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525	Cell[BoxData[
526	\(\(\(\[IndentingNewLine]\)\(\(MyInte0[f_, xlow_, xup_, nev_] :=
527	Module[{xi, delta, sum, var}, \[IndentingNewLine]delta =
528	xup - xlow; \[IndentingNewLine]sum = 0; \[IndentingNewLine]var =
529	0; \[IndentingNewLine]Do[
530	xi = xlow + delta*Random[]; \[IndentingNewLine]sum =
531	sum + f[xi]; \[IndentingNewLine]var = var\ + f[xi]^2;, {i, 1,
532	nev}]; \[IndentingNewLine]\
533	sum = sum/nev*delta\ ; \[IndentingNewLine]\
534	var = var/nev*delta^2; \[IndentingNewLine]sig =
535	Sqrt[var - sum^2]; \[IndentingNewLine]Return[{sum,
536	sig}]];\)\(\[IndentingNewLine]\)
537	\)\)\)], "Input"],
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539	Cell[CellGroupData[{
540
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542	\(\(f[x_] := Cos[Pi/2\ x];\)\), "\[IndentingNewLine]",
543	\(\(xlow = 0;\)\), "\[IndentingNewLine]",
544	\(\(xup = 1;\)\), "\[IndentingNewLine]",
545	\(\(events = 10000;\)\), "\[IndentingNewLine]",
546	\(\(Print["\<Analytic Result : \>",
547	Integrate[f[x], {x, xlow, xup}] // N];\)\), "\[IndentingNewLine]",
548	\(\({res, sig} =
549	MyInte0[f, xlow, xup, events];\)\), "\[IndentingNewLine]",
550	\(\(Print["\<MC result :\>", res, \ "\< +- \>", sig, "\</Sqrt[\>",
551	events, "\<]\>"];\)\)}], "Input"],
552
553	Cell[BoxData[
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559
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563	\[InvisibleSpace]" +- \
564	"\[InvisibleSpace]0.30650274228326685`\[InvisibleSpace]"/Sqrt["\
565	\[InvisibleSpace]10000\[InvisibleSpace]"]"\),
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567	"MC result :", 0.63709339768428408, " +- ", 0.30650274228326685,
568	"/Sqrt[", 10000, "]"],
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570	}, Open ]]
571	}, Closed]],
572
573	Cell[CellGroupData[{
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575	Cell[TextData[StyleBox["Apply an analytic transformation : importance \
576	sampling",
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578
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1200	\>"],
1201	ImageRangeCache->{{{0, 357.75}, {357.75, 0}} -> {-0.0681302, -0.0543801, \
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1203
1204	Cell[BoxData[
1205	FormBox[
1206	TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\),
1207	False,
1208	Editable->False], TraditionalForm]], "Output"]
1209	}, Open ]],
1210
1211	Cell["\<\
1212
1213	To implement the transformation it is necessary to know:
1214	a. the primitive G of the \"enclosing\" function g
1215	b. the inverse of the primitive Gi
1216	This is what is obtained in the following:\
1217	\>", "Text",
1218	FontSize->16],
1219
1220	Cell[CellGroupData[{
1221
1222	Cell[BoxData[{
1223	\(\(g[x_] = 1 - x^2;\)\), "\[IndentingNewLine]",
1224	\(\(G[x_] = Integrate[g[z], {z, 0, x}];\)\), "\[IndentingNewLine]",
1225	\(sub = \(\(Solve[rho \[Equal] G[x], x]\)[\([3]\)]\)[\([1]\)] //
1226	Simplify\), "\n",
1227	\(Gi[rho_] = x /. sub\), "\[IndentingNewLine]",
1228	\(rhomin = G[0]\), "\[IndentingNewLine]",
1229	\(rhomax = G[1]\)}], "Input"],
1230
1231	Cell[BoxData[
1232	\(TraditionalForm\`x \[Rule] \(\[ImaginaryI]\ \((2\ \((\[ImaginaryI] + \
1233	\@3)\) - \@2\%3\ \((\(-\[ImaginaryI]\) + \@3)\)\ \((\@\(9\ rho\^2 - 4\) - 3\ \
1234	rho)\)\^\(2/3\))\)\)\/\(2\ 2\^\(2/3\)\ \@\(\@\(9\ rho\^2 - 4\) - 3\ \
1235	rho\)\%3\)\)], "Output"],
1236
1237	Cell[BoxData[
1238	\(TraditionalForm\`\(\[ImaginaryI]\ \((2\ \((\[ImaginaryI] + \@3)\) - \@2\
1239	\%3\ \((\(-\[ImaginaryI]\) + \@3)\)\ \((\@\(9\ rho\^2 - 4\) - 3\ \
1240	rho)\)\^\(2/3\))\)\)\/\(2\ 2\^\(2/3\)\ \@\(\@\(9\ rho\^2 - 4\) - 3\ \
1241	rho\)\%3\)\)], "Output"],
1242
1243	Cell[BoxData[
1244	\(TraditionalForm\`0\)], "Output"],
1245
1246	Cell[BoxData[
1247	\(TraditionalForm\`2\/3\)], "Output"]
1248	}, Closed]],
1249
1250	Cell[TextData[StyleBox["The new function to integrate is f/g[ro]:",
1251	FontSize->16]], "Text"],
1252
1253	Cell[BoxData[
1254	\(\(f[x_] = Cos[Pi/2\ x]/\((1 - x^2)\)\ ;\)\)], "Input"],
1255
1256	Cell[BoxData[
1257	\(\(MyInte1[f_, nev_] :=
1258	Module[{var, xi, sum}, \[IndentingNewLine]sum =
1259	0; \[IndentingNewLine]var = 0; \[IndentingNewLine]delta =
1260	rhomax - rhomin; \[IndentingNewLine]Do[
1261	rho = rhomin + delta*Random[]; \[IndentingNewLine]xi =
1262	Re[Gi[rho]]; \[IndentingNewLine]sum =
1263	sum + f[xi]\ ; \[IndentingNewLine]var = var + f[xi]^2\ , {i, 1,
1264	nev}]; \[IndentingNewLine]sum =
1265	sum/nev\ \ delta; \[IndentingNewLine]var =
1266	var/nev\ delta^2; \[IndentingNewLine]sig =
1267	Sqrt[var - sum^2]; \[IndentingNewLine]Return[{sum,
1268	sig}]];\)\)], "Input"],
1269
1270	Cell[CellGroupData[{
1271
1272	Cell[BoxData[{
1273	\(\({res, sig} = MyInte1[f\ , 10000];\)\), "\[IndentingNewLine]",
1274	\(\(Print["\<MC result :\>", res, \ "\< +- \>", sig, "\</Sqrt[\>",
1275	events, "\<]\>"];\)\)}], "Input"],
1276
1277	Cell[BoxData[
1278	FormBox[
1279	InterpretationBox[\("MC result :"\[InvisibleSpace]0.6368129794154503`\
1280	\[InvisibleSpace]" +- \
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1282	\[InvisibleSpace]events\[InvisibleSpace]"]"\),
1283	SequenceForm[
1284	"MC result :", 0.63681297941545034, " +- ", 0.031136105543578044,
1285	"/Sqrt[", events, "]"],
1286	Editable->False], TraditionalForm]], "Print"]
1287	}, Open ]]
1288	}, Closed]],
1289
1290	Cell[CellGroupData[{
1291
1292	Cell[TextData[StyleBox["Von Neumann's rejection method : plain vanilla",
1293	FontSize->16]], "Section"],
1294
1295	Cell[CellGroupData[{
1296
1297	Cell[BoxData[{
1298	\(\(<< Graphics`Colors`;\)\), "\[IndentingNewLine]",
1299	\(\(ob1 =
1300	Plot[{Cos[Pi/2\ x], 1}, {x, 0, 1}, \
1301	PlotStyle \[Rule] {{Blue, Thickness[0.007]}, {Red,
1302	Thickness[0.007]}}];\)\), "\[IndentingNewLine]",
1303	\(\(ob2 =
1304	Graphics[{Thickness[0.007], Red,
1305	Line[{{1, 0}, {1, 1}}]}];\)\), "\[IndentingNewLine]",
1306	\(Show[ob1, ob2, AspectRatio \[Rule] 1]\)}], "Input"],
1307
1308	Cell[GraphicsData["PostScript", "\<\
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2751	\>"],
2752	ImageRangeCache->{{{0, 371.188}, {229, 0}} -> {-0.0660376, -0.0680253, \
2753	0.00293933, 0.00504922}}],
2754
2755	Cell[BoxData[
2756	FormBox[
2757	TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\),
2758	False,
2759	Editable->False], TraditionalForm]], "Output"]
2760	}, Open ]],
2761
2762	Cell[BoxData[
2763	\(\(MyInte3[f_, xlims_, ylims_, nev_] :=
2764	Module[{var, jacobian, xi, sum,
2765	max}, \[IndentingNewLine] (*do\ the\ unweighting\
2766	*) \[IndentingNewLine]sum = 0; \[IndentingNewLine]var =
2767	0; \[IndentingNewLine]count = 0; \[IndentingNewLine]noin =
2768	0; \[IndentingNewLine]deltax =
2769	Abs[xlims[\([2]\)] - xlims[\([1]\)]]; \[IndentingNewLine]deltay =
2770	Abs[ylims[\([2]\)] - ylims[\([1]\)]]; \[IndentingNewLine]While[
2771	count < nev, \[IndentingNewLine]xi =
2772	deltax*Random[]; \[IndentingNewLine]yi =
2773	deltay\ *Random[]; \[IndentingNewLine]If[
2774	f[xi] > yi, \[IndentingNewLine]sum =
2775	sum + 1, \[IndentingNewLine]noin =
2776	noin + 1]; \[IndentingNewLine]count =
2777	count + 1]; \[IndentingNewLine]sum =
2778	sum/nev; \[IndentingNewLine]sig = sum; \[IndentingNewLine]eff =
2779	count/\((count + noin)\) //
2780	N; \[IndentingNewLine]Return[{sum // N, sig // N,
2781	eff}]];\)\)], "Input"],
2782
2783	Cell[CellGroupData[{
2784
2785	Cell[BoxData[{
2786	\(\(f[x_] := Cos[Pi/2\ x];\)\), "\[IndentingNewLine]",
2787	\(\(xlow = 0;\)\), "\[IndentingNewLine]",
2788	\(\(xup = 1;\)\), "\[IndentingNewLine]",
2789	\(\(events = 10000;\)\), "\[IndentingNewLine]",
2790	\(\(Print["\<Analytic Result : \>",
2791	Integrate[f[x], {x, xlow, xup}] // N];\)\), "\[IndentingNewLine]",
2792	\({res, sig, eff} =
2793	MyInte3[f, {xlow, xup}, {0, 1}, events]\), "\[IndentingNewLine]",
2794	\(\(Print["\<MC result :\>", res, \ "\< +- \>", sig, "\</Sqrt[\>",
2795	events, "\<], eff = \>", eff];\)\)}], "Input"],
2796
2797	Cell[BoxData[
2798	FormBox[
2799	InterpretationBox[\("Analytic Result : \
2800	"\[InvisibleSpace]0.6366197723675814`\),
2801	SequenceForm[ "Analytic Result : ", 0.63661977236758138],
2802	Editable->False], TraditionalForm]], "Print"],
2803
2804	Cell[BoxData[
2805	\(TraditionalForm\`{0.6347`, 0.6347`, 0.7324397568300007`}\)], "Output"],
2806
2807	Cell[BoxData[
2808	FormBox[
2809	InterpretationBox[\("MC result :"\[InvisibleSpace]0.6347`\
2810	\[InvisibleSpace]" +- "\[InvisibleSpace]0.6347`\[InvisibleSpace]"/Sqrt["\
2811	\[InvisibleSpace]10000\[InvisibleSpace]"], eff = \
2812	"\[InvisibleSpace]0.7324397568300007`\),
2813	SequenceForm[
2814	"MC result :", 0.63470000000000004, " +- ", 0.63470000000000004,
2815	"/Sqrt[", 10000, "], eff = ", 0.73243975683000073],
2816	Editable->False], TraditionalForm]], "Print"]
2817	}, Open ]]
2818	}, Closed]],
2819
2820	Cell[CellGroupData[{
2821
2822	Cell[TextData[StyleBox["Von Neumann's rejection method : improved ",
2823	FontSize->16]], "Section"],
2824
2825	Cell[CellGroupData[{
2826
2827	Cell[BoxData[{
2828	\(\(<< Graphics`Colors`;\)\), "\[IndentingNewLine]",
2829	\(ob1 =
2830	Plot[{Cos[Pi/2\ x], 1 - x^2}, {x, 0, 1}, \
2831	PlotStyle \[Rule] {Blue, Red}]\), "\[IndentingNewLine]",
2832	\(\)}], "Input"],
2833
2834	Cell[GraphicsData["PostScript", "\<\
2835	%!
