MCSummerCERN13: mc101.nb

File mc101.nb, 453.9 KB (added by Fabio Maltoni, 11 years ago)
Line 
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58This is a set of solved exercises (borrowed from M. Seymour) for \
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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"],
538
539Cell[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
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1211Cell["\<\
1212
1213To implement the transformation it is necessary to know:
1214a. the primitive G of the \"enclosing\" function g
1215b. the inverse of the primitive Gi
1216This is what is obtained in the following:\
1217\>", "Text",
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1221
1222Cell[BoxData[{
1223 \(\(g[x_] = 1 - x^2;\)\), "\[IndentingNewLine]",
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1225 \(sub = \(\(Solve[rho \[Equal] G[x], x]\)[\([3]\)]\)[\([1]\)] //
1226 Simplify\), "\n",
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1234rho)\)\^\(2/3\))\)\)\/\(2\ 2\^\(2/3\)\ \@\(\@\(9\ rho\^2 - 4\) - 3\ \
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1254 \(\(f[x_] = Cos[Pi/2\ x]/\((1 - x^2)\)\ ;\)\)], "Input"],
1255
1256Cell[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
1270Cell[CellGroupData[{
1271
1272Cell[BoxData[{
1273 \(\({res, sig} = MyInte1[f\ , 10000];\)\), "\[IndentingNewLine]",
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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/\((count + noin)\) //
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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]",
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3537 nev_] := \[IndentingNewLine]Module[{sig, eff, noin, count, xi, sum,
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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
3558Cell[CellGroupData[{
3559
3560Cell[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
3572Cell[BoxData[
3573 FormBox[
3574 InterpretationBox[\("Analytic Result : \
3575"\[InvisibleSpace]0.6366197723675814`\),
3576 SequenceForm[ "Analytic Result : ", 0.63661977236758138],
3577 Editable->False], TraditionalForm]], "Print"],
3578
3579Cell[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
3592Cell[CellGroupData[{
3593
3594Cell["Dimensionality of the phase space of 1-> n particles", "Section"],
3595
3596Cell[BoxData[
3597 \(\(Ndim = Nparticles*3 - 4;\)\)], "Input"]
3598}, Closed]],
3599
3600Cell[CellGroupData[{
3601
3602Cell["Useful functions", "Section"],
3603
3604Cell[CellGroupData[{
3605
3606Cell["Scalar product between four momenta {E,px,py,pz}.", "Subsubsection"],
3607
3608Cell[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
3617Cell[CellGroupData[{
3618
3619Cell["\<\
3620Cos(theta), where theta is the angle between two spatial components \
3621of two four momenta:\
3622\>", "Subsubsection"],
3623
3624Cell[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
3637Cell[CellGroupData[{
3638
3639Cell["\<\
3640Boost: it boosts the four momentum P from the frame S to the frame \
3641S' of Q. The frame S is where Q is of the form {E,0,0,0}. It returns the four \
3642momentum P'.\
3643\>", "Subsubsection"],
3644
3645Cell[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
3665Cell[CellGroupData[{
3666
3667Cell["\<\
3668MO2CX: It calculates the two four-momenta P1 and P2, of masses m1 \
3669and m2, with total energy esum, in a given direction (theta,phi), in their \
3670rest frame. \
3671\>", "Subsubsection"],
3672
3673Cell[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
3691Cell[CellGroupData[{
3692
3693Cell["\<\
3694Transpole: given x distributed uniform in [0,1] it returns a \
3695Breit-Wigner distribution, centered in \"pole\" with \"width\".
