{"2", " ",
6256	RowBox[{"(",
6257	RowBox[{
6258	FormBox[
6259	OverscriptBox[
6260	FormBox["e2",
6261	TraditionalForm], "_"],
6262	TraditionalForm],
6263	FormBox["\<\"\[CenterDot]\"\>",
6264	TraditionalForm],
6265	FormBox[
6266	OverscriptBox[
6267	FormBox["p1",
6268	TraditionalForm], "_"],
6269	TraditionalForm]}], ")"}], " ",
6270	RowBox[{"(",
6271	RowBox[{
6272	FormBox[
6273	OverscriptBox[
6274	FormBox["e3",
6275	TraditionalForm], "_"],
6276	TraditionalForm],
6277	FormBox["\<\"\[CenterDot]\"\>",
6278	TraditionalForm],
6279	FormBox[
6280	OverscriptBox[
6281	FormBox["e4",
6282	TraditionalForm], "_"],
6283	TraditionalForm]}], ")"}], " ",
6284	SuperscriptBox["f",
6285	RowBox[{
6286	FormBox[
6287	FormBox["a",
6288	TraditionalForm],
6289	TraditionalForm],
6290	FormBox[
6291	FormBox["a1",
6292	TraditionalForm],
6293	TraditionalForm],
6294	FormBox[
6295	FormBox["a3",
6296	TraditionalForm],
6297	TraditionalForm]}]], " ",
6298	SuperscriptBox["f",
6299	RowBox[{
6300	FormBox[
6301	FormBox["a",
6302	TraditionalForm],
6303	TraditionalForm],
6304	FormBox[
6305	FormBox["a2",
6306	TraditionalForm],
6307	TraditionalForm],
6308	FormBox[
6309	FormBox["a4",
6310	TraditionalForm],
6311	TraditionalForm]}]]}], "-",
6312	RowBox[{
6313	RowBox[{"(",
6314	RowBox[{
6315	FormBox[
6316	OverscriptBox[
6317	FormBox["e2",
6318	TraditionalForm], "_"],
6319	TraditionalForm],
6320	FormBox["\<\"\[CenterDot]\"\>",
6321	TraditionalForm],
6322	FormBox[
6323	OverscriptBox[
6324	FormBox["p1",
6325	TraditionalForm], "_"],
6326	TraditionalForm]}], ")"}], " ",
6327	RowBox[{"(",
6328	RowBox[{
6329	FormBox[
6330	OverscriptBox[
6331	FormBox["e3",
6332	TraditionalForm], "_"],
6333	TraditionalForm],
6334	FormBox["\<\"\[CenterDot]\"\>",
6335	TraditionalForm],
6336	FormBox[
6337	OverscriptBox[
6338	FormBox["e4",
6339	TraditionalForm], "_"],
6340	TraditionalForm]}], ")"}], " ",
6341	SuperscriptBox["f",
6342	RowBox[{
6343	FormBox[
6344	FormBox["a",
6345	TraditionalForm],
6346	TraditionalForm],
6347	FormBox[
6348	FormBox["a1",
6349	TraditionalForm],
6350	TraditionalForm],
6351	FormBox[
6352	FormBox["a2",
6353	TraditionalForm],
6354	TraditionalForm]}]], " ",
6355	SuperscriptBox["f",
6356	RowBox[{
6357	FormBox[
6358	FormBox["a",
6359	TraditionalForm],
6360	TraditionalForm],
6361	FormBox[
6362	FormBox["a3",
6363	TraditionalForm],
6364	TraditionalForm],
6365	FormBox[
6366	FormBox["a4",
6367	TraditionalForm],
6368	TraditionalForm]}]]}], "-",
6369	RowBox[{"2", " ",
6370	RowBox[{"(",
6371	RowBox[{
6372	FormBox[
6373	OverscriptBox[
6374	FormBox["e2",
6375	TraditionalForm], "_"],
6376	TraditionalForm],
6377	FormBox["\<\"\[CenterDot]\"\>",
6378	TraditionalForm],
6379	FormBox[
6380	OverscriptBox[
6381	FormBox["e3",
6382	TraditionalForm], "_"],
6383	TraditionalForm]}], ")"}], " ",
6384	RowBox[{"(",
6385	RowBox[{
6386	FormBox[
6387	OverscriptBox[
6388	FormBox["e4",
6389	TraditionalForm], "_"],
6390	TraditionalForm],
6391	FormBox["\<\"\[CenterDot]\"\>",
6392	TraditionalForm],
6393	FormBox[
6394	OverscriptBox[
6395	FormBox["p1",
6396	TraditionalForm], "_"],
6397	TraditionalForm]}], ")"}], " ",
6398	SuperscriptBox["f",
6399	RowBox[{
6400	FormBox[
6401	FormBox["a",
6402	TraditionalForm],
6403	TraditionalForm],
6404	FormBox[
6405	FormBox["a1",
6406	TraditionalForm],
6407	TraditionalForm],
6408	FormBox[
6409	FormBox["a2",
6410	TraditionalForm],
6411	TraditionalForm]}]], " ",
6412	SuperscriptBox["f",
6413	RowBox[{
6414	FormBox[
6415	FormBox["a",
6416	TraditionalForm],
6417	TraditionalForm],
6418	FormBox[
6419	FormBox["a3",
6420	TraditionalForm],
6421	TraditionalForm],
6422	FormBox[
6423	FormBox["a4",
6424	TraditionalForm],
6425	TraditionalForm]}]]}], "-",
6426	RowBox[{"2", " ",
6427	RowBox[{"(",
6428	RowBox[{
6429	FormBox[
6430	OverscriptBox[
6431	FormBox["e2",
6432	TraditionalForm], "_"],
6433	TraditionalForm],
6434	FormBox["\<\"\[CenterDot]\"\>",
6435	TraditionalForm],
6436	FormBox[
6437	OverscriptBox[
6438	FormBox["e3",
6439	TraditionalForm], "_"],
6440	TraditionalForm]}], ")"}], " ",
6441	RowBox[{"(",
6442	RowBox[{
6443	FormBox[
6444	OverscriptBox[
6445	FormBox["e4",
6446	TraditionalForm], "_"],
6447	TraditionalForm],
6448	FormBox["\<\"\[CenterDot]\"\>",
6449	TraditionalForm],
6450	FormBox[
6451	OverscriptBox[
6452	FormBox["p2",
6453	TraditionalForm], "_"],
6454	TraditionalForm]}], ")"}], " ",
6455	SuperscriptBox["f",
6456	RowBox[{
6457	FormBox[
6458	FormBox["a",
6459	TraditionalForm],
6460	TraditionalForm],
6461	FormBox[
6462	FormBox["a1",
6463	TraditionalForm],
6464	TraditionalForm],
6465	FormBox[
6466	FormBox["a4",
6467	TraditionalForm],
6468	TraditionalForm]}]], " ",
6469	SuperscriptBox["f",
6470	RowBox[{
6471	FormBox[
6472	FormBox["a",
6473	TraditionalForm],
6474	TraditionalForm],
6475	FormBox[
6476	FormBox["a2",
6477	TraditionalForm],
6478	TraditionalForm],
6479	FormBox[
6480	FormBox["a3",
6481	TraditionalForm],
6482	TraditionalForm]}]]}], "+",
6483	RowBox[{"2", " ",
6484	RowBox[{"(",
6485	RowBox[{
6486	FormBox[
6487	OverscriptBox[
6488	FormBox["e2",
6489	TraditionalForm], "_"],
6490	TraditionalForm],
6491	FormBox["\<\"\[CenterDot]\"\>",
6492	TraditionalForm],
6493	FormBox[
6494	OverscriptBox[
6495	FormBox["e3",
6496	TraditionalForm], "_"],
6497	TraditionalForm]}], ")"}], " ",
6498	RowBox[{"(",
6499	RowBox[{
6500	FormBox[
6501	OverscriptBox[
6502	FormBox["e4",
6503	TraditionalForm], "_"],