2836	%%Creator: Mathematica
2837	%%AspectRatio: .61803
2838	MathPictureStart
2839	/Mabs {
2840	Mgmatrix idtransform
2841	Mtmatrix dtransform
2842	} bind def
2843	/Mabsadd { Mabs
2844	3 -1 roll add
2845	3 1 roll add
2846	exch } bind def
2847	%% Graphics
2848	%%IncludeResource: font Courier
2849	%%IncludeFont: Courier
2850	/Courier findfont 10 scalefont setfont
2851	% Scaling calculations
2852	0.0238095 0.952381 0.0147151 0.588604 [
2853	[.21429 .00222 -9 -9 ]
2854	[.21429 .00222 9 0 ]
2855	[.40476 .00222 -9 -9 ]
2856	[.40476 .00222 9 0 ]
2857	[.59524 .00222 -9 -9 ]
2858	[.59524 .00222 9 0 ]
2859	[.78571 .00222 -9 -9 ]
2860	[.78571 .00222 9 0 ]
2861	[.97619 .00222 -3 -9 ]
2862	[.97619 .00222 3 0 ]
2863	[.01131 .13244 -18 -4.5 ]
2864	[.01131 .13244 0 4.5 ]
2865	[.01131 .25016 -18 -4.5 ]
2866	[.01131 .25016 0 4.5 ]
2867	[.01131 .36788 -18 -4.5 ]
2868	[.01131 .36788 0 4.5 ]
2869	[.01131 .4856 -18 -4.5 ]
2870	[.01131 .4856 0 4.5 ]
2871	[.01131 .60332 -6 -4.5 ]
2872	[.01131 .60332 0 4.5 ]
2873	[ 0 0 0 0 ]
2874	[ 1 .61803 0 0 ]
2875	] MathScale
2876	% Start of Graphics
2877	1 setlinecap
2878	1 setlinejoin
2879	newpath
2880	0 g
2881	.25 Mabswid
2882	[ ] 0 setdash
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2884	.21429 .02097 L
2885	s
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2887	.40476 .01472 m
2888	.40476 .02097 L
2889	s
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2891	.59524 .01472 m
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2893	s
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2895	.78571 .01472 m
2896	.78571 .02097 L
2897	s
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2899	.97619 .01472 m
2900	.97619 .02097 L
2901	s
2902	[(1)] .97619 .00222 0 1 Mshowa
2903	.125 Mabswid
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2906	s
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2948	s
2949	.25 Mabswid
2950	0 .01472 m
2951	1 .01472 L
2952	s
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2955	s
2956	[(0.2)] .01131 .13244 1 0 Mshowa
2957	.02381 .25016 m
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2959	s
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2961	.02381 .36788 m
2962	.03006 .36788 L
2963	s
2964	[(0.6)] .01131 .36788 1 0 Mshowa
2965	.02381 .4856 m
2966	.03006 .4856 L
2967	s
2968	[(0.8)] .01131 .4856 1 0 Mshowa
2969	.02381 .60332 m
2970	.03006 .60332 L
2971	s
2972	[(1)] .01131 .60332 1 0 Mshowa
2973	.125 Mabswid
2974	.02381 .04415 m
2975	.02756 .04415 L
2976	s
2977	.02381 .07358 m
2978	.02756 .07358 L
2979	s
2980	.02381 .10301 m
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2982	s
2983	.02381 .16187 m
2984	.02756 .16187 L
2985	s
2986	.02381 .1913 m
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2988	s
2989	.02381 .22073 m
2990	.02756 .22073 L
2991	s
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2994	s
2995	.02381 .30902 m
2996	.02756 .30902 L
2997	s
2998	.02381 .33845 m
2999	.02756 .33845 L
3000	s
3001	.02381 .39731 m
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3003	s
3004	.02381 .42674 m
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3006	s
3007	.02381 .45617 m
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3009	s
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3015	s
3016	.02381 .57389 m
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3018	s
3019	.25 Mabswid
3020	.02381 0 m
3021	.02381 .61803 L
3022	s
3023	0 0 m
3024	1 0 L
3025	1 .61803 L
3026	0 .61803 L
3027	closepath
3028	clip
3029	newpath
3030	0 0 1 r
3031	.5 Mabswid
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3071	s
3072	1 0 0 r
3073	.02381 .60332 m
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3076	.02729 .60331 L
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3100	.54206 .42903 L
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3110	.93905 .05973 L
3111	.97619 .01472 L
3112	s
3113	% End of Graphics
3114	MathPictureEnd
3115	\
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3118	ImageMargins->{{43, 0}, {0, 0}},
3119	ImageRegion->{{0, 1}, {0, 1}},
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3514	oooocOooool001Wooooo00?o0000ooooooooool0;?ooool2o`00o`_oo`00oooooooEooooo`003Ooo
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3518	oooo00?o0000ooooooooool02?ooool9ool00?ooooooooooool9ooooo`003_ooool00ol0003ooooo
3519	ooooo`08ooooo`03o`000?oooooooooo0?ooooooooooool?ooooo`003Oooool2o`0000[ooooo00?o
3520	0000ooooooooool0ooooooooooooo`oooooo000>ooooo`03o`000?oooooooooo00Sooooo00?o0000
3521	ooooooooool0ooooooooooooo`oooooo000Iooooo`03o`000?oooooooooo0?ooooooooooool?oooo
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3523	ooooooooooooo`oooooo000Iooooo`03o`000?oooooooooo0?ooooooooooool?ooooo`00\
3524	\>"],
3525	ImageRangeCache->{{{0, 552}, {340.625, 0}} -> {-0.0497591, -0.0496274, \
3526	0.00194981, 0.00315486}}],
3527
3528	Cell[BoxData[
3529	FormBox[
3530	TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\),
3531	False,
3532	Editable->False], TraditionalForm]], "Output"]
3533	}, Open ]],
3534
3535	Cell[BoxData[
3536	\(\(MyInte4[f_, xlims_, ylims_,
3537	nev_] := \[IndentingNewLine]Module[{sig, eff, noin, count, xi, sum,
3538	max}, \[IndentingNewLine] (*\
3539	do\ the\ unweighting\ *) \[IndentingNewLine]sum =
3540	0; \[IndentingNewLine]count = 0; \[IndentingNewLine]noin =
3541	0; \[IndentingNewLine]deltarho = 2/3; \[IndentingNewLine]deltax =
3542	Abs[xlims[\([2]\)] - xlims[\([1]\)]]; \[IndentingNewLine]deltay =
3543	Abs[ylims[\([2]\)] - ylims[\([1]\)]]; \[IndentingNewLine]While[
3544	count < nev, \[IndentingNewLine]rhoi =
3545	deltarho*Random[]; \[IndentingNewLine]\
3546	xi = Chop[Gi[rhoi]]; \ \[IndentingNewLine]yi =
3547	Random[]; \[IndentingNewLine]If[
3548	f[xi] > yi*g[xi], \[IndentingNewLine]sum =
3549	sum + 1, \[IndentingNewLine]noin =
3550	noin + 1]; \[IndentingNewLine]\ \ count =
3551	count + 1]; \[IndentingNewLine]sum =
3552	sum/nev*Integrate[g[x], {x, 0, 1}]; \[IndentingNewLine]sig =
3553	sum; \[IndentingNewLine]eff = \
3554	count/\((count + noin)\) //
3555	N; \[IndentingNewLine]Return[{sum // N, sig // N,
3556	eff}]];\)\)], "Input"],
3557
3558	Cell[CellGroupData[{
3559
3560	Cell[BoxData[{
3561	\(\(f[x_] := Cos[Pi/2\ x];\)\), "\[IndentingNewLine]",
3562	\(\(xlow = 0;\)\), "\[IndentingNewLine]",
3563	\(\(xup = 1;\)\), "\[IndentingNewLine]",
3564	\(\(Print["\<Analytic Result : \>",
3565	Integrate[f[x], {x, xlow, xup}] // N];\)\), "\[IndentingNewLine]",
3566	\(\(events = 10000;\)\), "\[IndentingNewLine]",
3567	\(\({res, sig, eff} =
3568	MyInte4[f, {xlow, xup}, {0, 1}, events];\)\), "\[IndentingNewLine]",
3569	\(\(Print["\<MC result :\>", res, \ "\< +- \>", sig, "\</Sqrt[\>",
3570	events, "\<], eff = \>", eff];\)\)}], "Input"],