3696It also returns the jacobian of the transformation.\
3697\>", "Subsubsection"],
3698
3699Cell[CellGroupData[{
3700
3701Cell[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
3712Cell[BoxData[
3713 FormBox[
3714 RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
3715error: new symbol name \\\"\\!\\(TraditionalForm\\`Transpole\\)\\\" is \
3716similar to existing symbol \
3717\\\"\\!\\(TraditionalForm\\`Transpose\\)\\\".\"\>"}],
3718 TraditionalForm]], "Message"]
3719}, Open ]]
3720}, Closed]],
3721
3722Cell[CellGroupData[{
3723
3724Cell["\<\
3725Phase space maeasure for two particle decay, with masses ma and \
3726mb\
3727\>", "Subsubsection"],
3728
3729Cell[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
3736Cell[CellGroupData[{
3737
3738Cell["Install Vegas", "Section"],
3739
3740Cell["\<\
3741I 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 \
3743Vegas\
3744\>", "Text"],
3745
3746Cell[CellGroupData[{
3747
3748Cell[BoxData[
3749 \(Install["\</Users/fabiomaltoni/Physics/Codes/CUBA/VegasX\>"]\)], "Input"],
3750
3751Cell[BoxData[
3752 \(TraditionalForm\`LinkObject[
3753 "/Users/fabiomaltoni/Physics/Codes/CUBA/VegasX", 2, 2]\)], "Output"]
3754}, Open ]]
3755}, Closed]],
3756
3757Cell[CellGroupData[{
3758
3759Cell["Top decay ", "Section"],
3760
3761Cell[CellGroupData[{
3762
3763Cell["\<\
3764Generates the phase space of 3 particles, by iteration of the 1->2 \
3765decay:
3766t-> w b, w-> lv.
3767
3768INPUT:
3769x: an array of length 5 which contains the random numbers in [0,1].
3770p1: four momentum of the decaying particle (top).
3771params: an array which contains the masses of the final state particles and \
3772the width of the W {mw,gw,mb,ml,mv}.
3773BW: is a logical value. If true the Breit-Wigner is \"flattened out\" through \
3774an analytic transformation realized
3775by Transpole. If false, the invariant mass of the W is generated uniformly.
3776
3777OUTPUT:
3778The 3 momenta of the final particles and the phase space weight\
3779\>", \
3780"Subsubsection"],
3781
3782Cell[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 jacpole*s;\[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 2*Pi*x[\([5]\)]; \[IndentingNewLine]jac =
3821 jac*4\ *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\ \ \ \ (*\(/\)\((2*rs)\)*) ; \[IndentingNewLine]xa2 =
3832 s2/s; \[IndentingNewLine]xb2 = mb^2/s; \[IndentingNewLine]pswgt =
3833 pswgt/2*Pi*
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\ \
3840jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
3841 pswgt = pswgt*
3842 jac; \[IndentingNewLine]\ \[IndentingNewLine]\ \ Return[{pb,
3843 pl, pv, pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], \
3844"Input"]
3845}, Closed]],
3846
3847Cell[CellGroupData[{
3848
3849Cell[TextData[{
3850 "Generates the phase space of 3 particles, by iteration of the 1->2 decay:\n\
3851t-> w b, w-> lv, assuming the w to be exactly on shell. This is the so-called \
3852narrow-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 \
3858passes from 5 to 4\n\nINPUT:\nx: an array of length 4 which contains the \
3859random numbers in [0,1].\np1: four momentum of the decaying particle (top).\n\
3860params: an array which contains the masses of the final state particles and \
3861the width of the W {mw,mb,ml,mv}.\n\nOUTPUT:\nThe 3 momenta of the final \
3862particles and the phase space weight."
3863}], "Subsubsection"],
3864
3865Cell[CellGroupData[{
3866
3867Cell[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 2*Pi*x[\([4]\)]; \[IndentingNewLine]jac =
3895 jac*4\ *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/2*Pi*
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\ \
3911jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
3912 pswgt = pswgt*jac; \[IndentingNewLine]\ \[IndentingNewLine]\
3913 Return[{pb, pl, pv,
3914 pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"],
3915
3916Cell[BoxData[
3917 FormBox[
3918 RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
3919error: new symbol name \\\"\\!\\(TraditionalForm\\`Phi3NW\\)\\\" is similar \
3920to existing symbol \\\"\\!\\(TraditionalForm\\`Phi3BW\\)\\\".\"\>"}],
3921 TraditionalForm]], "Message"]
3922}, Open ]]
3923}, Closed]],
3924
3925Cell[CellGroupData[{
3926
3927Cell[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\
3936method=\"NW\" uses the narrow width approximation for the phase space (in \
3937this case the value of x5 is ignored). \nThe value of the matrix element \
3938squared can be found on the ESW."