6504	TraditionalForm],
6505	FormBox["\<\"\[CenterDot]\"\>",
6506	TraditionalForm],
6507	FormBox[
6508	OverscriptBox[
6509	FormBox["p2",
6510	TraditionalForm], "_"],
6511	TraditionalForm]}], ")"}], " ",
6512	SuperscriptBox["f",
6513	RowBox[{
6514	FormBox[
6515	FormBox["a",
6516	TraditionalForm],
6517	TraditionalForm],
6518	FormBox[
6519	FormBox["a1",
6520	TraditionalForm],
6521	TraditionalForm],
6522	FormBox[
6523	FormBox["a3",
6524	TraditionalForm],
6525	TraditionalForm]}]], " ",
6526	SuperscriptBox["f",
6527	RowBox[{
6528	FormBox[
6529	FormBox["a",
6530	TraditionalForm],
6531	TraditionalForm],
6532	FormBox[
6533	FormBox["a2",
6534	TraditionalForm],
6535	TraditionalForm],
6536	FormBox[
6537	FormBox["a4",
6538	TraditionalForm],
6539	TraditionalForm]}]]}], "-",
6540	RowBox[{"2", " ",
6541	RowBox[{"(",
6542	RowBox[{
6543	FormBox[
6544	OverscriptBox[
6545	FormBox["e2",
6546	TraditionalForm], "_"],
6547	TraditionalForm],
6548	FormBox["\<\"\[CenterDot]\"\>",
6549	TraditionalForm],
6550	FormBox[
6551	OverscriptBox[
6552	FormBox["e3",
6553	TraditionalForm], "_"],
6554	TraditionalForm]}], ")"}], " ",
6555	RowBox[{"(",
6556	RowBox[{
6557	FormBox[
6558	OverscriptBox[
6559	FormBox["e4",
6560	TraditionalForm], "_"],
6561	TraditionalForm],
6562	FormBox["\<\"\[CenterDot]\"\>",
6563	TraditionalForm],
6564	FormBox[
6565	OverscriptBox[
6566	FormBox["p2",
6567	TraditionalForm], "_"],
6568	TraditionalForm]}], ")"}], " ",
6569	SuperscriptBox["f",
6570	RowBox[{
6571	FormBox[
6572	FormBox["a",
6573	TraditionalForm],
6574	TraditionalForm],
6575	FormBox[
6576	FormBox["a1",
6577	TraditionalForm],
6578	TraditionalForm],
6579	FormBox[
6580	FormBox["a2",
6581	TraditionalForm],
6582	TraditionalForm]}]], " ",
6583	SuperscriptBox["f",
6584	RowBox[{
6585	FormBox[
6586	FormBox["a",
6587	TraditionalForm],
6588	TraditionalForm],
6589	FormBox[
6590	FormBox["a3",
6591	TraditionalForm],
6592	TraditionalForm],
6593	FormBox[
6594	FormBox["a4",
6595	TraditionalForm],
6596	TraditionalForm]}]]}], "+",
6597	RowBox[{"2", " ",
6598	RowBox[{"(",
6599	RowBox[{
6600	FormBox[
6601	OverscriptBox[
6602	FormBox["e2",
6603	TraditionalForm], "_"],
6604	TraditionalForm],
6605	FormBox["\<\"\[CenterDot]\"\>",
6606	TraditionalForm],
6607	FormBox[
6608	OverscriptBox[
6609	FormBox["p3",
6610	TraditionalForm], "_"],
6611	TraditionalForm]}], ")"}], " ",
6612	RowBox[{"(",
6613	RowBox[{
6614	FormBox[
6615	OverscriptBox[
6616	FormBox["e3",
6617	TraditionalForm], "_"],
6618	TraditionalForm],
6619	FormBox["\<\"\[CenterDot]\"\>",
6620	TraditionalForm],
6621	FormBox[
6622	OverscriptBox[
6623	FormBox["e4",
6624	TraditionalForm], "_"],
6625	TraditionalForm]}], ")"}], " ",
6626	RowBox[{"(",
6627	RowBox[{
6628	RowBox[{
6629	SuperscriptBox["f",
6630	RowBox[{
6631	FormBox[
6632	FormBox["a",
6633	TraditionalForm],
6634	TraditionalForm],
6635	FormBox[
6636	FormBox["a1",
6637	TraditionalForm],
6638	TraditionalForm],
6639	FormBox[
6640	FormBox["a4",
6641	TraditionalForm],
6642	TraditionalForm]}]], " ",
6643	SuperscriptBox["f",
6644	RowBox[{
6645	FormBox[
6646	FormBox["a",
6647	TraditionalForm],
6648	TraditionalForm],
6649	FormBox[
6650	FormBox["a2",
6651	TraditionalForm],
6652	TraditionalForm],
6653	FormBox[
6654	FormBox["a3",
6655	TraditionalForm],
6656	TraditionalForm]}]]}], "-",
6657	RowBox[{
6658	SuperscriptBox["f",
6659	RowBox[{
6660	FormBox[
6661	FormBox["a",
6662	TraditionalForm],
6663	TraditionalForm],
6664	FormBox[
6665	FormBox["a1",
6666	TraditionalForm],
6667	TraditionalForm],
6668	FormBox[
6669	FormBox["a3",
6670	TraditionalForm],
6671	TraditionalForm]}]], " ",
6672	SuperscriptBox["f",
6673	RowBox[{
6674	FormBox[
6675	FormBox["a",
6676	TraditionalForm],
6677	TraditionalForm],
6678	FormBox[
6679	FormBox["a2",
6680	TraditionalForm],
6681	TraditionalForm],
6682	FormBox[
6683	FormBox["a4",
6684	TraditionalForm],
6685	TraditionalForm]}]]}], "+",
6686	RowBox[{
6687	SuperscriptBox["f",
6688	RowBox[{
6689	FormBox[
6690	FormBox["a",
6691	TraditionalForm],
6692	TraditionalForm],
6693	FormBox[
6694	FormBox["a1",
6695	TraditionalForm],
6696	TraditionalForm],
6697	FormBox[
6698	FormBox["a2",
6699	TraditionalForm],
6700	TraditionalForm]}]], " ",
6701	SuperscriptBox["f",
6702	RowBox[{
6703	FormBox[
6704	FormBox["a",
6705	TraditionalForm],
6706	TraditionalForm],
6707	FormBox[
6708	FormBox["a3",
6709	TraditionalForm],
6710	TraditionalForm],
6711	FormBox[
6712	FormBox["a4",
6713	TraditionalForm],
6714	TraditionalForm]}]]}]}], ")"}]}]}], ")"}]}],
6715	TraditionalForm]], "Output",
6716	CellChangeTimes->{
6717	3.692616420713436^9, {3.692616483637841^9, 3.6926165124354*^9},
6718	3.692616679772766^9, 3.692616732506304^9, 3.69261690086381*^9,
6719	3.692616964589859^9, 3.692617068653575^9, 3.6926172433546877`*^9,
6720	3.692617320819441^9, 3.692617361848568^9, {3.692617490454344*^9,
6721	3.6926174954059343`^9}, 3.6926175264730387`^9, {3.69261755752717*^9,
6722	3.6926175646190166`^9}, 3.6926179371921^9, 3.69261907434236*^9, {
6723	3.692619545855597^9, 3.692619555544076^9}, 3.692622063183185*^9,
6724	3.69294073604916^9, 3.69294145595831^9, {3.693124173367807*^9,
6725	3.6931241822202597`*^9}}]
6726	}, Open ]],
6727
6728	Cell[BoxData[""], "Input",
6729	CellChangeTimes->{{3.692618068627322^9, 3.692618076577537^9}}],
6730
6731	Cell[CellGroupData[{
6732
6733	Cell[BoxData[
6734	RowBox[{"SUNSimplify", "[",