3571
3572	Cell[BoxData[
3573	FormBox[
3574	InterpretationBox[\("Analytic Result : \
3575	"\[InvisibleSpace]0.6366197723675814`\),
3576	SequenceForm[ "Analytic Result : ", 0.63661977236758138],
3577	Editable->False], TraditionalForm]], "Print"],
3578
3579	Cell[BoxData[
3580	FormBox[
3581	InterpretationBox[\("MC result :"\[InvisibleSpace]0.636`\
3582	\[InvisibleSpace]" +- "\[InvisibleSpace]0.636`\[InvisibleSpace]"/Sqrt["\
3583	\[InvisibleSpace]10000\[InvisibleSpace]"], eff = \
3584	"\[InvisibleSpace]0.9560229445506692`\),
3585	SequenceForm[
3586	"MC result :", 0.63600000000000001, " +- ", 0.63600000000000001,
3587	"/Sqrt[", 10000, "], eff = ", 0.95602294455066916],
3588	Editable->False], TraditionalForm]], "Print"]
3589	}, Open ]]
3590	}, Closed]],
3591
3592	Cell[CellGroupData[{
3593
3594	Cell["Dimensionality of the phase space of 1-> n particles", "Section"],
3595
3596	Cell[BoxData[
3597	\(\(Ndim = Nparticles*3 - 4;\)\)], "Input"]
3598	}, Closed]],
3599
3600	Cell[CellGroupData[{
3601
3602	Cell["Useful functions", "Section"],
3603
3604	Cell[CellGroupData[{
3605
3606	Cell["Scalar product between four momenta {E,px,py,pz}.", "Subsubsection"],
3607
3608	Cell[BoxData[
3609	\(\(SP[p1_, p2_] :=
3610	Module[{res}, \[IndentingNewLine]res =
3611	p1[\([1]\)]p2[\([1]\)] - p1[\([2]\)]p2[\([2]\)] -
3612	p1[\([3]\)]*p2[\([3]\)] -
3613	p1[\([4]\)]*p2[\([4]\)]; \[IndentingNewLine]Return[
3614	res]];\)\)], "Input"]
3615	}, Closed]],
3616
3617	Cell[CellGroupData[{
3618
3619	Cell["\<\
3620	Cos(theta), where theta is the angle between two spatial components \
3621	of two four momenta:\
3622	\>", "Subsubsection"],
3623
3624	Cell[BoxData[
3625	\(\(CosTheta[p1_, p2_] :=
3626	Module[{res, l1, l2}, \[IndentingNewLine]res =
3627	p1[\([2]\)]p2[\([2]\)] + p1[\([3]\)]p2[\([3]\)] +
3628	p1[\([4]\)]*p2[\([4]\)]; \[IndentingNewLine]l1 =
3629	Sqrt[p1[\([2]\)]p1[\([2]\)] + p1[\([3]\)]p1[\([3]\)] +
3630	p1[\([4]\)]*p1[\([4]\)]]; \[IndentingNewLine]l2 =
3631	Sqrt[p2[\([2]\)]p2[\([2]\)] + p2[\([3]\)]p2[\([3]\)] +
3632	p2[\([4]\)]*p2[\([4]\)]]; \[IndentingNewLine]If[
3633	l1*l2 =!= 0, \[IndentingNewLine]res = \(res/l1\)/l2,
3634	res = 0]; \[IndentingNewLine]Return[res]];\)\)], "Input"]
3635	}, Closed]],
3636
3637	Cell[CellGroupData[{
3638
3639	Cell["\<\
3640	Boost: it boosts the four momentum P from the frame S to the frame \
3641	S' of Q. The frame S is where Q is of the form {E,0,0,0}. It returns the four \
3642	momentum P'.\
3643	\>", "Subsubsection"],
3644
3645	Cell[BoxData[
3646	\(\(BOOST[P_, Q_] :=
3647	Module[{Q2, QP, M, TEMP, POUT}, \[IndentingNewLine]POUT =
3648	Table[0, {i, 1, 4}]; \[IndentingNewLine]Q2 =
3649	Q[\([2]\)]^2 + Q[\([3]\)]^2 +
3650	Q[\([4]\)]^2; \[IndentingNewLine]QP =
3651	Q[\([2]\)]P[\([2]\)] + Q[\([3]\)]P[\([3]\)] +
3652	Q[\([4]\)]*P[\([4]\)]; \[IndentingNewLine]M =
3653	Sqrt[Q[\([1]\)]^2 -
3654	Q2]; \[IndentingNewLine]POUT[\([1]\)] = \((Q[\([1]\)]*
3655	P[\([1]\)] + QP)\)/M; \[IndentingNewLine]If[
3656	Q2 =!= 0, \[IndentingNewLine]\(TEMP = \((P[\([1]\)] + \
3657	\((Q[\([1]\)] - M)\)\ QP/Q2)\)/M;\), \[IndentingNewLine]\(TEMP =
3658	0;\)]; \[IndentingNewLine]POUT[\([2]\)] =
3659	TEMP\ Q[\([2]\)] + P[\([2]\)]; \[IndentingNewLine]POUT[\([3]\)] =
3660	TEMP\ Q[\([3]\)] + P[\([3]\)]; \[IndentingNewLine]POUT[\([4]\)] =
3661	TEMP\ Q[\([4]\)] + P[\([4]\)]; \[IndentingNewLine]Return[
3662	POUT]];\)\)], "Input"]
3663	}, Closed]],
3664
3665	Cell[CellGroupData[{
3666
3667	Cell["\<\
3668	MO2CX: It calculates the two four-momenta P1 and P2, of masses m1 \
3669	and m2, with total energy esum, in a given direction (theta,phi), in their \
3670	rest frame. \
3671	\>", "Subsubsection"],
3672
3673	Cell[BoxData[
3674	\(\(MOM2CX[esum_, m1_, m2_, costh1_, phi1_] :=
3675	Module[{E1, E2, P, SIN1, P1, P2}, \[IndentingNewLine]E1 =
3676	1/2\ esum\ \((1 + \ \((m1^2 - m2^2)\)/
3677	esum^2)\); \[IndentingNewLine]E2 =
3678	1/2\ esum\ \((1 - \ \((m1^2 - m2^2)\)/
3679	esum^2)\); \[IndentingNewLine]P\ \ =
3680	1/2\ esum\ Sqrt[\((1 -
3681	2\ \((m1^2 + m2^2)\)/esum^2 + \((m1^2 - m2^2)\)^2/
3682	esum^4)\)]; \[IndentingNewLine]SIN1 =
3683	Sqrt[1 - costh1^2]; \[IndentingNewLine]P1 = {E1,
3684	P\ SIN1\ Cos[phi1], \ P\ SIN1\ Sin[phi1], \
3685	P\ costh1}; \[IndentingNewLine]P2 = {E2, \(-P\)\ SIN1\ Cos[
3686	phi1], \ \(-P\)\ SIN1\ Sin[
3687	phi1], \ \(-P\)\ costh1}; \[IndentingNewLine]Return[{P1 //
3688	N, P2 // N}]];\)\)], "Input"]
3689	}, Closed]],
3690
3691	Cell[CellGroupData[{
3692
3693	Cell["\<\
3694	Transpole: given x distributed uniform in [0,1] it returns a \
3695	Breit-Wigner distribution, centered in \"pole\" with \"width\".
3696	It also returns the jacobian of the transformation.\
3697	\>", "Subsubsection"],
3698
3699	Cell[CellGroupData[{
3700
3701	Cell[BoxData[
3702	\(\(Transpole[pole_, width_, x_] :=
3703	Module[{zmin, zmax, z, y, jac}, \[IndentingNewLine]zmin =
3704	ArcTan[\((\(-pole\))\)/width]/width; \[IndentingNewLine]zmax =
3705	ArcTan[\((1 - pole)\)/width]/width; \[IndentingNewLine]z =
3706	zmin + \((zmax - zmin)\)*x; \[IndentingNewLine]y =
3707	pole + width*
3708	Tan[width*z]; \[IndentingNewLine]jac = \((width/
3709	Cos[width*z])\)^2 \((zmax -
3710	zmin)\); \[IndentingNewLine]Return[{y, jac}]];\)\)], "Input"],
3711
3712	Cell[BoxData[
3713	FormBox[
3714	RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
3715	error: new symbol name \\\"\\!\\(TraditionalForm\\`Transpole\\)\\\" is \
3716	similar to existing symbol \
3717	\\\"\\!\\(TraditionalForm\\`Transpose\\)\\\".\"\>"}],
3718	TraditionalForm]], "Message"]
3719	}, Open ]]
3720	}, Closed]],
3721
3722	Cell[CellGroupData[{
3723
3724	Cell["\<\
3725	Phase space maeasure for two particle decay, with masses ma and \
3726	mb\
3727	\>", "Subsubsection"],
3728
3729	Cell[BoxData[
3730	\(\(lambda[s_, ma2_, mb2_] :=
3731	s^2 + ma2^2 + mb2^2 - 2 s\ ma2 - 2\ ma2\ mb2 -
3732	2\ s\ mb2;\)\)], "Input"]
3733	}, Closed]]
3734	}, Closed]],
3735
3736	Cell[CellGroupData[{
3737
3738	Cell["Install Vegas", "Section"],
3739
3740	Cell["\<\
3741	I use the Vegas package of CUBA library of T. Hahn \
3742	(hep-ph/0404043). If you do not want to use it, use NIntegrate instead of \
3743	Vegas\
3744	\>", "Text"],
3745
3746	Cell[CellGroupData[{
3747
3748	Cell[BoxData[
3749	\(Install["\</Users/fabiomaltoni/Physics/Codes/CUBA/VegasX\>"]\)], "Input"],
3750
3751	Cell[BoxData[
3752	\(TraditionalForm\`LinkObject[
3753	"/Users/fabiomaltoni/Physics/Codes/CUBA/VegasX", 2, 2]\)], "Output"]
3754	}, Open ]]
3755	}, Closed]],
3756
3757	Cell[CellGroupData[{
3758
3759	Cell["Top decay ", "Section"],
3760
3761	Cell[CellGroupData[{
3762
3763	Cell["\<\
3764	Generates the phase space of 3 particles, by iteration of the 1->2 \
3765	decay:
3766	t-> w b, w-> lv.