3939}], "Subsubsection"],
3940
3941Cell[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\ \
3947the\ 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
3988Cell[CellGroupData[{
3989
3990Cell["Parameters entering the top decay", "Subsection"],
3991
3992Cell[BoxData[{
3993 \(\(MB = 0;\)\), "\[IndentingNewLine]",
3994 \(\(MW = 80.419;\)\), "\[IndentingNewLine]",
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4822\>"],
4823 ImageRangeCache->{{{0, 472.562}, {291.688, 0}} -> {-0.0670434, -0.109275, \
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4825
4826Cell[BoxData[
4827 FormBox[
4828 TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\),
4829 False,
4830 Editable->False], TraditionalForm]], "Output"]
4831}, Open ]]
4832}, Open ]]
4833}, Open ]],
4834
4835Cell[CellGroupData[{
4836
4837Cell["q1 q2~ \[Rule] tt~ production", "Section"],
4838
4839Cell[CellGroupData[{
4840
4841Cell["Parameters", "Subsubsection"],
4842
4843Cell[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
4853Cell[CellGroupData[{
4854
4855Cell["\<\
4856Matrix element squared summed and averaged for tt~ production, \
4857divided by the initial flux. The inputs are the four momenta. channel=\"qq\", \
4858returns the value for qq->tt~, channel=\"gg\" returns the value for gg>tt~.\
4859\
4860\>", "Subsubsection"],
4861
4862Cell[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
4885Cell[CellGroupData[{
4886
4887Cell["\<\
4888Phase space for two particles.
4889
4890INPUT:
4891x: an array of length 2 which contains the random numbers in [0,1].
4892ptot: four momentum of the total momentum.
4893params: an array which contains the masses of the final state particles \
4894{m1,m2}.
4895
4896OUTPUT:
4897The 3 momenta of the final particles and the phase space weight.\
4898\>", \
4899"Subsubsection"],
4900
4901Cell[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/2*Pi*
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
4935Cell[CellGroupData[{
4936
4937Cell[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
4947Cell[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 EmmeSq*pswgt*gevpb; \[IndentingNewLine]Return[
4954 gamma];\[IndentingNewLine]];\)\)], "Input"]
4955}, Closed]],
4956
4957Cell[CellGroupData[{
4958
4959Cell["Calculate the cross section", "Subsection"],
4960
4961Cell[CellGroupData[{
4962
4963Cell[BoxData[
4964 \(Vegas[ff[x1, x2, "\<qq\>"], {x1, 0, 1}, {x2, 0, 1}\ ,
4965 Compiled \[Rule] False]\)], "Input"],
4966
4967Cell[BoxData[
4968 \(TraditionalForm\`"\nIteration 1: 1000 integrand evaluations so far\n\
4969[1] 25.9015 +- 0.0318548 \tchisq -0 (0 df)"\)], "Print"],
4970
4971Cell[BoxData[
4972 \(TraditionalForm\`"\nIteration 2: 2500 integrand evaluations so far\n\
4973[1] 25.9042 +- 0.00955214 \tchisq 0.00781108 (1 df)"\)], "Print"],
4974
4975Cell[BoxData[
4976 FormBox[
4977 RowBox[{\(Vegas::"success"\), \(\(:\)\(\ \)\), "\<\"Needed \
4978\\!\\(TraditionalForm\\`2500\\) function evaluations.\"\>"}],
4979 TraditionalForm]], "Message"],
4980
4981Cell[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
4993Cell[CellGroupData[{
4994
4995Cell["\<\
4996q1 q2~ \[Rule] tt~ \[Rule](e+ ve b ) (mu- vm~ b~) (full decay \
4997matrix element)\
4998\>", "Section"],
4999
5000Cell[CellGroupData[{
5001
5002Cell["Parameters", "Subsubsection"],
5003
5004Cell[CellGroupData[{
5005
5006Cell[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
5025Cell[BoxData[
5026 \(TraditionalForm\`0.6101671292521702`\)], "Output"]
5027}, Open ]],
5028
5029Cell[BoxData[""], "Input"]
5030}, Open ]],
5031
5032Cell[CellGroupData[{