6735	RowBox[{"AmpSimp4g", ",", " ",
6736	RowBox[{"SUNFJacobi", "\[Rule]", "True"}]}], "]"}]], "Input",
6737	CellChangeTimes->{{3.692618068711218^9, 3.692618072106385^9}, {
6738	3.6926181657714863`^9, 3.692618190735327^9}, 3.6926182265137043`*^9, {
6739	3.692618256956256^9, 3.6926182652036867`^9}, {3.692618306876439*^9,
6740	3.692618311033248^9}, {3.692619385718408^9, 3.692619525792911*^9}, {
6741	3.692619561954356^9, 3.692619566992593^9}, {3.6926196948273973`*^9,
6742	3.6926197012913^9}, {3.692620386201343^9, 3.692620445251412*^9}, {
6743	3.692622047788809^9, 3.692622047868909^9}}],
6744
6745	Cell[BoxData[
6746	FormBox[
6747	RowBox[{
6748	RowBox[{"-", "\[ImaginaryI]"}], " ",
6749	SuperscriptBox["gs", "2"], " ",
6750	RowBox[{"(",
6751	RowBox[{
6752	RowBox[{
6753	RowBox[{"(",
6754	RowBox[{
6755	FormBox[
6756	OverscriptBox[
6757	FormBox["e2",
6758	TraditionalForm], "_"],
6759	TraditionalForm],
6760	FormBox["\<\"\[CenterDot]\"\>",
6761	TraditionalForm],
6762	FormBox[
6763	OverscriptBox[
6764	FormBox["p1",
6765	TraditionalForm], "_"],
6766	TraditionalForm]}], ")"}], " ",
6767	RowBox[{"(",
6768	RowBox[{
6769	FormBox[
6770	OverscriptBox[
6771	FormBox["e3",
6772	TraditionalForm], "_"],
6773	TraditionalForm],
6774	FormBox["\<\"\[CenterDot]\"\>",
6775	TraditionalForm],
6776	FormBox[
6777	OverscriptBox[
6778	FormBox["e4",
6779	TraditionalForm], "_"],
6780	TraditionalForm]}], ")"}], " ",
6781	SuperscriptBox["f",
6782	RowBox[{
6783	FormBox[
6784	FormBox["a",
6785	TraditionalForm],
6786	TraditionalForm],
6787	FormBox[
6788	FormBox["a1",
6789	TraditionalForm],
6790	TraditionalForm],
6791	FormBox[
6792	FormBox["a4",
6793	TraditionalForm],
6794	TraditionalForm]}]], " ",
6795	SuperscriptBox["f",
6796	RowBox[{
6797	FormBox[
6798	FormBox["a",
6799	TraditionalForm],
6800	TraditionalForm],
6801	FormBox[
6802	FormBox["a2",
6803	TraditionalForm],
6804	TraditionalForm],
6805	FormBox[
6806	FormBox["a3",
6807	TraditionalForm],
6808	TraditionalForm]}]]}], "+",
6809	RowBox[{
6810	RowBox[{"(",
6811	RowBox[{
6812	FormBox[
6813	OverscriptBox[
6814	FormBox["e2",
6815	TraditionalForm], "_"],
6816	TraditionalForm],
6817	FormBox["\<\"\[CenterDot]\"\>",
6818	TraditionalForm],
6819	FormBox[
6820	OverscriptBox[
6821	FormBox["e3",
6822	TraditionalForm], "_"],
6823	TraditionalForm]}], ")"}], " ",
6824	RowBox[{"(",
6825	RowBox[{
6826	FormBox[
6827	OverscriptBox[
6828	FormBox["e4",
6829	TraditionalForm], "_"],
6830	TraditionalForm],
6831	FormBox["\<\"\[CenterDot]\"\>",
6832	TraditionalForm],
6833	FormBox[
6834	OverscriptBox[
6835	FormBox["p1",
6836	TraditionalForm], "_"],
6837	TraditionalForm]}], ")"}], " ",
6838	SuperscriptBox["f",
6839	RowBox[{
6840	FormBox[
6841	FormBox["a",
6842	TraditionalForm],
6843	TraditionalForm],
6844	FormBox[
6845	FormBox["a1",
6846	TraditionalForm],
6847	TraditionalForm],
6848	FormBox[
6849	FormBox["a4",
6850	TraditionalForm],
6851	TraditionalForm]}]], " ",
6852	SuperscriptBox["f",
6853	RowBox[{
6854	FormBox[
6855	FormBox["a",
6856	TraditionalForm],
6857	TraditionalForm],
6858	FormBox[
6859	FormBox["a2",
6860	TraditionalForm],
6861	TraditionalForm],
6862	FormBox[
6863	FormBox["a3",
6864	TraditionalForm],
6865	TraditionalForm]}]]}], "+",
6866	RowBox[{
6867	RowBox[{"(",
6868	RowBox[{
6869	FormBox[
6870	OverscriptBox[
6871	FormBox["e2",
6872	TraditionalForm], "_"],
6873	TraditionalForm],
6874	FormBox["\<\"\[CenterDot]\"\>",
6875	TraditionalForm],
6876	FormBox[
6877	OverscriptBox[
6878	FormBox["p1",
6879	TraditionalForm], "_"],
6880	TraditionalForm]}], ")"}], " ",
6881	RowBox[{"(",
6882	RowBox[{
6883	FormBox[
6884	OverscriptBox[
6885	FormBox["e3",
6886	TraditionalForm], "_"],
6887	TraditionalForm],
6888	FormBox["\<\"\[CenterDot]\"\>",
6889	TraditionalForm],
6890	FormBox[
6891	OverscriptBox[
6892	FormBox["e4",
6893	TraditionalForm], "_"],
6894	TraditionalForm]}], ")"}], " ",
6895	SuperscriptBox["f",
6896	RowBox[{
6897	FormBox[
6898	FormBox["a",
6899	TraditionalForm],
6900	TraditionalForm],
6901	FormBox[
6902	FormBox["a1",
6903	TraditionalForm],
6904	TraditionalForm],
6905	FormBox[
6906	FormBox["a3",
6907	TraditionalForm],
6908	TraditionalForm]}]], " ",
6909	SuperscriptBox["f",
6910	RowBox[{
6911	FormBox[
6912	FormBox["a",
6913	TraditionalForm],
6914	TraditionalForm],
6915	FormBox[
6916	FormBox["a2",
6917	TraditionalForm],
6918	TraditionalForm],
6919	FormBox[
6920	FormBox["a4",
6921	TraditionalForm],
6922	TraditionalForm]}]]}], "-",
6923	RowBox[{
6924	RowBox[{"(",
6925	RowBox[{
6926	FormBox[
6927	OverscriptBox[
6928	FormBox["e2",
6929	TraditionalForm], "_"],
6930	TraditionalForm],
6931	FormBox["\<\"\[CenterDot]\"\>",
6932	TraditionalForm],
6933	FormBox[
6934	OverscriptBox[
6935	FormBox["e4",
6936	TraditionalForm], "_"],
6937	TraditionalForm]}], ")"}], " ",
6938	RowBox[{"(",
6939	RowBox[{
6940	FormBox[
6941	OverscriptBox[
6942	FormBox["e3",
6943	TraditionalForm], "_"],
6944	TraditionalForm],
6945	FormBox["\<\"\[CenterDot]\"\>",
6946	TraditionalForm],
6947	FormBox[
6948	OverscriptBox[
6949	FormBox["p1",
6950	TraditionalForm], "_"],
6951	TraditionalForm]}], ")"}], " ",
6952	SuperscriptBox["f",
6953	RowBox[{
6954	FormBox[
6955	FormBox["a",
6956	TraditionalForm],
6957	TraditionalForm],
6958	FormBox[
6959	FormBox["a1",
6960	TraditionalForm],
6961	TraditionalForm],
6962	FormBox[
6963	FormBox["a3",
6964	TraditionalForm],
6965	TraditionalForm]}]], " ",