3767
3768	INPUT:
3769	x: an array of length 5 which contains the random numbers in [0,1].
3770	p1: four momentum of the decaying particle (top).
3771	params: an array which contains the masses of the final state particles and \
3772	the width of the W {mw,gw,mb,ml,mv}.
3773	BW: is a logical value. If true the Breit-Wigner is \"flattened out\" through \
3774	an analytic transformation realized
3775	by Transpole. If false, the invariant mass of the W is generated uniformly.
3776
3777	OUTPUT:
3778	The 3 momenta of the final particles and the phase space weight\
3779	\>", \
3780	"Subsubsection"],
3781
3782	Cell[BoxData[
3783	\(\(Phi3BW[x_, p1_, params_, BW_] :=
3784	Module[{s, rs, s2, rs2, jac, jacpole, pswgt, costh, phi, costh1,
3785	phi1, mb, ml, mv, mw, gw, pw, pb, pl, pv, xa2,
3786	xb2}, \[IndentingNewLine]\[IndentingNewLine] (*initialization\
3787	*) \[IndentingNewLine]mw = params[\([1]\)]; \[IndentingNewLine]gw =
3788	params[\([2]\)]; \[IndentingNewLine]mb =
3789	params[\([3]\)]; \[IndentingNewLine]ml =
3790	params[\([4]\)]; \[IndentingNewLine]mv =
3791	params[\([5]\)]; \[IndentingNewLine]pw =
3792	Table[0, {i, 1, 4}]; \[IndentingNewLine]pb =
3793	pw; \[IndentingNewLine]pl = pw; \[IndentingNewLine]pv =
3794	pw; \[IndentingNewLine]\[IndentingNewLine] (*\
3795	start\ *) \[IndentingNewLine]s =
3796	SP[p1, p1]; \[IndentingNewLine]rs =
3797	Sqrt[s]; \[IndentingNewLine]If[
3798	BW, \[IndentingNewLine] (*\
3799	make\ the\ analytic\ transformation\ *) \[IndentingNewLine]{s2,
3800	jacpole} =
3801	Transpole[mw^2/s, mw\ gw/s,
3802	x[\([3]\)]]; \[IndentingNewLine]s2 =
3803	s2*s; \[IndentingNewLine]rs2 =
3804	Sqrt[s2]; \[IndentingNewLine]jac =
3805	jacpoles;\[IndentingNewLine], \[IndentingNewLine]\ (\
3806	generate\ it\ flat\ ) \[IndentingNewLine]s2 = \((s - mb^2)\)
3807	x[\([3]\)] + mb^2; \[IndentingNewLine]rs2 =
3808	Sqrt[s2]; \[IndentingNewLine]jac = \((s -
3809	mb^2)\);\[IndentingNewLine]]; \[IndentingNewLine]\
3810	\[IndentingNewLine] (*\
3811	check\ kinematic\ boundaries\ *) \[IndentingNewLine]If[
3812	rs2 > \((rs - mb)\)\ \|\| \
3813	rs2 < \((ml + mv)\), \[IndentingNewLine]\(Return[
3814	0];\)]; \[IndentingNewLine]\[IndentingNewLine] (*\
3815	generate\ the\ angles\ *) \[IndentingNewLine]costh = \(-1\) +
3816	2\ x[\([1]\)]; \[IndentingNewLine]phi = \
3817	2\ Pi\ x[\([2]\)]; \[IndentingNewLine]jac =
3818	jac\ 4\ Pi; \[IndentingNewLine]costh1 = \(-1\) +
3819	2\ x[\([4]\)]; \[IndentingNewLine]phi1 = \
3820	2Pix[\([5]\)]; \[IndentingNewLine]jac =
3821	jac4\ Pi; \[IndentingNewLine]\[IndentingNewLine] (*\
3822	generate\ the\ four - momenta\ *) \[IndentingNewLine]{pw, pb} =
3823	MOM2CX[rs, rs2, mb, costh, phi]; \[IndentingNewLine]{pl, pv} =
3824	MOM2CX[rs2, ml, mv, costh1, phi1]; \[IndentingNewLine]pb =
3825	BOOST[pb, p1]; \[IndentingNewLine]pw =
3826	BOOST[pw, p1]; \[IndentingNewLine]pl =
3827	BOOST[pl, pw]; \[IndentingNewLine]pv =
3828	BOOST[pv, pw]; \[IndentingNewLine]\[IndentingNewLine] (*\
3829	phase\ space\ factors\ *) \[IndentingNewLine]jac =
3830	jac/\((2\ Pi)\)^5; \[IndentingNewLine]pswgt =
3831	1\ \ \ \ (\(/\)\((2rs)\)*) ; \[IndentingNewLine]xa2 =
3832	s2/s; \[IndentingNewLine]xb2 = mb^2/s; \[IndentingNewLine]pswgt =
3833	pswgt/2Pi
3834	Sqrt[lambda[1, xa2, xb2]]/\((4\ Pi)\); \[IndentingNewLine]xa2 =
3835	ml^2/s; \[IndentingNewLine]xb2 =
3836	mv^2/s; \[IndentingNewLine]pswgt =
3837	pswgt/2*Pi\ Sqrt[
3838	lambda[1, xa2,
3839	xb2]]/\((4\ Pi)\); \n\ \ \ \ \[IndentingNewLine] (*final\ \
3840	jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
3841	pswgt = pswgt*
3842	jac; \[IndentingNewLine]\ \[IndentingNewLine]\ \ Return[{pb,
3843	pl, pv, pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], \
3844	"Input"]
3845	}, Closed]],
3846
3847	Cell[CellGroupData[{
3848
3849	Cell[TextData[{
3850	"Generates the phase space of 3 particles, by iteration of the 1->2 decay:\n\
3851	t-> w b, w-> lv, assuming the w to be exactly on shell. This is the so-called \
3852	narrow-width approximation:\n\n",
3853	Cell[BoxData[
3854	\(1\/\((\((q\^2 - \ mW\^2)\)\^2 - \ \(\[CapitalGamma]\^2\) mW\^2)\)\ \
3855	\[Rule] \ \[CapitalPi]\/\(mW\ \[CapitalGamma]\)\ \[Delta] \((q\^2 -
3856	mW\^2)\)\)]],
3857	"\n\nUsing this approximation the number of dimensions of the phase space \
3858	passes from 5 to 4\n\nINPUT:\nx: an array of length 4 which contains the \
3859	random numbers in [0,1].\np1: four momentum of the decaying particle (top).\n\
3860	params: an array which contains the masses of the final state particles and \
3861	the width of the W {mw,mb,ml,mv}.\n\nOUTPUT:\nThe 3 momenta of the final \
3862	particles and the phase space weight."
3863	}], "Subsubsection"],
3864
3865	Cell[CellGroupData[{
3866
3867	Cell[BoxData[
3868	\(\(Phi3NW[x_, p1_, params_] :=
3869	Module[{s, rs, s2, rs2, jac, jacpole, pswgt, costh, phi, costh1,
3870	phi1, mb, ml, mv, mw, gw, pw, pb, pl, pv, xa2,
3871	xb2}, \[IndentingNewLine]\[IndentingNewLine] (*initialization\
3872	*) \[IndentingNewLine]\[IndentingNewLine]mw =
3873	params[\([1]\)]; \[IndentingNewLine]mb =
3874	params[\([2]\)]; \[IndentingNewLine]ml =
3875	params[\([3]\)]; \[IndentingNewLine]mv =
3876	params[\([4]\)]; \[IndentingNewLine]pw =
3877	Table[0, {i, 1, 4}]; \[IndentingNewLine]pb =
3878	pw; \[IndentingNewLine]pl = pw; \[IndentingNewLine]pv =
3879	pw; \[IndentingNewLine]\[IndentingNewLine] (*\
3880	start\ *) \[IndentingNewLine]\[IndentingNewLine]s =
3881	SP[p1, p1]; \[IndentingNewLine]rs =
3882	Sqrt[s]; \[IndentingNewLine]\[IndentingNewLine] (*\
3883	generate\ the\ angles\ and\ momenta\ for\ the\ t\ \[Rule] \
3884	b\ W\ decay\ *) \[IndentingNewLine]\[IndentingNewLine]costh = \
3885	\(-1\) + 2\ x[\([1]\)]; \[IndentingNewLine]phi = \
3886	2\ Pi\ x[\([2]\)]; \[IndentingNewLine]jac =
3887	4\ *Pi; \[IndentingNewLine]{pw, pb} =
3888	MOM2CX[rs, mw, mb, costh, phi]; \[IndentingNewLine]pb =
3889	BOOST[pb, p1]; \[IndentingNewLine]pw =
3890	BOOST[pw, p1]; \[IndentingNewLine]\[IndentingNewLine] (*\
3891	generate\ the\ angles\ and\ momenta\ for\ the\ W\ \[Rule] \
3892	l\ ve\ decay\ *) \[IndentingNewLine]\[IndentingNewLine]costh1 = \
3893	\(-1\) + 2\ x[\([3]\)]; \[IndentingNewLine]phi1 = \
3894	2Pix[\([4]\)]; \[IndentingNewLine]jac =
3895	jac4\ Pi; \[IndentingNewLine]{pl, pv} =
3896	MOM2CX[mw, ml, mv, costh1, phi1]; \[IndentingNewLine]pl =
3897	BOOST[pl, pw]; \[IndentingNewLine]pv =
3898	BOOST[pv, pw]; \[IndentingNewLine]\[IndentingNewLine] (*\
3899	phase\ space\ factors\ \
3900	*) \[IndentingNewLine]\[IndentingNewLine]jac =
3901	jac/\((2\ Pi)\)^5; \[IndentingNewLine]pswgt =
3902	1\ ; \[IndentingNewLine]xa2 = mw^2/s; \[IndentingNewLine]xb2 =
3903	mb^2/s; \[IndentingNewLine]pswgt =
3904	pswgt/2Pi
3905	Sqrt[lambda[1, xa2, xb2]]/\((4\ Pi)\); \[IndentingNewLine]xa2 =
3906	ml^2/s; \[IndentingNewLine]xb2 =
3907	mv^2/s; \[IndentingNewLine]pswgt =
3908	pswgt/2*Pi\ Sqrt[
3909	lambda[1, xa2,
3910	xb2]]/\((4\ Pi)\); \n\ \ \ \ \[IndentingNewLine] (*final\ \
3911	jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
3912	pswgt = pswgt*jac; \[IndentingNewLine]\ \[IndentingNewLine]\
3913	Return[{pb, pl, pv,
3914	pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"],
3915
3916	Cell[BoxData[
3917	FormBox[
3918	RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
3919	error: new symbol name \\\"\\!\\(TraditionalForm\\`Phi3NW\\)\\\" is similar \
3920	to existing symbol \\\"\\!\\(TraditionalForm\\`Phi3BW\\)\\\".\"\>"}],
3921	TraditionalForm]], "Message"]
3922	}, Open ]]
3923	}, Closed]],
3924
3925	Cell[CellGroupData[{
3926
3927	Cell[TextData[{
3928	"Function which gives the top width when integrated in the volume ",
3929	Cell[BoxData[
3930	\(TraditionalForm\`\([0, 1]\)\^5\)]],
3931	".The ",
3932	Cell[BoxData[
3933	\(TraditionalForm\`x\_i\)]],
3934	"are the inputs in [0,1]. \nmethod=\"Plain\", it does not apply the \
3935	\"flattening\" of the BW. method=\"BW\" applies the analytic transformation.\n\
3936	method=\"NW\" uses the narrow width approximation for the phase space (in \
3937	this case the value of x5 is ignored). \nThe value of the matrix element \
3938	squared can be found on the ESW."