5033
5034Cell["\<\
5035Matrix element squared summed and averaged for tt~ production, \
5036divided by the initial flux. Momenta must be given in the ZMF for the \
5037calculation of cmq,ceq,cqt,sqt,cmt,cet. ceb and cmb are to be calculated in \
5038the respective rest frames of the W+ and W-\
5039\>", "Subsubsection"],
5040
5041Cell[CellGroupData[{
5042
5043Cell[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\ \
5057ZMF*) \[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, \
5085pve] \((1 + ceb)\)\^2\)\)\/\(\((2\ SP[pe, pve] - mw\^2)\)\^2 + \ \((Gammaw\ \
5086mw)\)\^2\); \[IndentingNewLine]tab =
5087 pre \((mt^2 -
5088 2\ SP[pm,
5089 pvm])\) \(\(\ \)\(mt^2 \((1 - cmb\^2)\) + \ 2\ SP[pm, \
5090pvm] \((1 + cmb)\)\^2\)\)\/\(\((2\ SP[pm, pvm] - mw\^2)\)\^2 + \ \((Gammaw\ \
5091mw)\)\^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
5102Cell[BoxData[
5103 FormBox[
5104 RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5105error: new symbol name \\\"\\!\\(TraditionalForm\\`beta\\)\\\" is similar to \
5106existing symbol \\\"\\!\\(TraditionalForm\\`Beta\\)\\\".\"\>"}],
5107 TraditionalForm]], "Message"],
5108
5109Cell[BoxData[
5110 FormBox[
5111 RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5112error: new symbol name \\\"\\!\\(TraditionalForm\\`gamma\\)\\\" is similar to \
5113existing symbol \\\"\\!\\(TraditionalForm\\`Gamma\\)\\\".\"\>"}],
5114 TraditionalForm]], "Message"],
5115
5116Cell[BoxData[
5117 FormBox[
5118 RowBox[{\(General::"spell"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5119error: new symbol name \\\"\\!\\(TraditionalForm\\`pbbw\\)\\\" is similar to \
5120existing symbols \\!\\(TraditionalForm\\`\\({pbb, pbw}\\)\\).\"\>"}],
5121 TraditionalForm]], "Message"],
5122
5123Cell[BoxData[
5124 FormBox[
5125 RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
5126error: new symbol name \\\"\\!\\(TraditionalForm\\`tab\\)\\\" is similar to \
5127existing symbol \\\"\\!\\(TraditionalForm\\`Tab\\)\\\".\"\>"}],
5128 TraditionalForm]], "Message"],
5129
5130Cell[BoxData[
5131 FormBox[
5132 RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output of \
5133\\!\\(TraditionalForm\\`\\(General :: \\\"spell1\\\"\\)\\) will be suppressed \
5134during this calculation.\"\>"}], TraditionalForm]], "Message"]
5135}, Open ]],
5136
5137Cell["check the matrix element against MadGraph", "Text"],
5138
5139Cell[CellGroupData[{
5140
5141Cell[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 \(MsqTT[Pq, Pqb, Pb, Pe, Pve, Pbb, Pm, Pvm]*2*SP[psum, psum]/dsiguu*
5160 gevpb\)}], "Input"],
5161
5162Cell[BoxData[
5163 \(TraditionalForm\`5915.84796`\)], "Output"],
5164
5165Cell[BoxData[
5166 \(TraditionalForm\`6.358735901081007`\)], "Output"]
5167}, Open ]],
5168
5169Cell[CellGroupData[{
5170
5171Cell[BoxData[
5172 \(SP[Pm, Pm] // Sqrt\)], "Input"],
5173
5174Cell[BoxData[
5175 \(TraditionalForm\`7.152557373046875`*^-7\)], "Output"]
5176}, Open ]]
5177}, Open ]],
5178
5179Cell[CellGroupData[{
5180
5181Cell["\<\
5182Generates the phase space of 3 particles, by iteration of the 1->2 \
5183decay:
5184t-> w b, w-> lv.
5185
5186INPUT:
5187x: an array of length 5 which contains the random numbers in [0,1].
5188p1: four momentum of the decaying particle (top).
5189params: an array which contains the masses of the final state particles and \
5190the width of the W {mw,gw,mb,ml,mv}.
5191BW: is a logical value. If true the Breit-Wigner is \"flattened out\" through \
5192an analytic transformation realized
5193by Transpole. If false, the invariant mass of the W is generated uniformly.
