GIFSchool: HiggsGG-NLO.nb

File HiggsGG-NLO.nb, 38.8 KB (added by anonymous, 6 years ago)
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98
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477Cell["\<\
478-----I take all momenta outgoing 
479
480     p1 + p2 + p3 + p4 = 0     
481     
482     p1^2=0
483     p2^2=0
484     p3^3=0
485     p4^2=mh^2
486     
487-----invariants
488
489     s  = (p1 + p2)^2 = (p3 + p4)^2=2 p1.p2=mh^2+2 p3.p4
490     t  = (p1 + p3)^2 = (p2+  p4)^2 =2 p1.p3=mh^2+2 p2.p4
491     u  = (p2 + p3)^2 = (p1 + p4)^2=2 p1.p4+mh^2=+2 p2.p3
492   
493     s + t + u = mh^2
494   
495     \[Sigma]+\[Tau]+\[Upsilon]=1
496       
497 -----scalar products   
498     
499     p1.p2=s/2
500     p1.p3=(t)/2
501     p1.p4= (u-mh^2)/2
502     p2.p3= (u)/2
503     p2.p4=(t-mh^2)/2
504     p3.p4=(s-mh^2)/2
505   
506-----physical region for production   p1+p2=-p3-p4
507
508      s>(mh)^2 ;
509      t<0;
510      u<0;
511         
512\
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518    \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]",
519    \(\(ScalarProduct[p3, p3] = 0;\)\), "\[IndentingNewLine]",
520    \(\(ScalarProduct[q, p3] = u/2;\)\  (*q =
521        p2 + p3*) \), "\[IndentingNewLine]",
522    \(\(ScalarProduct[q, p2] = u/2;\)\  (*q =
523        p2 + p3*) \), "\[IndentingNewLine]",
524    \(\(ScalarProduct[q, q] = u;\)\  (*q =
525        p2 + p3*) \), "\[IndentingNewLine]",
526    \(\(ScalarProduct[p4, p4] = mh2;\)\), "\[IndentingNewLine]",
527    \(\(ScalarProduct[p1, p2] = s/2;\)\), "\[IndentingNewLine]",
528    \(\(ScalarProduct[p1, p3] = t/2;\)\), "\[IndentingNewLine]",
529    \(\(ScalarProduct[p1, p4] = \((u - mh2)\)/2;\)\), "\[IndentingNewLine]",
530    \(\(ScalarProduct[p2, p3] = u/2;\)\), "\[IndentingNewLine]",
531    \(\(ScalarProduct[p2, p4] = \ \((t - mh2)\)/
532          2;\)\), "\[IndentingNewLine]",
533    \(\(ScalarProduct[p3, p4] = \((s - mh2)\)/
534          2\ \ ;\)\), "\[IndentingNewLine]",
535    \(\(ScalarProduct[p3, e3] = 0\ ;\)\), "\[IndentingNewLine]",
536    \(\(ScalarProduct[p1, e1] = 0\ ;\)\), "\[IndentingNewLine]",
537    \(\(ScalarProduct[p2, e2] = 0\ ;\)\), "\[IndentingNewLine]",
538    \(\(s13 = t;\)\), "\[IndentingNewLine]",
539    \(\(s23 = u;\)\)}], "Input",
540  PageWidth->PaperWidth]
541}, Open  ]],
542
543Cell[CellGroupData[{
544
545Cell["Verteces and Propagators", "Subsection",
546  PageWidth->PaperWidth],
547
548Cell[TextData[{
549  StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \
550outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ",
551    FontSize->14],
552  Cell[BoxData[
553      \(TraditionalForm\`g\_s\)],
554    FontSize->14],
555  StyleBox[" ) (\[ImaginaryI] ",
556    FontSize->14],
557  Cell[BoxData[
558      \(TraditionalForm\`f\^abc\)],
559    FontSize->14],
560  StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ",
561    FontSize->14],
562  Cell[BoxData[
563      \(TraditionalForm\`g\_s\)],
564    FontSize->14],
565  ")",
566  StyleBox[" ",
567    FontSize->14],
568  Cell[BoxData[
569      \(TraditionalForm\`\((T\^a)\)\_ij\)],
570    FontSize->14],