3939	}], "Subsubsection"],
3940
3941	Cell[BoxData[
3942	\(\(\(ff[x1_, x2_, x3_, x4_, x5_, method_] :=
3943	Module[{xx,
3944	aa}, \[IndentingNewLine]\[IndentingNewLine]Which[\
3945	\[IndentingNewLine]\[IndentingNewLine] (*\
3946	does\ not\ make\ the\ analytic\ transformation\ to\ flatten\ \
3947	the\ BW\ *) \[IndentingNewLine]\[IndentingNewLine]\
3948	method\ === "\<Plain\>", \[IndentingNewLine]xx = {x1, x2, x3,
3949	x4, x5}; \n\ \ \ \ \ \ \ \ \ \ aa =
3950	Phi3BW[xx, ptop, {MW, GW, MB, 0, 0},
3951	False]; \[IndentingNewLine]If[
3952	aa =!= 0, \[IndentingNewLine]{pb, pl, pv, pswgt} =
3953	aa; \n\ \ \ \ \ \ EmmeSq = \(1/2\)/mt\ \ *\
3954	1/2\ \((8\ Pi\ aw\ )\)^2\ \ SP[ptop, pl]\ *
3955	SP[pb, pv]/\((\((SP[pl + pv, pl + pv] - MW^2)\)^2 +
3956	GW^2\ MW^2)\); \n\ \ \ \ \ \ \ gamma = EmmeSq*pswgt;,
3957	gamma = 0];\[IndentingNewLine], \[IndentingNewLine]\
3958	\[IndentingNewLine] (*\
3959	make\ the\ analytic\ transformation\ to\ flatten\ the\ BW\ \
3960	*) \[IndentingNewLine]\[IndentingNewLine]method === "\<BW\>", \
3961	\[IndentingNewLine]xx = {x1, x2, x3, x4, x5}; \n\ \ \ \ \ \ \ \ \ \ aa =
3962	Phi3BW[xx, ptop, {MW, GW, MB, 0, 0},
3963	True]; \[IndentingNewLine]If[
3964	aa =!= 0, \[IndentingNewLine]{pb, pl, pv, pswgt} =
3965	aa; \n\ \ \ \ \ \ EmmeSq = \(1/2\)/mt\ *\
3966	1/2\ \((8\ Pi\ aw\ )\)^2\ \ SP[ptop, pl]\ *
3967	SP[pb, pv]/\((\((SP[pl + pv, pl + pv] - MW^2)\)^2 +
3968	GW^2\ MW^2)\); \n\ \ \ \ \ \ \ gamma = EmmeSq*pswgt;,
3969	gamma = 0];\[IndentingNewLine], \[IndentingNewLine]\
3970	\[IndentingNewLine] (*\
3971	narrow\ width\ approximation\ 1/\((\((q\^2 - \ mW\^2)\)\^2 - \ \
3972	\(\[CapitalGamma]\^2\)
3973	mW\^2)\)\ \[Rule] \ \[CapitalPi]\/\(mW\ \
3974	\[CapitalGamma]\)\ *) \[IndentingNewLine]\[IndentingNewLine]method\ === \ \
3975	"\<NW\>", \[IndentingNewLine]xx = {x1, x2, x3,
3976	x4}; \n\ \ \ \ \ \ \ \ \ \ aa =
3977	Phi3NW[xx, ptop, {MW, MB, 0, 0}]; \[IndentingNewLine]\ \ If[
3978	aa =!= 0, \[IndentingNewLine]{pb, pl, pv, pswgt} =
3979	aa; \n\ \ \ \ \ \ \ \ \ \ EmmeSq = \(1/2\)/mt\ *1/
3980	2\ \((8\ Pi\ aw\ )\)^2\ \ SP[ptop, pl]\ *
3981	SP[pb, pv] \(Pi/MW\)/GW; \n\ \ \ \ \ \ \ \ \ gamma =
3982	EmmeSq*pswgt;,
3983	gamma = 0];\[IndentingNewLine]]; \[IndentingNewLine]Return[
3984	gamma];\[IndentingNewLine]];\)\(\[IndentingNewLine]\)
3985	\)\)], "Input"]
3986	}, Open ]],
3987
3988	Cell[CellGroupData[{
3989
3990	Cell["Parameters entering the top decay", "Subsection"],
3991
3992	Cell[BoxData[{
3993	\(\(MB = 0;\)\), "\[IndentingNewLine]",
3994	\(\(MW = 80.419;\)\), "\[IndentingNewLine]",
3995	\(\(GW = 2.0479;\)\), "\[IndentingNewLine]",
3996	\(\(MT =
3997	174.3;\)\[IndentingNewLine] (random\ momentum\ for\ the\ top) \), "\
3998	\[IndentingNewLine]",
3999	\(\(ptop = {174.700891, \ 4.98437434, \ 8.1448859\ ,
4000	6.98055029};\)\ \), "\[IndentingNewLine]",
4001	\(\(gw = 0.6532;\)\), "\[IndentingNewLine]",
4002	\(\(aw = gw^2/\((4\ Pi)\);\)\)}], "Input"]
4003	}, Open ]],
4004
4005	Cell[CellGroupData[{
4006
4007	Cell["Calculate the top width", "Subsection"],
4008
4009	Cell["\<\
4010	If you do not have installed the Vegas package of CUBA library of \
4011	T. Hahn (hep-ph/0404043), use NIntegrate instead\
4012	\>", "Text"],
4013
4014	Cell[CellGroupData[{
4015
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4020	NStart \[Rule] 5000, \ MaxPoints \[Rule] \ 5000]\)], "Input"],
4021
4022	Cell[BoxData[
4023	\(TraditionalForm\`"\nIteration 1: 5000 integrand evaluations so far\n\
4024	[1] 0.168057 +- 0.00086253 \tchisq -0 (0 df)"\)], "Print"],
4025
4026	Cell[BoxData[
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4028	RowBox[{\(Vegas::"accuracy"\), \(\(:\)\(\ \)\), "\<\"Desired accuracy \
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4037	ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}],
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4039	}, Open ]],
4040
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4045	NStart \[Rule] 5000, \ MaxPoints \[Rule] \ 5000]\)], "Input"],
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4047	Cell[BoxData[
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4058	PlotRange \[Rule] {{0, 1}, {0, 2}},
4059	PlotStyle \[Rule] {{Blue, Thickness[0.005]}, {Red,
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4822	\>"],
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4825
4826	Cell[BoxData[
4827	FormBox[
4828	TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\),
4829	False,
4830	Editable->False], TraditionalForm]], "Output"]
4831	}, Open ]]
4832	}, Open ]]
4833	}, Open ]],
4834
4835	Cell[CellGroupData[{
4836
4837	Cell["q1 q2~ \[Rule] tt~ production", "Section"],
4838
4839	Cell[CellGroupData[{
4840
4841	Cell["Parameters", "Subsubsection"],
4842
4843	Cell[BoxData[{
4844	\(\(Ebeam = 200;\)\), "\[IndentingNewLine]",
4845	\(\(p1\ = {Ebeam, 0, 0, Ebeam};\)\), "\[IndentingNewLine]",
4846	\(\(p2 = \ {Ebeam, 0, 0, \(-Ebeam\)};\)\), "\[IndentingNewLine]",
4847	\(\(psum = p1 + p2;\)\), "\[IndentingNewLine]",
4848	\(\(mt = 174.3;\)\), "\[IndentingNewLine]",
4849	\(\(as = 0.130;\)\), "\[IndentingNewLine]",
4850	\(\(gevpb = 389379.66*1000;\)\)}], "Input"]
4851	}, Closed]],
4852
4853	Cell[CellGroupData[{
4854
4855	Cell["\<\
4856	Matrix element squared summed and averaged for tt~ production, \
4857	divided by the initial flux. The inputs are the four momenta. channel=\"qq\", \
4858	returns the value for qq->tt~, channel=\"gg\" returns the value for gg>tt~.\
4859	\
4860	\>", "Subsubsection"],
4861
4862	Cell[BoxData[
4863	\(\(Msq[p1_, p2_, p3_, p4_, channel_] := \
4864	Module[{m2, s, tau1, tau2, ro}, \[IndentingNewLine]s =
4865	2 SP[p1, p2]; \[IndentingNewLine]tau1 =
4866	2\ SP[p1, p3]/s; \[IndentingNewLine]tau2 =
4867	2\ SP[p2, p3]/s; \[IndentingNewLine]ro =
4868	4 mt^2/s; \[IndentingNewLine]\[IndentingNewLine] (*\
4869	matrix\ elements\ for\ qq \[Rule] \ \(tt\ and\ gg\ \[Rule] \
4870	tt\), \ from\ ESW, \
4871	pag . \ 348\ *) \[IndentingNewLine]\[IndentingNewLine]Which[
4872	channel == "\<qq\>", \[IndentingNewLine]\(m2 =
4873	4\/9\ \((4\ Pi\ as)\)\^2\ \((tau1\^2 + tau2\^2 + \
4874	ro\/2)\);\), \[IndentingNewLine]channel == "\<gg\>", \
4875	\[IndentingNewLine]\(m2\ = \((4\ Pi\ as)\)\^2\ \ \((1\/\(6 tau1\ tau2\) -
4876	3\/8)\)\ \ \((tau1\^2 + tau2\^2 + \ ro -
4877	ro\^2\/\(4\ tau1\ tau2\))\);\)]; \[IndentingNewLine]\
4878	\[IndentingNewLine] (*\
4879	inclusion\ of\ the\ initial\ flux\ *) \[IndentingNewLine]\
4880	\[IndentingNewLine]m2 = \(m2/2\)/
4881	s; \[IndentingNewLine]\[IndentingNewLine]Return[
4882	m2];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"]
4883	}, Open ]],
4884
4885	Cell[CellGroupData[{
4886
4887	Cell["\<\
4888	Phase space for two particles.
4889
4890	INPUT:
4891	x: an array of length 2 which contains the random numbers in [0,1].