6966	SuperscriptBox["f",
6967	RowBox[{
6968	FormBox[
6969	FormBox["a",
6970	TraditionalForm],
6971	TraditionalForm],
6972	FormBox[
6973	FormBox["a2",
6974	TraditionalForm],
6975	TraditionalForm],
6976	FormBox[
6977	FormBox["a4",
6978	TraditionalForm],
6979	TraditionalForm]}]]}], "-",
6980	RowBox[{"2", " ",
6981	RowBox[{"(",
6982	RowBox[{
6983	FormBox[
6984	OverscriptBox[
6985	FormBox["e2",
6986	TraditionalForm], "_"],
6987	TraditionalForm],
6988	FormBox["\<\"\[CenterDot]\"\>",
6989	TraditionalForm],
6990	FormBox[
6991	OverscriptBox[
6992	FormBox["e3",
6993	TraditionalForm], "_"],
6994	TraditionalForm]}], ")"}], " ",
6995	RowBox[{"(",
6996	RowBox[{
6997	FormBox[
6998	OverscriptBox[
6999	FormBox["e4",
7000	TraditionalForm], "_"],
7001	TraditionalForm],
7002	FormBox["\<\"\[CenterDot]\"\>",
7003	TraditionalForm],
7004	FormBox[
7005	OverscriptBox[
7006	FormBox["p1",
7007	TraditionalForm], "_"],
7008	TraditionalForm]}], ")"}], " ",
7009	SuperscriptBox["f",
7010	RowBox[{
7011	FormBox[
7012	FormBox["a",
7013	TraditionalForm],
7014	TraditionalForm],
7015	FormBox[
7016	FormBox["a1",
7017	TraditionalForm],
7018	TraditionalForm],
7019	FormBox[
7020	FormBox["a3",
7021	TraditionalForm],
7022	TraditionalForm]}]], " ",
7023	SuperscriptBox["f",
7024	RowBox[{
7025	FormBox[
7026	FormBox["a",
7027	TraditionalForm],
7028	TraditionalForm],
7029	FormBox[
7030	FormBox["a2",
7031	TraditionalForm],
7032	TraditionalForm],
7033	FormBox[
7034	FormBox["a4",
7035	TraditionalForm],
7036	TraditionalForm]}]]}], "+",
7037	RowBox[{"2", " ",
7038	RowBox[{"(",
7039	RowBox[{
7040	FormBox[
7041	OverscriptBox[
7042	FormBox["e2",
7043	TraditionalForm], "_"],
7044	TraditionalForm],
7045	FormBox["\<\"\[CenterDot]\"\>",
7046	TraditionalForm],
7047	FormBox[
7048	OverscriptBox[
7049	FormBox["e4",
7050	TraditionalForm], "_"],
7051	TraditionalForm]}], ")"}], " ",
7052	RowBox[{"(",
7053	RowBox[{
7054	FormBox[
7055	OverscriptBox[
7056	FormBox["e3",
7057	TraditionalForm], "_"],
7058	TraditionalForm],
7059	FormBox["\<\"\[CenterDot]\"\>",
7060	TraditionalForm],
7061	FormBox[
7062	OverscriptBox[
7063	FormBox["p1",
7064	TraditionalForm], "_"],
7065	TraditionalForm]}], ")"}], " ",
7066	SuperscriptBox["f",
7067	RowBox[{
7068	FormBox[
7069	FormBox["a",
7070	TraditionalForm],
7071	TraditionalForm],
7072	FormBox[
7073	FormBox["a1",
7074	TraditionalForm],
7075	TraditionalForm],
7076	FormBox[
7077	FormBox["a2",
7078	TraditionalForm],
7079	TraditionalForm]}]], " ",
7080	SuperscriptBox["f",
7081	RowBox[{
7082	FormBox[
7083	FormBox["a",
7084	TraditionalForm],
7085	TraditionalForm],
7086	FormBox[
7087	FormBox["a3",
7088	TraditionalForm],
7089	TraditionalForm],
7090	FormBox[
7091	FormBox["a4",
7092	TraditionalForm],
7093	TraditionalForm]}]]}], "+",
7094	RowBox[{"2", " ",
7095	RowBox[{"(",
7096	RowBox[{
7097	FormBox[
7098	OverscriptBox[
7099	FormBox["e2",
7100	TraditionalForm], "_"],
7101	TraditionalForm],
7102	FormBox["\<\"\[CenterDot]\"\>",
7103	TraditionalForm],
7104	FormBox[
7105	OverscriptBox[
7106	FormBox["e4",
7107	TraditionalForm], "_"],
7108	TraditionalForm]}], ")"}], " ",
7109	RowBox[{"(",
7110	RowBox[{
7111	FormBox[
7112	OverscriptBox[
7113	FormBox["e3",
7114	TraditionalForm], "_"],
7115	TraditionalForm],
7116	FormBox["\<\"\[CenterDot]\"\>",
7117	TraditionalForm],
7118	FormBox[
7119	OverscriptBox[
7120	FormBox["p2",
7121	TraditionalForm], "_"],
7122	TraditionalForm]}], ")"}], " ",
7123	SuperscriptBox["f",
7124	RowBox[{
7125	FormBox[
7126	FormBox["a",
7127	TraditionalForm],
7128	TraditionalForm],
7129	FormBox[
7130	FormBox["a1",
7131	TraditionalForm],
7132	TraditionalForm],
7133	FormBox[
7134	FormBox["a4",
7135	TraditionalForm],
7136	TraditionalForm]}]], " ",
7137	SuperscriptBox["f",
7138	RowBox[{
7139	FormBox[
7140	FormBox["a",
7141	TraditionalForm],
7142	TraditionalForm],
7143	FormBox[
7144	FormBox["a2",
7145	TraditionalForm],
7146	TraditionalForm],
7147	FormBox[
7148	FormBox["a3",
7149	TraditionalForm],
7150	TraditionalForm]}]]}], "-",
7151	RowBox[{"2", " ",
7152	RowBox[{"(",
7153	RowBox[{
7154	FormBox[
7155	OverscriptBox[
7156	FormBox["e2",
7157	TraditionalForm], "_"],
7158	TraditionalForm],
7159	FormBox["\<\"\[CenterDot]\"\>",
7160	TraditionalForm],
7161	FormBox[
7162	OverscriptBox[
7163	FormBox["e4",
7164	TraditionalForm], "_"],
7165	TraditionalForm]}], ")"}], " ",
7166	RowBox[{"(",
7167	RowBox[{
7168	FormBox[
7169	OverscriptBox[
7170	FormBox["e3",
7171	TraditionalForm], "_"],
7172	TraditionalForm],
7173	FormBox["\<\"\[CenterDot]\"\>",
7174	TraditionalForm],
7175	FormBox[
7176	OverscriptBox[
7177	FormBox["p2",
7178	TraditionalForm], "_"],
7179	TraditionalForm]}], ")"}], " ",
7180	SuperscriptBox["f",
7181	RowBox[{
7182	FormBox[
7183	FormBox["a",
7184	TraditionalForm],
7185	TraditionalForm],
7186	FormBox[
7187	FormBox["a1",
7188	TraditionalForm],
7189	TraditionalForm],
7190	FormBox[
7191	FormBox["a3",
7192	TraditionalForm],
7193	TraditionalForm]}]], " ",
7194	SuperscriptBox["f",
7195	RowBox[{
7196	FormBox[
7197	FormBox["a",
7198	TraditionalForm],
7199	TraditionalForm],
7200	FormBox[
7201	FormBox["a2",
7202	TraditionalForm],
7203	TraditionalForm],
7204	FormBox[
7205	FormBox["a4",
7206	TraditionalForm],
7207	TraditionalForm]}]]}], "+",
7208	RowBox[{"2", " ",
7209	RowBox[{"(",
7210	RowBox[{
7211	FormBox[
7212	OverscriptBox[
7213	FormBox["e2",
7214	TraditionalForm], "_"],
