bo2020: phenHiggs.nb

File phenHiggs.nb, 110.9 KB (added by fabiomaltoni, 5 months ago)
Line 
1(* Content-type: application/vnd.wolfram.mathematica *)
2
3(*** Wolfram Notebook File ***)
4(* http://www.wolfram.com/nb *)
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22Cell[CellGroupData[{
23Cell["Phenomenology of pp\[Rule]H+X at NLO", "Title"],
24
25Cell[CellGroupData[{
26
27Cell["Introduction", "Section"],
28
29Cell[TextData[{
30 StyleBox["In this notebook we calculate the inclusive cross section for \
31Higgs production at hadron colliders, at NLO in the strong coupling. We use \
32the analytic results obtained in a previous notebook corresponding to the \
33original calculation by Sally Dawson (Nuclear Physics B (1991) 283). To get \
34useful numbers we use a modern set of PDF, i.e. the MRST as implemented in ",
35  FontSize->16],
36 StyleBox["Mathematica",
37  FontSize->16,
38  FontSlant->"Italic"],
39 StyleBox[" by J.Andersen (many thanks!).  A description of the calculation, \
40including the formulas used here for the numerical results, is given in the \
41notes.",
42  FontSize->16]
43}], "Text"]
44}, Open  ]],
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51Off[General::spell];
52Off[General::spell1];
53Clear[\"Global`*\"];\
54\>", "Input"],
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56Cell[CellGroupData[{
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58Cell["Install Vegas", "Subsection"],
59
60Cell["\<\
61I use the  Vegas package of CUBA library of T. Hahn (hep-ph/0404043). If you \
62do not want to use it, use NIntegrate instead of Vegas\
63\>", "Text"],
64
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70  "Install", "[", "\"\</Users/fabiomaltoni/Physics/Codes/CUBA/VegasX\>\"",
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72  RowBox[{
73  "Install", "[", "\"\</Users/fabiomaltoni/Physics/Codes/CUBA/SuaveX\>\"",
74   "]"}]}]], "Input"],
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121<< Graphics`Colors`;
122<< Graphics`Graphics`;\
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261Cell["\<\
262Wrapper to call the pdf. Notation is self-explanatory. Just notice that all \
263parton distribution codes usually return x f(x). To avoid confusion, at \
264expense of a couple more floating point operations, I divide all the values \
265by the corresponding x.  pdfcall calculates the parton-parton luminoties, gg \
266qg, qq~\
267\>", "Subsubsection"],
268
269Cell["\<\
270(* MRST *)
271
272pdfMRST[X1_,X2_,q_]:=Module[
273{Q,xgDF,xdDF,xdbDF,xuDF,xubDF,xsDF,xcDF,xbDF,pd1,pd2,xgg,xqg,xqq},
274Q=q*1.;
275xgDF =f[3,X1,Q]/X1;
276xdbDF=f[8,X1,Q]/X1;
277xdDF =f[2,X1,Q]/X1+xdbDF;
278xubDF=f[4,X1,Q]/X1;
279xuDF =f[1,X1,Q]/X1+xubDF;
280xsDF =f[6,X1,Q]/X1;
281xcDF =f[5,X1,Q]/X1;
282xbDF =f[7,X1,Q]/X1;
283
284pd1={xgDF,xdDF,xdbDF,xuDF,xubDF,xsDF,xcDF,xbDF};