4892	ptot: four momentum of the total momentum.
4893	params: an array which contains the masses of the final state particles \
4894	{m1,m2}.
4895
4896	OUTPUT:
4897	The 3 momenta of the final particles and the phase space weight.\
4898	\>", \
4899	"Subsubsection"],
4900
4901	Cell[BoxData[
4902	\(\(Phi2[x_, ptot_, params_] :=
4903	Module[{s, rs, jac, pswgt, costh, phi, xa2, xb2, m1, m2, p1,
4904	p2}, \[IndentingNewLine]\[IndentingNewLine] (*initialization\
4905	*) \[IndentingNewLine]\[IndentingNewLine]m1 =
4906	params[\([1]\)]; \[IndentingNewLine]m2 =
4907	params[\([2]\)]; \[IndentingNewLine]p1 =
4908	Table[0, {i, 1, 4}]; \[IndentingNewLine]p2 =
4909	p1; \[IndentingNewLine]\[IndentingNewLine] (*\
4910	start\ *) \[IndentingNewLine]\[IndentingNewLine]s =
4911	SP[ptot, ptot]; \[IndentingNewLine]rs =
4912	Sqrt[s]; \[IndentingNewLine]\[IndentingNewLine] (*\
4913	generate\ the\ angles\ and\ momenta\ for\ ptot \[Rule] \
4914	1\ 2\ *) \[IndentingNewLine]\[IndentingNewLine]costh = \(-1\) +
4915	2\ x[\([1]\)]; \[IndentingNewLine]phi = \
4916	2\ Pi\ x[\([2]\)]; \[IndentingNewLine]jac =
4917	4\ *Pi; \[IndentingNewLine]{p1, p2} =
4918	MOM2CX[rs, m1, m2, costh, phi]; \[IndentingNewLine]p1 =
4919	BOOST[p1, ptot]; \[IndentingNewLine]p2 =
4920	BOOST[p2, ptot]; \[IndentingNewLine]\[IndentingNewLine] (*\
4921	phase\ space\ factors\ \
4922	*) \[IndentingNewLine]\[IndentingNewLine]jac =
4923	jac/\((2\ Pi)\)^2; \[IndentingNewLine]pswgt =
4924	1; \[IndentingNewLine]xa2 = m1^2/s; \[IndentingNewLine]xb2 =
4925	m2^2/s; \[IndentingNewLine]pswgt =
4926	pswgt/2Pi
4927	Sqrt[lambda[1, xa2,
4928	xb2]]/\((4\ Pi)\); \[IndentingNewLine]\[IndentingNewLine] \
4929	(final\ jacobian) \[IndentingNewLine]\[IndentingNewLine]\
4930	pswgt = pswgt*jac; \[IndentingNewLine]\ \[IndentingNewLine]\
4931	Return[{p1, p2,
4932	pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"]
4933	}, Closed]],
4934
4935	Cell[CellGroupData[{
4936
4937	Cell[TextData[{
4938	"Function which gives the cross section when integrated in the volume ",
4939	Cell[BoxData[
4940	\(TraditionalForm\`\([0, 1]\)\^2\)]],
4941	".The ",
4942	Cell[BoxData[
4943	\(TraditionalForm\`x\_i\)]],
4944	"are the inputs in [0,1]. \nchannel=\"gg\" or \"qq\"."
4945	}], "Subsubsection"],
4946
4947	Cell[BoxData[
4948	\(\(ff[x1_, x2_, channel_] :=
4949	Module[{xx,
4950	aa}, \[IndentingNewLine]xx = {x1, x2}; \n\ \ \ \ \ \ {p3, p4,
4951	pswgt} = Phi2[xx, psum, {mt, mt}]; \n\ \ \ \ \ \ EmmeSq =
4952	Msq[p1, p2, p3, p4, channel]; \n\ \ \ \ \ \ gamma =
4953	EmmeSqpswgtgevpb; \[IndentingNewLine]Return[
4954	gamma];\[IndentingNewLine]];\)\)], "Input"]
4955	}, Closed]],
4956
4957	Cell[CellGroupData[{
4958
4959	Cell["Calculate the cross section", "Subsection"],
4960
4961	Cell[CellGroupData[{
4962
4963	Cell[BoxData[
4964	\(Vegas[ff[x1, x2, "\<qq\>"], {x1, 0, 1}, {x2, 0, 1}\ ,
4965	Compiled \[Rule] False]\)], "Input"],
4966
4967	Cell[BoxData[
4968	\(TraditionalForm\`"\nIteration 1: 1000 integrand evaluations so far\n\
4969	[1] 25.9015 +- 0.0318548 \tchisq -0 (0 df)"\)], "Print"],
4970
4971	Cell[BoxData[
4972	\(TraditionalForm\`"\nIteration 2: 2500 integrand evaluations so far\n\
4973	[1] 25.9042 +- 0.00955214 \tchisq 0.00781108 (1 df)"\)], "Print"],
4974
4975	Cell[BoxData[
4976	FormBox[
4977	RowBox[{\(Vegas::"success"\), \(\(:\)\(\ \)\), "\<\"Needed \
4978	\\!\\(TraditionalForm\\`2500\\) function evaluations.\"\>"}],
4979	TraditionalForm]], "Message"],
4980
4981	Cell[BoxData[
4982	FormBox[
4983	RowBox[{"(", "\[NoBreak]", GridBox[{
4984	{"25.90415039401596`", "0.009552138320792365`",
4985	"0.07042558482847515`"}
4986	},
4987	ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}],
4988	TraditionalForm]], "Output"]
4989	}, Open ]]
4990	}, Closed]]
4991	}, Closed]],
4992
4993	Cell[CellGroupData[{
4994
4995	Cell["\<\
4996	q1 q2~ \[Rule] tt~ \[Rule](e+ ve b ) (mu- vm~ b~) (full decay \
4997	matrix element)\
4998	\>", "Section"],
4999
5000	Cell[CellGroupData[{
5001
5002	Cell["Parameters", "Subsubsection"],
5003
5004	Cell[CellGroupData[{
5005
5006	Cell[BoxData[{
5007	\(\(Ebeam = 220;\)\[IndentingNewLine] (*\
5008	these\ momenta\ define\ the\ zero\ momentum\ frame\ \((ZMF)\)\ *) \), "\
5009	\[IndentingNewLine]",
5010	\(\(pq\ = {Ebeam, 0, 0, Ebeam};\)\), "\[IndentingNewLine]",
5011	\(\(pqb = \ {Ebeam, 0, 0, \(-Ebeam\)};\)\), "\[IndentingNewLine]",
5012	\(\(psum = pq + pqb;\)\), "\[IndentingNewLine]",
5013	\(\(mt = 174.3;\)\), "\[IndentingNewLine]",
5014	\(\(mb = 0;\)\), "\[IndentingNewLine]",
5015	\(\(mw = 80.419;\)\), "\[IndentingNewLine]",
5016	\(\(Gammaw = 2.0479;\)\), "\[IndentingNewLine]",
5017	\(\(Gammat = 0.168*9;\)\), "\[IndentingNewLine]",
5018	\(\(gw = 0.6532;\)\), "\[IndentingNewLine]",
5019	\(\(aw = gw^2/\((4\ Pi)\);\)\), "\[IndentingNewLine]",
5020	\(\(as = 0.130;\)\), "\[IndentingNewLine]",
5021	\(\(gevpb = 389379.66*1000;\)\), "\[IndentingNewLine]",
5022	\(Sqrt[1 - 4 mt^2/SP[psum, psum]]\), "\[IndentingNewLine]",
5023	\(\)}], "Input"],
5024
5025	Cell[BoxData[
5026	\(TraditionalForm\`0.6101671292521702`\)], "Output"]
5027	}, Open ]],
5028
5029	Cell[BoxData[""], "Input"]
5030	}, Open ]],
5031
5032	Cell[CellGroupData[{
5033
5034	Cell["\<\
5035	Matrix element squared summed and averaged for tt~ production, \
5036	divided by the initial flux. Momenta must be given in the ZMF for the \
5037	calculation of cmq,ceq,cqt,sqt,cmt,cet. ceb and cmb are to be calculated in \
5038	the respective rest frames of the W+ and W-\
5039	\>", "Subsubsection"],
5040
5041	Cell[CellGroupData[{
5042
5043	Cell[BoxData[
5044	\(\(MsqTT[pq_, pqb_, pb_, pe_, pve_, pbb_, pm_, pvm_] := \
5045	Module[\[IndentingNewLine]{pt, ptb, beta, gamma, m2, pew, pmw, pbw,
5046	pbbw, s, ro, cebhat, ceb, cmb, cmt, cet, cqt, ceq, cmq, cem,
5047	sqt}, \[IndentingNewLine]\[IndentingNewLine]pt =
5048	pb + pe + pve; \[IndentingNewLine]ptb =
5049	pbb + pm + pvm; \[IndentingNewLine]pbw = {0, 0, 0,
5050	0}; \[IndentingNewLine]pbbw = {0, 0, 0,
5051	0}; \[IndentingNewLine]s =
5052	2 SP[pq, pqb]; \[IndentingNewLine]ro =
5053	4 mt^2/s; \[IndentingNewLine]beta =
5054	Sqrt[1 - ro]; \[IndentingNewLine]gamma =
5055	1/Sqrt[ro]; \[IndentingNewLine]\[IndentingNewLine] (*\
5056	first\ calculate\ the\ angles\ in\ the\ \