5194
5195OUTPUT:
5196The 3 momenta of the final particles and the phase space weight\
5197\>", \
5198"Subsubsection"],
5199
5200Cell[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 jacpole*s;\[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 2*Pi*x[\([5]\)]; \[IndentingNewLine]jac =
5239 jac*4\ *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\ \ \ \ (*\(/\)\((2*rs)\)*) ; \[IndentingNewLine]xa2 =
5250 s2/s; \[IndentingNewLine]xb2 = mb^2/s; \[IndentingNewLine]pswgt =
5251 pswgt/2*Pi*
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\ \
5258jacobian*) \[IndentingNewLine]\[IndentingNewLine]\
5259 pswgt = pswgt*
5260 jac; \[IndentingNewLine]\ \[IndentingNewLine]\ \ Return[{pb,
5261 pl, pv, pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], \
5262"Input"]
5263}, Closed]],
5264
5265Cell[CellGroupData[{
5266
5267Cell["\<\
5268Phase space for two particles.
5269
5270INPUT:
5271x: an array of length 2 which contains the random numbers in [0,1].
5272ptot: four momentum of the total momentum.
5273params: an array which contains the masses of the final state particles \
5274{m1,m2}.
5275
5276OUTPUT:
5277The 3 momenta of the final particles and the phase space weight.\
5278\>", \
5279"Subsubsection"],
5280
5281Cell[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/2*Pi*
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,
5312 pswgt}];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"]
5313}, Closed]],
5314
5315Cell[CellGroupData[{
5316
5317Cell[TextData[{
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
5327Cell[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 EmmeSq*wgt*gevpb; \[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"]
5364}, Open ]],
5365
5366Cell[CellGroupData[{
5367
5368Cell["Calculate the cross section", "Subsection"],
5369
5370Cell[CellGroupData[{
5371
5372Cell[BoxData[
5373 \(\(\(\[IndentingNewLine]\)\(\(ciccio =
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5375 \(WriteString[ciccio, "\<histo={\>"];\)\n
5376 \(Vegas[
5377 ff[x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12], {x1, 0,
5378 1}, {x2, 0, 1}\ , {x3, 0, 1}, {x4, 0, 1}, {x5, 0, 1}, {x6, 0,
5379 1}, {x7, 0, 1}, {x8, 0, 1}, {x9, 0, 1}, {x10, 0, 1}, {x11, 0,
5380 1}, {x12, 0, 1}, Compiled \[Rule] False,
5381 MaxPoints \[Rule] \ 10000];\)\n
5382 WriteString[ciccio, "\<{1,0}};\>"]; Close[ciccio]\)\)\)], "Input"],
5383
5384Cell[BoxData[
5385 \(TraditionalForm\`"\nIteration 1: 1000 integrand evaluations so far\n\
5386[1] 0.315829 +- 0.00648571 \tchisq -0 (0 df)"\)], "Print"],
5387
5388Cell[BoxData[
5389 \(TraditionalForm\`"\nIteration 2: 2500 integrand evaluations so far\n\
5390[1] 0.317021 +- 0.00428438 \tchisq 0.0600085 (1 df)"\)], "Print"],
5391
5392Cell[BoxData[
5393 \(TraditionalForm\`"\nIteration 3: 4500 integrand evaluations so far\n\
5394[1] 0.316365 +- 0.00325219 \tchisq 0.115435 (2 df)"\)], "Print"],
5395
5396Cell[BoxData[
5397 \(TraditionalForm\`"\nIteration 4: 7000 integrand evaluations so far\n\
5398[1] 0.314562 +- 0.00256801 \tchisq 0.931906 (3 df)"\)], "Print"],
5399
5400Cell[BoxData[
5401 \(TraditionalForm\`"\nIteration 5: 10000 integrand evaluations so far\n\
5402[1] 0.313353 +- 0.00212722 \tchisq 1.63756 (4 df)"\)], "Print"],
5403
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5409
5410Cell[BoxData[
5411 \(TraditionalForm\`"prova"\)], "Output"]
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5415
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5434 \(\(Do[\[IndentingNewLine]temp =
5435 histo[\([i]\)]; \[IndentingNewLine]Do[\[IndentingNewLine]If[
5436 temp[\([1]\)] > delta*\((n - 1)\)\ && temp[\([1]\)] < delta*n,
5437 histos[\([n + 10]\)] =
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5440
5441Cell[BoxData[
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5462Cell[CellGroupData[{
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5465 \(dafit =
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9799}
9800]
9801*)
9802
9803
9804
9805(*******************************************************************
9806End of Mathematica Notebook file.
9807*******************************************************************)
9808