285xgDF =f[3,X2,Q]/X2;
286xdbDF=f[8,X2,Q]/X2;
287xdDF =f[2,X2,Q]/X2+xdbDF;
288xubDF=f[4,X2,Q]/X2;
289xuDF =f[1,X2,Q]/X2+xubDF;
290xsDF =f[6,X2,Q]/X2;
291xcDF =f[5,X2,Q]/X2;
292xbDF =f[7,X2,Q]/X2;
293pd2={xgDF,xdDF,xdbDF,xuDF,xubDF,xsDF,xcDF,xbDF};
294
295
296xgg=pd1[[1]]*pd2[[1]];
297xqg=pd1[[1]]*(pd2[[2]]+pd2[[3]]+pd2[[4]]+pd2[[5]]+2(pd2[[6]]+pd2[[7]]+pd2[[8]]\
298))+
299   pd2[[1]]*(pd1[[2]]+pd1[[3]]+pd1[[4]]+pd1[[5]]+2(pd1[[6]]+pd1[[7]]+pd1[[8]])\
300);
301xqq=pd1[[2]]*pd2[[3]]+pd1[[4]]*pd2[[5]]+pd1[[3]]*pd2[[2]]+pd1[[5]]*pd2[[4]]+
302   2(pd1[[6]] pd2[[6]]+pd1[[7]] pd2[[7]]+pd1[[8]] pd2[[8]]);
303 
304  Return[{xgg,xqg,xqq}];
305];\
306\>", "Input"],
307
308Cell["\<\
309
310(* cteq5 *)\
311\>", "Input"],
312
313Cell["\<\
314pdfCTEQ[X1_,X2_,q_]:=Module[
315{Q,pd1,pd2,xgg,xqg,xqq},
316Q=q*1.;
317
318pd1={cteq5pdf[1,0,X1,Q],
319     cteq5pdf[1,2,X1,Q],
320     cteq5pdf[1,-2,X1,Q],
321     cteq5pdf[1,1,X1,Q],
322     cteq5pdf[1,-1,X1,Q],
323     cteq5pdf[1,3,X1,Q],
324     cteq5pdf[1,4,X1,Q],
325     cteq5pdf[1,5,X1,Q]};
326pd2={cteq5pdf[1,0,X2,Q],
327     cteq5pdf[1,2,X2,Q],
328     cteq5pdf[1,-2,X2,Q],
329     cteq5pdf[1,1,X2,Q],
330     cteq5pdf[1,-1,X2,Q],
331     cteq5pdf[1,3,X2,Q],
332     cteq5pdf[1,4,X2,Q],
333     cteq5pdf[1,5,X2,Q]};
334     
335xgg=pd1[[1]]*pd2[[1]];
336xqg=pd1[[1]]*(pd2[[2]]+pd2[[3]]+pd2[[4]]+pd2[[5]]+2(pd2[[6]]+pd2[[7]]+pd2[[8]]\
337))+
338   pd2[[1]]*(pd1[[2]]+pd1[[3]]+pd1[[4]]+pd1[[5]]+2(pd1[[6]]+pd1[[7]]+pd1[[8]])\
339);
340xqq=pd1[[2]]*pd2[[3]]+pd1[[4]]*pd2[[5]]+pd1[[3]]*pd2[[2]]+pd1[[5]]*pd2[[4]]+
341   2(pd1[[6]] pd2[[6]]+pd1[[7]] pd2[[7]]+pd1[[8]] pd2[[8]]);
342 
343  Return[{xgg,xqg,xqq}];
344];
345
346pdfCTEQLO[X1_,X2_,q_]:=Module[
347{Q,xgDF,xdDF,xdbDF,xuDF,xubDF,xsDF,xcDF,xbDF,pd1,pd2,xgg,xqg,xqq},
348Q=q*1.;
349
350pd1={cteq5pdf[3,0,X1,Q],
351     cteq5pdf[3,2,X1,Q],
352     cteq5pdf[3,-2,X1,Q],
353     cteq5pdf[3,1,X1,Q],
354     cteq5pdf[3,-1,X1,Q],
355     cteq5pdf[3,3,X1,Q],
356     cteq5pdf[3,4,X1,Q],
357     cteq5pdf[3,5,X1,Q]};
358pd2={cteq5pdf[3,0,X2,Q],
359     cteq5pdf[3,2,X2,Q],
360     cteq5pdf[3,-2,X2,Q],
361     cteq5pdf[3,1,X2,Q],
362     cteq5pdf[3,-1,X2,Q],
363     cteq5pdf[3,3,X2,Q],
364     cteq5pdf[3,4,X2,Q],
365     cteq5pdf[3,5,X2,Q]};
366     
367xgg=pd1[[1]]*pd2[[1]];
368xqg=pd1[[1]]*(pd2[[2]]+pd2[[3]]+pd2[[4]]+pd2[[5]]+2(pd2[[6]]+pd2[[7]]+pd2[[8]]\
369))+
370   pd2[[1]]*(pd1[[2]]+pd1[[3]]+pd1[[4]]+pd1[[5]]+2(pd1[[6]]+pd1[[7]]+pd1[[8]])\
371);
372xqq=pd1[[2]]*pd2[[3]]+pd1[[4]]*pd2[[5]]+pd1[[3]]*pd2[[2]]+pd1[[5]]*pd2[[4]]+
373   2(pd1[[6]] pd2[[6]]+pd1[[7]] pd2[[7]]+pd1[[8]] pd2[[8]]);
374 
375  Return[{xgg,xqg,xqq}];
376];\
377\>", "Input"]
378}, Open  ]],
379
380Cell[CellGroupData[{
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391Alpha_S: Very basic implementation of Alpha_S.