5057	ZMF*) \[IndentingNewLine]cqt = CosTheta[pq, pt]; \[IndentingNewLine]sqt =
5058	Sqrt[1 - cqt^2]; \[IndentingNewLine]\[IndentingNewLine]ceq =
5059	CosTheta[pe, pq]; \[IndentingNewLine]cet =
5060	CosTheta[pe, pt]; \[IndentingNewLine]\[IndentingNewLine]cmq =
5061	CosTheta[pm, pqb]; \[IndentingNewLine]cmt =
5062	CosTheta[pm, ptb]; \[IndentingNewLine]\[IndentingNewLine]cem =
5063	CosTheta[pe, pm]; \[IndentingNewLine]\[IndentingNewLine] (*\
5064	calculate\ the\ angles\ of\ the\ leptons\ in\ the\ W + \
5065	and\ W - \
5066	frames\ *) \[IndentingNewLine]pwp = \(-\((pe +
5067	pve)\)\); \[IndentingNewLine]pwm = \(-\((pm +
5068	pvm)\)\); \[IndentingNewLine]pwp[\([1]\)] = \(-pwp[\([1]\)]\
5069	\); \[IndentingNewLine]pwm[\([1]\)] = \(-pwm[\([1]\)]\); \[IndentingNewLine]\
5070	\[IndentingNewLine]\[IndentingNewLine]pbw =
5071	BOOST[pb, pwp]; \[IndentingNewLine]pew =
5072	BOOST[pe, pwp]; \[IndentingNewLine]\[IndentingNewLine]pbbw =
5073	BOOST[pbb, pwm]; \[IndentingNewLine]pmw = \
5074	BOOST[pm, pwm]; \[IndentingNewLine]\[IndentingNewLine]ceb =
5075	CosTheta[pbw, pew]; \[IndentingNewLine]cmb =
5076	CosTheta[pbbw, pmw]; \[IndentingNewLine]\[IndentingNewLine] (*\
5077	matrix\ elements\ for\ \(\(qq\)\(~\)\) \[Rule] \ tt, \
5078	hep - ph/
5079	9706304\ *) \[IndentingNewLine]\[IndentingNewLine]\
5080	\[IndentingNewLine]pre = \(\(gw^4\)\(\ \)\)\/\(4\ \((mt\ Gammat)\)\^2\); \
5081	\[IndentingNewLine]\[IndentingNewLine]ta =
5082	pre \((mt^2 -
5083	2\ SP[pe,
5084	pve])\) \(\(\ \)\(mt^2 \((1 - ceb\^2)\) + \ 2\ SP[pe, \
5085	pve] \((1 + ceb)\)\^2\)\)\/\(\((2\ SP[pe, pve] - mw\^2)\)\^2 + \ \((Gammaw\ \
5086	mw)\)\^2\); \[IndentingNewLine]tab =
5087	pre \((mt^2 -
5088	2\ SP[pm,
5089	pvm])\) \(\(\ \)\(mt^2 \((1 - cmb\^2)\) + \ 2\ SP[pm, \
5090	pvm] \((1 + cmb)\)\^2\)\)\/\(\((2\ SP[pm, pvm] - mw\^2)\)\^2 + \ \((Gammaw\ \
5091	mw)\)\^2\); \[IndentingNewLine]m2 = \ \(\(\(\((4\ Pi\ as)\)\^2\)\(\ \)\)\/9\)
5092	ta\ tab\((\((2 -
5093	beta\^2\ sqt\^2)\) - \(\((1 - ceq\ cmq)\) - beta\ \
5094	\((cmt\ + cet)\) + \ beta\ cqt\ \((ceq + cmq)\)\ + \ 1/2\ beta^2\ sqt^2\ \
5095	\((1 - cem)\)\)\/\(gamma^2\ \((1 - beta\ cet)\)\ \((1 - beta\ cmt)\)\))\); \
5096	\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine] (*\
5097	inclusion\ of\ the\ initial\ flux\ *) \[IndentingNewLine]\
5098	\[IndentingNewLine]m2 = \(m2/2\)/
5099	s; \[IndentingNewLine]\[IndentingNewLine]Return[
5100	m2];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"],
5101
5102	Cell[BoxData[
5103	FormBox[
5104	RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5105	error: new symbol name \\\"\\!\\(TraditionalForm\\`beta\\)\\\" is similar to \
5106	existing symbol \\\"\\!\\(TraditionalForm\\`Beta\\)\\\".\"\>"}],
5107	TraditionalForm]], "Message"],
5108
5109	Cell[BoxData[
5110	FormBox[
5111	RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5112	error: new symbol name \\\"\\!\\(TraditionalForm\\`gamma\\)\\\" is similar to \
5113	existing symbol \\\"\\!\\(TraditionalForm\\`Gamma\\)\\\".\"\>"}],
5114	TraditionalForm]], "Message"],
5115
5116	Cell[BoxData[
5117	FormBox[
5118	RowBox[{\(General::"spell"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5119	error: new symbol name \\\"\\!\\(TraditionalForm\\`pbbw\\)\\\" is similar to \
5120	existing symbols \\!\\(TraditionalForm\\`\\({pbb, pbw}\\)\\).\"\>"}],
5121	TraditionalForm]], "Message"],
5122
5123	Cell[BoxData[
5124	FormBox[
5125	RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5126	error: new symbol name \\\"\\!\\(TraditionalForm\\`tab\\)\\\" is similar to \
5127	existing symbol \\\"\\!\\(TraditionalForm\\`Tab\\)\\\".\"\>"}],
5128	TraditionalForm]], "Message"],
5129
5130	Cell[BoxData[
5131	FormBox[
5132	RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output of \
5133	\\!\\(TraditionalForm\\`\\(General :: \\\"spell1\\\"\\)\\) will be suppressed \
5134	during this calculation.\"\>"}], TraditionalForm]], "Message"]
5135	}, Open ]],
5136
5137	Cell["check the matrix element against MadGraph", "Text"],
5138
5139	Cell[CellGroupData[{
5140
5141	Cell[BoxData[{
5142	\(dsiguu = 5915.84796\), "\n",
5143	\(\(Pq = {200.000000000000000, 0.000000000000000, 0.000000000000000,
5144	200.000000000000000};\)\), "\n",
5145	\(\(Pqb = {200.000000000000000, 0.000000000000000,
5146	0.000000000000000, \(-200.000000000000000\)};\)\), "\n",
5147	\(\(Pb = {46.572642255938540, 34.897558228069109,
5148	28.833151059160027, \(-10.946270433705514\)};\)\), "\n",
5149	\(\(Pbb = {94.270778282554218, 88.724754117517620,
5150	3.024989803798522, \(-31.713515753720035\)};\)\), "\n",
5151	\(\(Pe = {111.800803917467817, \(-94.012734182798397\), \
5152	\(-29.615126770051678\), \(-52.763337976859873\)};\)\), "\n",
5153	\(\(Pm = {48.384357892271723, 2.735985567654750, \(-42.862627840873451\),
5154	22.279039615652231};\)\), "\n",
5155	\(\(Pve = {40.801322525238085, \(-30.692673191600964\),
5156	6.447102135143032, 26.098708899505258};\)\), "\n",
5157	\(\(Pvm = {58.170095126529681, \(-1.652890538842172\),
5158	34.172511612823548, 47.045375649127912};\)\n\n\), "\n",
5159	\(MsqTT[Pq, Pqb, Pb, Pe, Pve, Pbb, Pm, Pvm]2SP[psum, psum]/dsiguu*
5160	gevpb\)}], "Input"],
5161
5162	Cell[BoxData[
5163	\(TraditionalForm\`5915.84796`\)], "Output"],
5164
5165	Cell[BoxData[
5166	\(TraditionalForm\`6.358735901081007`\)], "Output"]
5167	}, Open ]],
5168
5169	Cell[CellGroupData[{
5170
5171	Cell[BoxData[
5172	\(SP[Pm, Pm] // Sqrt\)], "Input"],
5173
5174	Cell[BoxData[
5175	\(TraditionalForm\`7.152557373046875`*^-7\)], "Output"]
5176	}, Open ]]
5177	}, Open ]],
5178
5179	Cell[CellGroupData[{
5180
5181	Cell["\<\
5182	Generates the phase space of 3 particles, by iteration of the 1->2 \
5183	decay:
5184	t-> w b, w-> lv.
5185
5186	INPUT:
5187	x: an array of length 5 which contains the random numbers in [0,1].
5188	p1: four momentum of the decaying particle (top).
5189	params: an array which contains the masses of the final state particles and \
5190	the width of the W {mw,gw,mb,ml,mv}.
5191	BW: is a logical value. If true the Breit-Wigner is \"flattened out\" through \
5192	an analytic transformation realized
5193	by Transpole. If false, the invariant mass of the W is generated uniformly.