7215	TraditionalForm],
7216	FormBox["\<\"\[CenterDot]\"\>",
7217	TraditionalForm],
7218	FormBox[
7219	OverscriptBox[
7220	FormBox["e4",
7221	TraditionalForm], "_"],
7222	TraditionalForm]}], ")"}], " ",
7223	RowBox[{"(",
7224	RowBox[{
7225	FormBox[
7226	OverscriptBox[
7227	FormBox["e3",
7228	TraditionalForm], "_"],
7229	TraditionalForm],
7230	FormBox["\<\"\[CenterDot]\"\>",
7231	TraditionalForm],
7232	FormBox[
7233	OverscriptBox[
7234	FormBox["p2",
7235	TraditionalForm], "_"],
7236	TraditionalForm]}], ")"}], " ",
7237	SuperscriptBox["f",
7238	RowBox[{
7239	FormBox[
7240	FormBox["a",
7241	TraditionalForm],
7242	TraditionalForm],
7243	FormBox[
7244	FormBox["a1",
7245	TraditionalForm],
7246	TraditionalForm],
7247	FormBox[
7248	FormBox["a2",
7249	TraditionalForm],
7250	TraditionalForm]}]], " ",
7251	SuperscriptBox["f",
7252	RowBox[{
7253	FormBox[
7254	FormBox["a",
7255	TraditionalForm],
7256	TraditionalForm],
7257	FormBox[
7258	FormBox["a3",
7259	TraditionalForm],
7260	TraditionalForm],
7261	FormBox[
7262	FormBox["a4",
7263	TraditionalForm],
7264	TraditionalForm]}]]}], "+",
7265	RowBox[{"2", " ",
7266	RowBox[{"(",
7267	RowBox[{
7268	FormBox[
7269	OverscriptBox[
7270	FormBox["e2",
7271	TraditionalForm], "_"],
7272	TraditionalForm],
7273	FormBox["\<\"\[CenterDot]\"\>",
7274	TraditionalForm],
7275	FormBox[
7276	OverscriptBox[
7277	FormBox["p3",
7278	TraditionalForm], "_"],
7279	TraditionalForm]}], ")"}], " ",
7280	RowBox[{"(",
7281	RowBox[{
7282	FormBox[
7283	OverscriptBox[
7284	FormBox["e3",
7285	TraditionalForm], "_"],
7286	TraditionalForm],
7287	FormBox["\<\"\[CenterDot]\"\>",
7288	TraditionalForm],
7289	FormBox[
7290	OverscriptBox[
7291	FormBox["e4",
7292	TraditionalForm], "_"],
7293	TraditionalForm]}], ")"}], " ",
7294	SuperscriptBox["f",
7295	RowBox[{
7296	FormBox[
7297	FormBox["a",
7298	TraditionalForm],
7299	TraditionalForm],
7300	FormBox[
7301	FormBox["a1",
7302	TraditionalForm],
7303	TraditionalForm],
7304	FormBox[
7305	FormBox["a4",
7306	TraditionalForm],
7307	TraditionalForm]}]], " ",
7308	SuperscriptBox["f",
7309	RowBox[{
7310	FormBox[
7311	FormBox["a",
7312	TraditionalForm],
7313	TraditionalForm],
7314	FormBox[
7315	FormBox["a2",
7316	TraditionalForm],
7317	TraditionalForm],
7318	FormBox[
7319	FormBox["a3",
7320	TraditionalForm],
7321	TraditionalForm]}]]}], "-",
7322	RowBox[{"2", " ",
7323	RowBox[{"(",
7324	RowBox[{
7325	FormBox[
7326	OverscriptBox[
7327	FormBox["e2",
7328	TraditionalForm], "_"],
7329	TraditionalForm],
7330	FormBox["\<\"\[CenterDot]\"\>",
7331	TraditionalForm],
7332	FormBox[
7333	OverscriptBox[
7334	FormBox["p3",
7335	TraditionalForm], "_"],
7336	TraditionalForm]}], ")"}], " ",
7337	RowBox[{"(",
7338	RowBox[{
7339	FormBox[
7340	OverscriptBox[
7341	FormBox["e3",
7342	TraditionalForm], "_"],
7343	TraditionalForm],
7344	FormBox["\<\"\[CenterDot]\"\>",
7345	TraditionalForm],
7346	FormBox[
7347	OverscriptBox[
7348	FormBox["e4",
7349	TraditionalForm], "_"],
7350	TraditionalForm]}], ")"}], " ",
7351	SuperscriptBox["f",
7352	RowBox[{
7353	FormBox[
7354	FormBox["a",
7355	TraditionalForm],
7356	TraditionalForm],
7357	FormBox[
7358	FormBox["a1",
7359	TraditionalForm],
7360	TraditionalForm],
7361	FormBox[
7362	FormBox["a3",
7363	TraditionalForm],
7364	TraditionalForm]}]], " ",
7365	SuperscriptBox["f",
7366	RowBox[{
7367	FormBox[
7368	FormBox["a",
7369	TraditionalForm],
7370	TraditionalForm],
7371	FormBox[
7372	FormBox["a2",
7373	TraditionalForm],
7374	TraditionalForm],
7375	FormBox[
7376	FormBox["a4",
7377	TraditionalForm],
7378	TraditionalForm]}]]}], "+",
7379	RowBox[{"2", " ",
7380	RowBox[{"(",
7381	RowBox[{
7382	FormBox[
7383	OverscriptBox[
7384	FormBox["e2",
7385	TraditionalForm], "_"],
7386	TraditionalForm],
7387	FormBox["\<\"\[CenterDot]\"\>",
7388	TraditionalForm],
7389	FormBox[
7390	OverscriptBox[
7391	FormBox["p3",
7392	TraditionalForm], "_"],
7393	TraditionalForm]}], ")"}], " ",
7394	RowBox[{"(",
7395	RowBox[{
7396	FormBox[
7397	OverscriptBox[
7398	FormBox["e3",
7399	TraditionalForm], "_"],
7400	TraditionalForm],
7401	FormBox["\<\"\[CenterDot]\"\>",
7402	TraditionalForm],
7403	FormBox[
7404	OverscriptBox[
7405	FormBox["e4",
7406	TraditionalForm], "_"],
7407	TraditionalForm]}], ")"}], " ",
7408	SuperscriptBox["f",
7409	RowBox[{
7410	FormBox[
7411	FormBox["a",
7412	TraditionalForm],
7413	TraditionalForm],
7414	FormBox[
7415	FormBox["a1",
7416	TraditionalForm],
7417	TraditionalForm],
7418	FormBox[
7419	FormBox["a2",
7420	TraditionalForm],
7421	TraditionalForm]}]], " ",
7422	SuperscriptBox["f",
7423	RowBox[{
7424	FormBox[
7425	FormBox["a",
7426	TraditionalForm],
7427	TraditionalForm],
7428	FormBox[
7429	FormBox["a3",
7430	TraditionalForm],
7431	TraditionalForm],
7432	FormBox[
7433	FormBox["a4",
7434	TraditionalForm],
7435	TraditionalForm]}]]}]}], ")"}]}], TraditionalForm]], "Output",
7436	CellChangeTimes->{{3.692619453272781^9, 3.692619462919716^9}, {
7437	3.692619516470133^9, 3.692619520154277^9}, {3.6926195634654016`*^9,
7438	3.6926195676063843`^9}, {3.692620420743376^9, 3.692620445924453*^9},
7439	3.692622063444206^9, 3.692940736241342^9, 3.692941456119411*^9, {
7440	3.6931241734325323`^9, 3.6931241824094543`^9}}]
7441	}, Open ]]
7442	}, Open ]],
7443
7444	Cell["\<\
7445	The last two lines are zero because of Jacobi Id. Applying the Jacobi Id. \
7446	gives back the textbook result.