392Check that the value of Lamda_4 or Lambda_5 is consistent with that of the \
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394One simple, but indirect way to do it is to compare the value of alphas(MZ) \
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498\
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509Cell["\<\
510Integral of the LO loop. The sign of the Imaginary part is given by the usual \
511prescription mt^2-i eps which is equivalent to t+i eps in the notation below.
512t=mh^2/4/mt^2;\
513\>", "Subsubsection"],
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519Cell["\<\
520eps=0.00000001;
521inte01[t_]=Integrate[(1-4 x y)/(1-4 t x y),{x,0,1},{y,0,1-x}, \
522Assumptions->{t>0, t<1}]//Simplify
523integt1[t_]=Integrate[(1-4 x y)/(1-4 t x y),{x,0,1},{y,0,1-x}, \
524Assumptions->{Im[t]>0, Re[t]>1}]//Simplify
525inte[t_]:=3*If[ Re[t]<1, inte01[t], integt1[t]]
526Plot[{Re[inte[t+I eps]],Im[inte[t+I \
527eps]]},{t,0,10},PlotStyle\[Rule]{{Blue,Thickness[0.007]},{Red,Thickness[0.007]\
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846   FrameLabel->{{None, None}, {None, None}},
847   FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
848   GridLines->{None, None},
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851   ImagePadding->All,
852   Method->{
853    "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" ->
854     AbsolutePointSize[6], "ScalingFunctions" -> None},
855   PlotRange->{{0, 10}, {0., 1.7384634940975814`}},
856   PlotRangeClipping->True,
857   PlotRangePadding->{{
858      Scaled[0.02],
859      Scaled[0.02]}, {
860      Scaled[0.05],
861      Scaled[0.05]}},
862   Ticks->{Automatic, Automatic}], TraditionalForm]], "Output",
863 CellChangeTimes->{
864  3.692598901668117*^9, 3.692598959121788*^9, {3.692599021175414*^9,
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867}, Open  ]],
868
869Cell["\<\
870Plot of the real and imaginary part of 3*I(a) as defined in the notes (I use \
8713*I(a) so that the function goes to 1 as a->0.)\
872\>", "Text"]
873}, Open  ]],
874
875Cell[CellGroupData[{
876
877Cell[TextData[{
878 "Function to be integrated to get the LO cross section. In general, we use \
879the convention that all functions to be numerically integrated in the \
880hypercubes ",
881 Cell[BoxData[
882  FormBox[
883   SuperscriptBox[
884    RowBox[{"[",
885     RowBox[{"0", ",", "1"}], "]"}], "d"], TraditionalForm]]],
886 ", where d is the dimension of the integration space. dsigma depends on the \
887Higgs mass (mh), the total collider energy (sqrtS) and from the arbitrary \
888normalization and factorization scales (mur,muf)."
889}], "Subsubsection"],
890
891Cell["\<\
892dsigmaLO[zz_,mh_,sqrtS_,muf_,mur_]:=Module[
893{y,x1,x2,gg0,qg0,qq0,s0,ymin,ymax,JAC,v,S,tau0},
894Muf=muf*1.;
895v=246.;
896S=sqrtS^2;
897tau0=mh^2/S;
898ymax=-Log[Sqrt[tau0]];
899ymin=-ymax;
900y=ymin+(ymax-ymin)*zz;
901JAC=ymax-ymin;
902x1=Sqrt[tau0] Exp[y];
903x2=Sqrt[tau0] Exp[-y];
904{gg0,qg0,qq0}=pdfCTEQLO[x1,x2,Muf];
905s0=asLO[mur,5]^2/576/Pi/v^2*tau0;
906s0=s0*gg0;
907s0=s0*389379660; (*to picobarns*)
908s0=s0*JAC;
909Return[s0];
910];\
911\>", "Input"],
912
913Cell[CellGroupData[{
914
915Cell["sLO=NIntegrate[dsigmaLO[xx,100,14000,100,100],{xx,0,1}]", "Input"],
916
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921}, Open  ]]
922}, Open  ]],
923
924Cell[CellGroupData[{
925
926Cell["\<\
927Get the cross section for various Higgs masses and including the form factor \
928of the loop.