5194
5195	OUTPUT:
5196	The 3 momenta of the final particles and the phase space weight\
5197	\>", \
5198	"Subsubsection"],
5199
5200	Cell[BoxData[
5201	\(\(Phi3BW[x_, p1_, params_, BW_] :=
5202	Module[{s, rs, s2, rs2, jac, jacpole, pswgt, costh, phi, costh1,
5203	phi1, mb, ml, mv, mw, gw, pw, pb, pl, pv, xa2,
5204	xb2}, \[IndentingNewLine]\[IndentingNewLine] (*initialization\
5205	*) \[IndentingNewLine]mw = params[\([1]\)]; \[IndentingNewLine]gw =
5206	params[\([2]\)]; \[IndentingNewLine]mb =
5207	params[\([3]\)]; \[IndentingNewLine]ml =
5208	params[\([4]\)]; \[IndentingNewLine]mv =
5209	params[\([5]\)]; \[IndentingNewLine]pw =
5210	Table[0, {i, 1, 4}]; \[IndentingNewLine]pb =
5211	pw; \[IndentingNewLine]pl = pw; \[IndentingNewLine]pv =
5212	pw; \[IndentingNewLine]\[IndentingNewLine] (*\
5213	start\ *) \[IndentingNewLine]s =
5214	SP[p1, p1]; \[IndentingNewLine]rs =
5215	Sqrt[s]; \[IndentingNewLine]If[
5216	BW, \[IndentingNewLine] (*\
5217	make\ the\ analytic\ transformation\ *) \[IndentingNewLine]{s2,
5218	jacpole} =
5219	Transpole[mw^2/s, mw\ gw/s,
5220	x[\([3]\)]]; \[IndentingNewLine]s2 =
5221	s2*s; \[IndentingNewLine]rs2 =
5222	Sqrt[s2]; \[IndentingNewLine]jac =
5223	jacpoles;\[IndentingNewLine], \[IndentingNewLine]\ (\
5224	generate\ it\ flat\ ) \[IndentingNewLine]s2 = \((s - mb^2)\)
5225	x[\([3]\)] + mb^2; \[IndentingNewLine]rs2 =
5226	Sqrt[s2]; \[IndentingNewLine]jac = \((s -
5227	mb^2)\);\[IndentingNewLine]]; \[IndentingNewLine]\
5228	\[IndentingNewLine] (*\
5229	check\ kinematic\ boundaries\ *) \[IndentingNewLine]If[
5230	rs2 > \((rs - mb)\)\ \|\| \
5231	rs2 < \((ml + mv)\), \[IndentingNewLine]\(Return[
5232	0];\)]; \[IndentingNewLine]\[IndentingNewLine] (*\
5233	generate\ the\ angles\ *) \[IndentingNewLine]costh = \(-1\) +
5234	2\ x[\([1]\)]; \[IndentingNewLine]phi = \
5235	2\ Pi\ x[\([2]\)]; \[IndentingNewLine]jac =
5236	jac\ 4\ Pi; \[IndentingNewLine]costh1 = \(-1\) +
5237	2\ x[\([4]\)]; \[IndentingNewLine]phi1 = \
5238	2Pix[\([5]\)]; \[IndentingNewLine]jac =
5239	jac4\ Pi; \[IndentingNewLine]\[IndentingNewLine] (*\
5240	generate\ the\ four - momenta\ *) \[IndentingNewLine]{pw, pb} =
5241	MOM2CX[rs, rs2, mb, costh, phi]; \[IndentingNewLine]{pl, pv} =
5242	MOM2CX[rs2, ml, mv, costh1, phi1]; \[IndentingNewLine]pb =
5243	BOOST[pb, p1]; \[IndentingNewLine]pw =
5244	BOOST[pw, p1]; \[IndentingNewLine]pl =
5245	BOOST[pl, pw]; \[IndentingNewLine]pv =
5246	BOOST[pv, pw]; \[IndentingNewLine]\[IndentingNewLine] (*\
5247	phase\ space\ factors\ *) \[IndentingNewLine]jac =
5248	jac/\((2\ Pi)\)^5; \[IndentingNewLine]pswgt =
5249	1\ \ \ \ (\(/\)\((2rs)\)*) ; \[IndentingNewLine]xa2 =
5250	s2/s; \[IndentingNewLine]xb2 = mb^2/s; \[IndentingNewLine]pswgt =
5251	pswgt/2Pi
5252	Sqrt[lambda[1, xa2, xb2]]/\((4\ Pi)\); \[IndentingNewLine]xa2 =
5253	ml^2/s; \[IndentingNewLine]xb2 =
5254	mv^2/s; \[IndentingNewLine]pswgt =
5255	pswgt/2*Pi\ Sqrt[
5256	lambda[1, xa2,
5257	xb2]]/\((4\ Pi)\); \n\ \ \ \ \[IndentingNewLine] (*final\ \
5258	jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
5259	pswgt = pswgt*
5260	jac; \[IndentingNewLine]\ \[IndentingNewLine]\ \ Return[{pb,
5261	pl, pv, pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], \
5262	"Input"]
5263	}, Closed]],
5264
5265	Cell[CellGroupData[{
5266
5267	Cell["\<\
5268	Phase space for two particles.
5269
5270	INPUT:
5271	x: an array of length 2 which contains the random numbers in [0,1].
5272	ptot: four momentum of the total momentum.
5273	params: an array which contains the masses of the final state particles \
5274	{m1,m2}.
5275
5276	OUTPUT:
5277	The 3 momenta of the final particles and the phase space weight.\
5278	\>", \
5279	"Subsubsection"],
5280
5281	Cell[BoxData[
5282	\(\(Phi2[x_, ptot_, params_] :=
5283	Module[{s, rs, jac, pswgt, costh, phi, xa2, xb2, m1, m2, p1,
5284	p2}, \[IndentingNewLine]\[IndentingNewLine] (*initialization\
5285	*) \[IndentingNewLine]\[IndentingNewLine]m1 =
5286	params[\([1]\)]; \[IndentingNewLine]m2 =
5287	params[\([2]\)]; \[IndentingNewLine]p1 =
5288	Table[0, {i, 1, 4}]; \[IndentingNewLine]p2 =
5289	p1; \[IndentingNewLine]\[IndentingNewLine] (*\
5290	start\ *) \[IndentingNewLine]\[IndentingNewLine]s =
5291	SP[ptot, ptot]; \[IndentingNewLine]rs =
5292	Sqrt[s]; \[IndentingNewLine]\[IndentingNewLine] (*\
5293	generate\ the\ angles\ and\ momenta\ for\ ptot \[Rule] \
5294	1\ 2\ *) \[IndentingNewLine]\[IndentingNewLine]costh = \(-1\) +
5295	2\ x[\([1]\)]; \[IndentingNewLine]phi = \
5296	2\ Pi\ x[\([2]\)]; \[IndentingNewLine]jac =
5297	4\ *Pi; \[IndentingNewLine]{p1, p2} =
5298	MOM2CX[rs, m1, m2, costh, phi]; \[IndentingNewLine]p1 =
5299	BOOST[p1, ptot]; \[IndentingNewLine]p2 =
5300	BOOST[p2, ptot]; \[IndentingNewLine]\[IndentingNewLine] (*\
5301	phase\ space\ factors\ \
5302	*) \[IndentingNewLine]\[IndentingNewLine]jac =
5303	jac/\((2\ Pi)\)^2; \[IndentingNewLine]pswgt =
5304	1; \[IndentingNewLine]xa2 = m1^2/s; \[IndentingNewLine]xb2 =
5305	m2^2/s; \[IndentingNewLine]pswgt =
5306	pswgt/2Pi
5307	Sqrt[lambda[1, xa2,
5308	xb2]]/\((4\ Pi)\); \[IndentingNewLine]\[IndentingNewLine] \
5309	(final\ jacobian) \[IndentingNewLine]\[IndentingNewLine]\
5310	pswgt = pswgt*jac; \[IndentingNewLine]\ \[IndentingNewLine]\
5311	Return[{p1, p2,
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5313	}, Closed]],
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5318	"Function which gives the cross section when integrated in the volume ",
5319	Cell[BoxData[
5320	\(TraditionalForm\`\([0, 1]\)\^2\)]],
5321	".The ",
5322	Cell[BoxData[
5323	\(TraditionalForm\`x\_i\)]],
5324	"are the inputs in [0,1]. "
5325	}], "Subsubsection"],
5326
5327	Cell[BoxData[
5328	\(\(ff[x1_, x2_, x3_, x4_, x5_, x6_, x7_, x8_, x9_, x10_, x11_, x12_] :=
5329	Module[{cos, p1, p2, pe1, pe2, xx, aa, pswgt,
5330	wgt}, \[IndentingNewLine] (*1/\((Gammat\ mt)\)^2\ \[Rule] \ \(Pi/
5331	Gammat\)/mt*) \[IndentingNewLine]wgt =
5332	1/\((2\ Pi)\)^2*\(Pi^2/mt^2\)/
5333	Gammat^2\ *\((mt^2\ Gammat^2)\)^2; \[IndentingNewLine]xx = \
5334	{x1, x2}; \n\ \ \ \ \ \ {pt, ptb, pswgt} =
5335	Phi2[xx, psum, {mt, mt}]; \[IndentingNewLine]wgt =
5336	wgt*pswgt; \[IndentingNewLine]xx = {x3, x4, x5, x6,
5337	x7}; \[IndentingNewLine]aa =
5338	Phi3BW[xx, pt, {mw, Gammaw, mb, 0, 0},
5339	True]; \[IndentingNewLine]If[
5340	aa =!= 0, {pb, pe, pve, pswgt} = aa,
5341	Return[0]]; \[IndentingNewLine]wgt =
5342	wgt*pswgt; \[IndentingNewLine]xx = {x8, x9, x10, x11,
5343	x12}; \[IndentingNewLine]aa =
5344	Phi3BW[xx, ptb, {mw, Gammaw, mb, 0, 0},
5345	True]; \[IndentingNewLine]If[
5346	aa =!= 0, {pbb, pm, pvm, pswgt} = aa,
5347	Return[0]]; \[IndentingNewLine]wgt =
5348	wgt*pswgt; \[IndentingNewLine]EmmeSq =
5349	MsqTT[pq, pqb, pb, pe, pve, pbb, pm,
5350	pvm]; \n\ \ \ \ \ \ \ \ gamma =
5351	EmmeSqwgtgevpb; \[IndentingNewLine]\[IndentingNewLine] (*\
5352	calculate\ the\ angles\ between\ the\ leptons\ \
5353	*) \[IndentingNewLine]p1 = \(-pt\); \[IndentingNewLine]p2 = \(-ptb\); \
5354	\[IndentingNewLine]p1[\([1]\)] = \(-p1[\([1]\)]\); \
5355	\[IndentingNewLine]p2[\([1]\)] = \(-p2[\([1]\)]\); \[IndentingNewLine]\
5356	\[IndentingNewLine]\[IndentingNewLine]pe1 =
5357	BOOST[pe, p1]; \[IndentingNewLine]pe2 =
5358	BOOST[pm, p2]; \[IndentingNewLine]\[IndentingNewLine]cos =
5359	CosTheta[pe1, pe2]; \[IndentingNewLine]Write[
5360	ciccio, {cos, gamma}]; \[IndentingNewLine]WriteString[
5361	ciccio, "\<,\>"]; \[IndentingNewLine]\[IndentingNewLine]Return[
5362	gamma];\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]\
5363	\[IndentingNewLine]];\)\)], "Input"]
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5400	Cell[BoxData[
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7107	*)
7108
7109
7110
7111	(*******************************************************************
7112	End of Mathematica Notebook file.
7113	*******************************************************************)
7114

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