7447	Point b) follows from the form of the vertex\
7448	\>", "Subsection",
7449	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7450	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7451	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7452	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7453	3.6926219139309196`*^9}}]
7454	}, Open ]],
7455
7456	Cell[CellGroupData[{
7457
7458	Cell["2.2.4: Renormalisation schemes", "Chapter",
7459	CellChangeTimes->{{3.692613426285159^9, 3.6926134324672422`^9}, {
7460	3.6926219265062923`^9, 3.69262192669018^9}, {3.692625478990284*^9,
7461	3.692625491516202*^9}}],
7462
7463	Cell[CellGroupData[{
7464
7465	Cell["\<\
7466	a) We write the definition of beta in terms of a generic alpha\
7467	\>", "Subsection",
7468	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7469	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7470	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7471	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7472	3.6926219139309196`^9}, {3.692625501772744^9, 3.692625513899186*^9}, {
7473	3.6926261351061983`^9, 3.692626135479179^9}}],
7474
7475	Cell[CellGroupData[{
7476
7477	Cell[BoxData[{
7478	RowBox[{
7479	RowBox[{
7480	RowBox[{"BetaF", "[", "as_", "]"}], " ", ":=", " ",
7481	RowBox[{"b", " ",
7482	RowBox[{"as", "^", "2"}],
7483	RowBox[{"(",
7484	RowBox[{"1", "+",
7485	RowBox[{"b1", " ", "as"}], "+", " ",
7486	RowBox[{"b2", " ",
7487	RowBox[{"as", "^", "2"}]}]}], ")"}]}]}],
7488	";"}], "\[IndentingNewLine]",
7489	RowBox[{"Series", "[",
7490	RowBox[{
7491	RowBox[{"BetaF", "[", "asA", "]"}], ",",
7492	RowBox[{"{",
7493	RowBox[{"asA", ",", "0", ",", "4"}], "}"}]}], "]"}]}], "Input",
7494	CellChangeTimes->{{3.692624936489541^9, 3.692624987968774^9}, {
7495	3.69262552178106^9, 3.692625521948695^9}}],
7496
7497	Cell[BoxData[
7498	FormBox[
7499	InterpretationBox[
7500	RowBox[{
7501	RowBox[{
7502	SuperscriptBox["asA", "2"], " ", "b"}], "+",
7503	RowBox[{
7504	SuperscriptBox["asA", "3"], " ", "b", " ", "b1"}], "+",
7505	RowBox[{
7506	SuperscriptBox["asA", "4"], " ", "b", " ", "b2"}], "+",
7507	InterpretationBox[
7508	RowBox[{"O", "(",
7509	SuperscriptBox["asA", "5"], ")"}],
7510	SeriesData[$CellContext`asA, 0, {}, 2, 5, 1],
7511	Editable->False]}],
7512	SeriesData[$CellContext`asA,
7513	0, {$CellContext`b, $CellContext`b $CellContext`b1, $CellContext`b \
7514	$CellContext`b2}, 2, 5, 1],
7515	Editable->False], TraditionalForm]], "Output",
7516	CellChangeTimes->{3.692625522605631^9, 3.6929407363059473`^9,
7517	3.692941456171633^9, 3.693124182472146^9}]
7518	}, Open ]]
7519	}, Open ]],
7520
7521	Cell[CellGroupData[{
7522
7523	Cell["\<\
7524	We compute the beta function for alphaSA, by using the one of alphaSB and \
7525	changing the scheme\
7526	\>", "Subsection",
7527	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7528	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7529	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7530	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7531	3.6926219139309196`^9}, {3.692625501772744^9, 3.692625513899186*^9}, {
7532	3.6926255519733057`^9, 3.692625573706592^9}, {3.692625628463838*^9,
7533	3.692625642551269^9}, {3.6926260410004168`^9, 3.692626065045793*^9}}],
7534
7535	Cell[CellGroupData[{
7536
7537	Cell[BoxData[
7538	RowBox[{"betaA", " ", "=", " ",
7539	RowBox[{"Series", "[",
7540	RowBox[{
7541	RowBox[{
7542	RowBox[{"BetaF", "[", "asB", "]"}], " ", "/.",
7543	RowBox[{"{",
7544	RowBox[{"asB", "\[Rule]",
7545	RowBox[{"asA", "+",
7546	RowBox[{"c1", " ",
7547	RowBox[{"asA", "^", "2"}]}], "+",
7548	RowBox[{"c2", " ",
7549	RowBox[{"asA", "^", "3"}]}]}]}], "}"}]}], ",",
7550	RowBox[{"{",
7551	RowBox[{"asA", ",", "0", ",", "4"}], "}"}]}], "]"}]}]], "Input",
7552	CellChangeTimes->{{3.692625028415299^9, 3.6926250934626427`^9}, {
7553	3.6926259797415667`^9, 3.692625982171582^9}}],
7554
7555	Cell[BoxData[
7556	FormBox[
7557	InterpretationBox[
7558	RowBox[{
7559	RowBox[{
7560	SuperscriptBox["asA", "2"], " ", "b"}], "+",
7561	RowBox[{
7562	SuperscriptBox["asA", "3"], " ",
7563	RowBox[{"(",
7564	RowBox[{
7565	RowBox[{"b", " ", "b1"}], "+",
7566	RowBox[{"2", " ", "b", " ", "c1"}]}], ")"}]}], "+",
7567	RowBox[{
7568	SuperscriptBox["asA", "4"], " ",
7569	RowBox[{"(",
7570	RowBox[{
7571	RowBox[{"3", " ", "b", " ", "b1", " ", "c1"}], "+",
7572	RowBox[{"b", " ", "b2"}], "+",
7573	RowBox[{"b", " ",
7574	SuperscriptBox["c1", "2"]}], "+",
7575	RowBox[{"2", " ", "b", " ", "c2"}]}], ")"}]}], "+",
7576	InterpretationBox[
7577	RowBox[{"O", "(",
7578	SuperscriptBox["asA", "5"], ")"}],
7579	SeriesData[$CellContext`asA, 0, {}, 2, 5, 1],
7580	Editable->False]}],
7581	SeriesData[$CellContext`asA,
7582	0, {$CellContext`b, $CellContext`b $CellContext`b1 +
7583	2 $CellContext`b $CellContext`c1, $CellContext`b $CellContext`b2 +
7584	3 $CellContext`b $CellContext`b1 $CellContext`c1 + $CellContext`b \
7585	$CellContext`c1^2 + 2 $CellContext`b $CellContext`c2}, 2, 5, 1],
7586	Editable->False], TraditionalForm]], "Output",
7587	CellChangeTimes->{{3.692625072533464^9, 3.692625093872745^9},
7588	3.692625992027192^9, 3.6929407363439007`^9, 3.692941456222253*^9,
7589	3.693124182511261*^9}]
7590	}, Open ]]
7591	}, Open ]],
7592
7593	Cell[CellGroupData[{
7594
7595	Cell["\<\
7596	We compute the beta function for alphaSB, as
7597	dalphaSB/dmu = dalphaSB/dalphaSA * dalphaSA/dmu\
7598	\>", "Subsection",
7599	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7600	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7601	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7602	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7603	3.6926219139309196`^9}, {3.692625501772744^9, 3.692625513899186*^9}, {
7604	3.6926255519733057`^9, 3.692625573706592^9}, {3.692625628463838*^9,