929I first build a table with the results of the cross section in picobarns, and \
930then plot it.\
931\>", "Subsection"],
932
933Cell["\<\
934resEFT=Table[{i,NIntegrate[dsigmaLO[x,i*1.,14000.,i*1.,i*1.],{x,0,1}]},{i,20,\
935600,10}];\
936\>", "Input"],
937
938Cell["\<\
939resFULL=Table[{resEFT[[i]][[1]],Abs[inte[resEFT[[i]][[1]]^2/175^2/4]]^2*\
940resEFT[[i]][[2]]},{i,1,Length[resEFT]}];\
941\>", "Input"],
942
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1563Cross section (pb) as a function of the Higgs mass. This plot shows how well \
1564the EFT does. For a Higgs<200 GeV the approximation is very good.\
1565\>", "Text"]
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1575
1576uno=Integrate[x1*x2,{x1,t0,1},{x2,t0/x1,1}]//Expand
1577due=Integrate[ta,{ta,t0,1},{y,Log[Sqrt[ta]],-Log[Sqrt[ta]]}]//PowerExpand//\
1578Expand
1579tre=Integrate[t0/z \
1580t0/z^2,{z,t0,1},{y,Log[Sqrt[t0/z]],-Log[Sqrt[t0/z]]}]//PowerExpand//Expand\
1581\>", "Input"],
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1735Cell["NLO cross section", "Section"],
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1737Cell["\<\
1738We have integrated over the angular variables, so we are only left with two \
1739integrations. One is over z (=mh/S/x1/x2) and the other over the rapidity y \
1740of the partonic cms. For every point in the phase space we have to calculate \
1741an event and a corresponding counter-event with z=1 to implement the + \
1742distributions in the gg channel. Various contributions add to the final result:
1743
1744virtual: gg>h  at 1-loop + corrections to the effective lagrangian (UV and IR \
1745divergent)
1746real: qq~ >h g (finite)
1747           qg > qh (collinear divergent)
1748            gg>gh  (soft and collinear divergent)
1749\
1750\>", "Text",
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1758Cell[TextData[{
1759 "dsigmaBV[yy_,mh_,sqrtS_,muf_,mur_]:=Module[\n",
1760 StyleBox["(* local variables *)",
1761  FontColor->RGBColor[0, 1, 0]],
1762 "\n{y0,x10,x20,sig0,ymax0,JAC0,v,S,tau0,beta,gg0,qg0,qq0},\nMuf=muf 1.;\n\
1763Mur=mur 1.;\nv=246.;\nS=sqrtS^2;\ntau0=mh^2/S;\nbeta=Sqrt[1-tau0];\n\n",
1764 StyleBox["(* calculate quantities for z=1 *)",
1765  FontColor->RGBColor[1, 0, 0]],
1766 "\nymax0=-Log[Sqrt[tau0]];\ny0=-ymax0+2*ymax0*yy;\nJAC0=2*ymax0;\n\
1767x10=Sqrt[tau0] Exp[y0];\nx20=Sqrt[tau0] Exp[-y0];\n\
1768{gg0,qg0,qq0}=pdfcall[x10,x20,Muf];\n\n",
1769 StyleBox["(* sigma0 *)",
1770  FontColor->RGBColor[1, 0, 0]],
1771 "\nsig0=asNLO[Mur,5]^2/576/Pi/v^2*tau0;\nsig0=sig0+sig0*asNLO[Mur,5]/2/Pi*\n\
1772(11/3 CA+ 2 Pi^2 - 2 b0 2 Log[Muf/Mur]+\n16 CA Log[beta] Log[mh/Muf]+16 CA \
1773Log[beta]^2);\nsig0=sig0*gg0;\nsig0=sig0*389379660; (*to picobarns*)\n\
1774sig0=sig0*JAC0;\n\n\
1775(*Print[{muf,mur,v,S,tau0,beta,ymax0,y0,JAC0,x10,x20,gg0,qg0,qq0,sig0}];*)\n\
1776Return[sig0];\n];"