7605	3.6926257183046427`^9}, {3.692626067749858^9, 3.6926260763250427`*^9}}],
7606
7607	Cell[CellGroupData[{
7608
7609	Cell[BoxData[
7610	RowBox[{"betaB", " ", "=", " ",
7611	RowBox[{"Series", "[",
7612	RowBox[{
7613	RowBox[{
7614	RowBox[{"D", "[",
7615	RowBox[{
7616	RowBox[{"asA", "+",
7617	RowBox[{"c1", " ",
7618	RowBox[{"asA", "^", "2"}]}], "+",
7619	RowBox[{"c2", " ",
7620	RowBox[{"asA", "^", "3"}]}]}], ",", " ", "asA"}], "]"}], "*",
7621	RowBox[{"BetaF", "[", "asA", "]"}]}], ",", " ",
7622	RowBox[{"{",
7623	RowBox[{"asA", ",", " ", "0", ",", " ", "4"}], "}"}]}], "]"}]}]], "Input",\
7624
7625	CellChangeTimes->{{3.69262518933298^9, 3.692625254107587^9}, {
7626	3.692625414636454^9, 3.692625438913671^9}, {3.69262598496443*^9,
7627	3.6926259863398123`*^9}}],
7628
7629	Cell[BoxData[
7630	FormBox[
7631	InterpretationBox[
7632	RowBox[{
7633	RowBox[{
7634	SuperscriptBox["asA", "2"], " ", "b"}], "+",
7635	RowBox[{
7636	SuperscriptBox["asA", "3"], " ",
7637	RowBox[{"(",
7638	RowBox[{
7639	RowBox[{"b", " ", "b1"}], "+",
7640	RowBox[{"2", " ", "b", " ", "c1"}]}], ")"}]}], "+",
7641	RowBox[{
7642	SuperscriptBox["asA", "4"], " ",
7643	RowBox[{"(",
7644	RowBox[{
7645	RowBox[{"2", " ", "b", " ", "b1", " ", "c1"}], "+",
7646	RowBox[{"b", " ", "b2"}], "+",
7647	RowBox[{"3", " ", "b", " ", "c2"}]}], ")"}]}], "+",
7648	InterpretationBox[
7649	RowBox[{"O", "(",
7650	SuperscriptBox["asA", "5"], ")"}],
7651	SeriesData[$CellContext`asA, 0, {}, 2, 5, 1],
7652	Editable->False]}],
7653	SeriesData[$CellContext`asA,
7654	0, {$CellContext`b, $CellContext`b $CellContext`b1 +
7655	2 $CellContext`b $CellContext`c1, $CellContext`b $CellContext`b2 +
7656	2 $CellContext`b $CellContext`b1 $CellContext`c1 +
7657	3 $CellContext`b $CellContext`c2}, 2, 5, 1],
7658	Editable->False], TraditionalForm]], "Output",
7659	CellChangeTimes->{{3.6926252396542377`^9, 3.692625243401655^9},
7660	3.692625300194817^9, 3.6926254391660843`^9, {3.692625986876377*^9,
7661	3.692625992086767^9}, 3.692940736388238^9, 3.692941456271041*^9,
7662	3.6931241825717287`*^9}]
7663	}, Open ]]
7664	}, Open ]],
7665
7666	Cell[CellGroupData[{
7667
7668	Cell["The difference is", "Subsection",
7669	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7670	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7671	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7672	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7673	3.6926219139309196`^9}, {3.692625501772744^9, 3.692625513899186*^9}, {
7674	3.6926255519733057`^9, 3.692625573706592^9}, {3.692625628463838*^9,
7675	3.6926257183046427`^9}, {3.692626067749858^9, 3.692626092212718*^9}}],
7676
7677	Cell[CellGroupData[{
7678
7679	Cell[BoxData[
7680	RowBox[{
7681	RowBox[{"betaB", "-", "betaA"}], "//", "Simplify"}]], "Input",
7682	CellChangeTimes->{{3.6926259937683783`^9, 3.6926260157855883`^9}}],
7683
7684	Cell[BoxData[
7685	FormBox[
7686	InterpretationBox[
7687	RowBox[{
7688	RowBox[{"-",
7689	RowBox[{
7690	SuperscriptBox["asA", "4"], " ", "b", " ",
7691	RowBox[{"(",
7692	RowBox[{
7693	RowBox[{"b1", " ", "c1"}], "+",
7694	SuperscriptBox["c1", "2"], "-", "c2"}], ")"}]}]}], "+",
7695	InterpretationBox[
7696	RowBox[{"O", "(",
7697	SuperscriptBox["asA", "5"], ")"}],
7698	SeriesData[$CellContext`asA, 0, {}, 4, 5, 1],
7699	Editable->False]}],
7700	SeriesData[$CellContext`asA,
7701	0, {-$CellContext`b ($CellContext`b1 $CellContext`c1 + $CellContext`c1^2 - \
7702	$CellContext`c2)}, 4, 5, 1],
7703	Editable->False], TraditionalForm]], "Output",
7704	CellChangeTimes->{{3.692626001417677^9, 3.6926260161746807`^9},
7705	3.69294073643788^9, 3.6929414563218403`^9, 3.69312418263839*^9}]
7706	}, Open ]]
7707	}, Open ]],
7708
7709	Cell[CellGroupData[{
7710
7711	Cell["\<\
7712	b) using the definition for Lambda, the hint is easily verified. We do the \
7713	integration using only the first order of beta and of the asA -> asB change \
7714	of scheme, and we take the leading term of the result\
7715	\>", "Subsection",
7716	CellChangeTimes->{{3.692613597944758^9, 3.692613636916294^9}, {
7717	3.6926137866433907`^9, 3.69261380584085^9}, {3.692615800854991*^9,
7718	3.692615820540888^9}, {3.6926214826547203`^9, 3.692621498412581*^9}, {
7719	3.692621541883204^9, 3.69262158712852^9}, {3.6926219060614977`*^9,
7720	3.6926219139309196`^9}, {3.692625501772744^9, 3.692625513899186*^9}, {
7721	3.6926261351061983`^9, 3.692626201680035^9}, {3.692626355705297*^9,
7722	3.69262636603194^9}, {3.692626396373218^9, 3.692626408116714*^9}, {
7723	3.692626474532464^9, 3.692626480899115^9}}],
7724
7725	Cell[CellGroupData[{
7726
7727	Cell[BoxData[
7728	RowBox[{"Series", "[",
7729	RowBox[{
7730	RowBox[{
7731	RowBox[{"1", "/", "2"}], " ",
7732	RowBox[{"Integrate", "[",
7733	RowBox[{
7734	RowBox[{
7735	RowBox[{"1", "/", "b"}], " ", "/",
7736	RowBox[{"x", "^", "2"}]}], ",", " ",
7737	RowBox[{"{",
7738	RowBox[{"x", ",", "asA", ",", " ",
7739	RowBox[{"asA",
7740	RowBox[{"(",
7741	RowBox[{"1", "+",
7742	RowBox[{"c1", " ", "asA"}]}], ")"}]}]}], "}"}]}], "]"}]}], " ",
7743	",", " ",
7744	RowBox[{"{",
7745	RowBox[{"asA", ",", "0", ",", "0"}], "}"}]}], "]"}]], "Input",
7746	CellChangeTimes->{{3.692626218559719^9, 3.692626270897923^9}, {
7747	3.6926263357033587`^9, 3.692626339094391^9}, {3.692626369944179*^9,
7748	3.692626429028923^9}, {3.692626509889593^9, 3.692626512753497*^9}, {
7749	3.6926266452201633`^9, 3.692626651354129^9}, {3.692626695350904*^9,
7750	3.692626711212742*^9}}],
7751
7752	Cell[BoxData[
7753	FormBox[
7754	RowBox[{"ConditionalExpression", "[",
7755	RowBox[{
7756	InterpretationBox[
7757	RowBox[{
7758	FractionBox["c1",
7759	RowBox[{"2", " ", "b"}]], "+",
7760	InterpretationBox[
7761	RowBox[{"O", "(",
7762	SuperscriptBox["asA", "1"], ")"}],
7763	SeriesData[$CellContext`asA, 0, {}, 0, 1, 1],
7764	Editable->False]}],
7765	SeriesData[$CellContext`asA, 0, {