1777}], "Input"]
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1782Cell["Real contributions", "Subsection"],
1783
1784Cell[TextData[{
1785 "dsigmaR[xx_,yy_,mh_,sqrtS_,muf_,mur_]:=Module[\n",
1786 StyleBox["(* local variables *)",
1787  FontColor->RGBColor[0, 1, 0]],
1788 "\n{v,S,tau0,\n y,y0,z,tau,\n x1,x2,x10,x20,\n ymax0,ymax,\n JAC,JAC0,\n \
1789gg,qg,qq,gg0,qg0,qq0,\n qqterm,qgterm,ggterm,ggterm0,\n sig,sig0,\n Muf},\n\n\
1790v=246.;\nS=sqrtS^2;\ntau0=mh^2/S;\nMuf=muf*1.0;\n\n",
1791 StyleBox["(* calculate quantities for an event *)",
1792  FontColor->RGBColor[1, 0, 0]],
1793 "\nz=tau0+(1-tau0)*xx;\ntau=tau0/z;\nymax =-Log[Sqrt[tau]];\n\
1794y=-ymax+2*ymax*yy;\nJAC =2*ymax*(1-tau0)*tau0/z^2;\nx1=Sqrt[tau] Exp[y];\n\
1795x2=tau/x1;\n",
1796 StyleBox["(* call the pdf *)\n",
1797  FontColor->RGBColor[1, 0, 0]],
1798 "\n{gg,qg,qq}=pdfcall[x1,x2,Muf];\n\n",
1799 StyleBox["(* calculate quantities for counter-event *)",
1800  FontColor->RGBColor[1, 0, 0]],
1801 "\nymax0=-Log[Sqrt[tau0]];\ny0=-ymax0+2*ymax0*yy;\n\
1802JAC0=2*ymax0*(1-tau0)*tau0;\nx10=Sqrt[tau0] Exp[y0];\nx20=tau0/x10;\n",
1803 StyleBox["(* call the pdf at z=1 *)",
1804  FontColor->RGBColor[1, 0, 0]],
1805 "\n{gg0,qg0,qq0}=pdfcall[x10,x20,Muf];\n\n",
1806 StyleBox["(* sigma0 *)",
1807  FontColor->RGBColor[1, 0, 0]],
1808 "\nsig0=asNLO[mur,5]^2/576/Pi/v^2;\nsig0=sig0*asNLO[mur,5]/2/Pi;\n\n",
1809 StyleBox["(* qq channnel : no counter event *)",
1810  FontColor->RGBColor[1, 0, 0]],
1811 "\nqqterm=64/27*(1-z)^3;\nqqterm=qqterm*sig0*JAC*qq;\n\n",
1812 StyleBox["(* qg channnel : no counter event *)",
1813  FontColor->RGBColor[1, 0, 0]],
1814 "\nqgterm=CF*( (1+(1-z)^2)/z (2*Log[mh/muf]+2 Log[1-z]-Log[z])\n           \
1815+(z^2-3/2(1-z)^2)/z )*z;\nqgterm=qgterm*sig0*JAC*qg;\n\n",
1816 StyleBox["(* gg channnel *)",
1817  FontColor->RGBColor[1, 0, 0]],
1818 "\nggterm=CA*(2 (2 (z/(1-z)+(1-z)/z+z (1-z) )) * (2*Log[mh/muf])-\n       \
181911/3 (1-z)^3/z -\n       4 (1-z+z^2)^2/z/(1-z) Log[z]+\n       8 \
1820(1-z+z^2)^2/z Log[1-z]/(1-z) )*z;\nggterm=ggterm*sig0*JAC*gg;\n",
1821 StyleBox["(* gg counter-event *)",
1822  FontColor->RGBColor[1, 0, 0]],
1823 "\nggterm0=CA*(-4/(1-z) 2*Log[mh/muf] - 8*Log[1-z]/(1-z) );\n\
1824ggterm0=ggterm0*sig0*JAC0*gg0;\n\n\n",
1825 StyleBox["(* total *)",
1826  FontColor->RGBColor[1, 0, 0]],
1827 "\nsig=0;\nsig=sig+qqterm;\nsig=sig+qgterm;\nsig=sig+ggterm+ggterm0;\n\
1828sig=sig*389379660; (*to picobarns*)\n\nReturn[sig];\n\n];\n"
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1835virt\
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1844eps=0.0000000001;
1845(* real=Vegas[dsigmaR[xvar,yvar,100,14000,100,100],{xvar,eps,1-eps},{yvar,eps,\
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1879integrand evaluations. NIntegrate obtained \
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