7766	Rational[1, 2] $CellContext`b^(-1) $CellContext`c1}, 0, 1, 1],
7767	Editable->False], ",",
7768	RowBox[{"\[Not]",
7769	RowBox[{"(",
7770	RowBox[{
7771	RowBox[{
7772	FractionBox["1",
7773	RowBox[{"asA", " ", "c1"}]], "\[Element]",
7774	TagBox["\[DoubleStruckCapitalR]",
7775	Function[{}, Reals]]}], "\[And]",
7776	RowBox[{
7777	RowBox[{"-", "1"}], "\[LessEqual]",
7778	RowBox[{"Re", "(",
7779	FractionBox["1",
7780	RowBox[{"asA", " ", "c1"}]], ")"}], "<", "0"}], "\[And]",
7781	RowBox[{"\[Not]",
7782	RowBox[{"(",
7783	RowBox[{
7784	RowBox[{
7785	FractionBox["1",
7786	RowBox[{"asA", " ", "c1"}]], "\[GreaterEqual]", "0"}], "\[Or]",
7787	RowBox[{
7788	FractionBox["1",
7789	RowBox[{"asA", " ", "c1"}]], "<",
7790	RowBox[{"-", "1"}]}]}], ")"}]}]}], ")"}]}]}], "]"}],
7791	TraditionalForm]], "Output",
7792	CellChangeTimes->{
7793	3.692626261713792^9, 3.692626339690404^9, {3.692626373197838*^9,
7794	3.692626382524496^9}, 3.6926264434469137`^9, 3.692626514725747*^9, {
7795	3.692626646864758^9, 3.692626652948028^9}, 3.692626715795827*^9,
7796	3.692940741196389^9, 3.692941461357312^9, 3.693124188169265*^9}]
7797	}, Open ]]
7798	}, Open ]]
7799	}, Open ]]
7800	}, Open ]]
7801	},
7802	WindowSize->{808, 597},
7803	WindowMargins->{{Automatic, 608}, {328, Automatic}},
7804	FrontEndVersion->"10.4 for Mac OS X x86 (32-bit, 64-bit Kernel) (February 25, \
7805	2016)",
7806	StyleDefinitions->"Default.nb"
7807	]
7808	(* End of Notebook Content *)
7809
7810	(* Internal cache information *)
7811	(*CellTagsOutline
7812	CellTagsIndex->{}
7813	*)
7814	(*CellTagsIndex
7815	CellTagsIndex->{}
7816	*)
7817	(*NotebookFileOutline
7818	Notebook[{
7819	Cell[558, 20, 85, 1, 28, "Input"],
7820	Cell[CellGroupData[{
7821	Cell[668, 25, 189, 2, 148, "Title"],
7822	Cell[CellGroupData[{
7823	Cell[882, 31, 156, 2, 44, "Subsection"],
7824	Cell[1041, 35, 182, 4, 28, "Input"],
7825	Cell[1226, 41, 109, 2, 28, "Input"]
7826	}, Open ]],
7827	Cell[CellGroupData[{
7828	Cell[1372, 48, 158, 2, 65, "Chapter"],
7829	Cell[CellGroupData[{
7830	Cell[1555, 54, 70, 1, 44, "Subsection"],
7831	Cell[1628, 57, 1432, 59, 170, "Text"],
7832	Cell[3063, 118, 1933, 58, 165, "Input"]
7833	}, Open ]],
7834	Cell[CellGroupData[{
7835	Cell[5033, 181, 119, 1, 44, "Subsection"],
7836	Cell[5155, 184, 2909, 84, 284, "Input"],
7837	Cell[8067, 270, 231, 6, 28, "Input"],
7838	Cell[CellGroupData[{
7839	Cell[8323, 280, 299, 7, 28, "Input"],
7840	Cell[8625, 289, 703, 12, 28, "Output"]
7841	}, Open ]]
7842	}, Open ]],
7843	Cell[CellGroupData[{
7844	Cell[9377, 307, 345, 6, 69, "Subsection"],
7845	Cell[CellGroupData[{
7846	Cell[9747, 317, 1411, 38, 80, "Input"],
7847	Cell[11161, 357, 1818, 68, 52, "Output"],
7848	Cell[12982, 427, 1816, 68, 52, "Output"]
7849	}, Open ]]
7850	}, Open ]],
7851	Cell[CellGroupData[{
7852	Cell[14847, 501, 550, 9, 69, "Subsection"],
7853	Cell[CellGroupData[{
7854	Cell[15422, 514, 485, 14, 28, "Input"],
7855	Cell[15910, 530, 2592, 96, 35, "Output"]
7856	}, Open ]]
7857	}, Open ]],
7858	Cell[CellGroupData[{
7859	Cell[18551, 632, 486, 8, 69, "Subsection"],
7860	Cell[CellGroupData[{
7861	Cell[19062, 644, 1548, 38, 63, "Input"],
7862	Cell[20613, 684, 1340, 42, 34, "Output"]
7863	}, Open ]]
7864	}, Open ]],
7865	Cell[CellGroupData[{
7866	Cell[22002, 732, 525, 9, 94, "Subsection"],
7867	Cell[CellGroupData[{
7868	Cell[22552, 745, 2127, 56, 80, "Input"],
7869	Cell[24682, 803, 3779, 141, 53, "Output"]
7870	}, Open ]]
7871	}, Open ]],
7872	Cell[28488, 948, 533, 8, 69, "Subsection"],
7873	Cell[CellGroupData[{
7874	Cell[29046, 960, 304, 6, 61, "Subsection"],
7875	Cell[29353, 968, 2269, 62, 165, "Input"],
7876	Cell[31625, 1032, 3608, 94, 216, "Input"]
7877	}, Open ]],
7878	Cell[CellGroupData[{
7879	Cell[35270, 1131, 282, 6, 69, "Subsection"],
7880	Cell[CellGroupData[{
7881	Cell[35577, 1141, 145, 2, 28, "Input"],
7882	Cell[35725, 1145, 60148, 2248, 325, "Output"]
7883	}, Open ]],
7884	Cell[CellGroupData[{
7885	Cell[95910, 3398, 955, 22, 46, "Input"],
7886	Cell[96868, 3422, 81685, 2556, 441, "Output"]
7887	}, Open ]],
7888	Cell[CellGroupData[{
7889	Cell[178590, 5983, 417, 9, 28, "Input"],
7890	Cell[179010, 5994, 19837, 730, 104, "Output"]
7891	}, Open ]],
7892	Cell[198862, 6727, 92, 1, 28, "Input"],
7893	Cell[CellGroupData[{
7894	Cell[198979, 6732, 628, 10, 28, "Input"],
7895	Cell[199610, 6744, 17875, 695, 104, "Output"]
7896	}, Open ]]
7897	}, Open ]],
7898	Cell[217512, 7443, 496, 9, 94, "Subsection"]
7899	}, Open ]],
7900	Cell[CellGroupData[{
7901	Cell[218045, 7457, 216, 3, 65, "Chapter"],
7902	Cell[CellGroupData[{
7903	Cell[218286, 7464, 503, 8, 44, "Subsection"],
7904	Cell[CellGroupData[{
7905	Cell[218814, 7476, 618, 18, 46, "Input"],
7906	Cell[219435, 7496, 731, 20, 35, "Output"]
7907	}, Open ]]
7908	}, Open ]],
7909	Cell[CellGroupData[{
7910	Cell[220215, 7522, 633, 10, 69, "Subsection"],
7911	Cell[CellGroupData[{
7912	Cell[220873, 7536, 595, 16, 28, "Input"],
7913	Cell[221471, 7554, 1299, 34, 35, "Output"]
7914	}, Open ]]
7915	}, Open ]],
7916	Cell[CellGroupData[{
7917	Cell[222819, 7594, 633, 10, 69, "Subsection"],
7918	Cell[CellGroupData[{
7919	Cell[223477, 7608, 655, 18, 28, "Input"],
7920	Cell[224135, 7628, 1275, 33, 35, "Output"]
7921	}, Open ]]
7922	}, Open ]],
7923	Cell[CellGroupData[{
7924	Cell[225459, 7667, 547, 7, 44, "Subsection"],
7925	Cell[CellGroupData[{
7926	Cell[226031, 7678, 158, 3, 28, "Input"],
7927	Cell[226192, 7683, 775, 21, 35, "Output"]
7928	}, Open ]]
7929	}, Open ]],
7930	Cell[CellGroupData[{
7931	Cell[227016, 7710, 792, 12, 94, "Subsection"],
7932	Cell[CellGroupData[{
7933	Cell[227833, 7726, 869, 23, 28, "Input"],
7934	Cell[228705, 7751, 1598, 44, 48, "Output"]
7935	}, Open ]]
7936	}, Open ]]
7937	}, Open ]]
7938	}, Open ]]
7939	}
7940	]
7941	*)
7942

Download in other formats:

Original Format