571  StyleBox[" ",
572    FontSize->14],
573  Cell[BoxData[
574      \(TraditionalForm\`\[Gamma]\^\[Mu]\)],
575    FontSize->14],
576  StyleBox["\nGluon Propagator= ",
577    FontSize->14],
578  Cell[BoxData[
579      FormBox[
580        FractionBox[
581          StyleBox[\(\(-\[ImaginaryI]\)\ g\^\[Mu]\[Nu]\),
582            FontSize->16], \(p\^2\)], TraditionalForm]],
583    FontSize->14],
584  "\nQuark ",
585  StyleBox["Propagator= ",
586    FontSize->14],
587  Cell[BoxData[
588      FormBox[
589        FractionBox[
590          StyleBox[\(\(\[ImaginaryI]\)\(\ \)\),
591            FontSize->16], \(p\&^\)], TraditionalForm]],
592    FontSize->14]
593}], "Text",
594  PageWidth->PaperWidth],
595
596Cell[BoxData[{
597    \(\(GGGD[p1_, p2_, p3_, m1_, m2_, m3_] :=
598        FourVector[p1 - p2, m3, \ Dimension\  \[Rule] D]\ MTD[m1, m2] +
599          FourVector[p2 - p3, m1, \ Dimension\  \[Rule] \ D]\ MTD[m2, m3] +
600          FourVector[p3 - p1, m2, \ Dimension\  \[Rule] \ D]\ MTD[m1,
601              m3];\)\), "\[IndentingNewLine]",
602    \(\(H[p1_, p2_, m1_, m2_] :=
603        MTD[m1, m2]\ SPD[p1, p2]\  - \
604          FourVector[p1, m2, \ Dimension \[Rule] D]*\[IndentingNewLine]\
605            FourVector[p2, m1, \
606              Dimension\  \[Rule] D];\)\), "\[IndentingNewLine]",
607    \(\(PropQuark = I;\)\), "\[IndentingNewLine]",
608    \(\(PropGluon\  = \(-I\);\)\), "\[IndentingNewLine]",
609    \(\(vtx = \(-I\)\ gs;\)\)}], "Input",
610  PageWidth->PaperWidth]
611}, Open  ]],
612
613Cell[CellGroupData[{
614
615Cell["Sum over the  four Feynman diagrams", "Subsection"],
616
617Cell["\<\
618
619uno= - gs (-f123) GGGD[p1,p2,-p1-p2,m1,m2,mu]  (-I MTD[mu,nu]/s) (I A) \
620H[p3,p1+p2,m3,nu]//Contract;
621tre= - gs (-f123) GGGD[p3,p1,-p3-p1,m3,m1,mu] (-I MTD[mu,nu]/t) (I A)  \
622H[p2,p1+p3,m2,nu]//Contract;
623qua= - gs (-f123) GGGD[p2,p3,-p3-p2,m2,m3,mu] (-I MTD[mu,nu]/u) (I A)  \
624H[p1,p2+p3,m1,nu]//Contract;
625due= -  A gs (-f123) GGGD[p1,p2,p3,m1,m2,m3];
626res=uno+due+tre+qua//ExpandScalarProduct;\
627\>", "Input"]
628}, Open  ]],
629
630Cell[CellGroupData[{
631
632Cell["\<\
633Now I have to square the amplitude.  The sum is performed over the physical \
634polarizations  of the gluons:\
635\>", "Subsection"],
636
637Cell["\<\
638res1=res;
639res2=res /. m1->m1p /. m2->m2p /. m3-> m3p;
640tot=res1*(-MTD[m1,m1p]+(FVD[p1,m1] FVD[p2,m1p]+FVD[p1,m1p] \
641FVD[p2,m1])/SPD[p1,p2])//Contract;
642tot=tot*(-MTD[m2,m2p]+(FVD[p2,m2] FVD[p3,m2p]+FVD[p2,m2p] \
643FVD[p3,m2])/SPD[p3,p2])//Contract;
644tot=tot*(-MTD[m3,m3p]+(FVD[p1,m3] FVD[p3,m3p]+FVD[p1,m3p] \
645FVD[p3,m3])/SPD[p3,p1])//Contract;
646tot=tot*res2//Contract//Expand;\
647\>", "Input"],
648
649Cell[CellGroupData[{
650
651Cell["\<\
652Amp2=((tot//Contract//Factor)/. A-> as/3/Pi/v /. gs^2-> 4 Pi as/. f123^2-> 24 \
653//Simplify)//Factor\
654\>", "Input"],
655
656Cell[BoxData[
657    \(TraditionalForm\`\(\(1\/\(3\ \[Pi]\ s\ t\ u\ v\^2\)\)\((32\ as\^3\ \((D\
658\ s\^4 - 2\ s\^4 + 2\ D\ t\ s\^3 - 4\ t\ s\^3 + 2\ D\ u\ s\^3 - 4\ u\ s\^3 +
659            3\ D\ t\^2\ s\^2 - 6\ t\^2\ s\^2 + 3\ D\ u\^2\ s\^2 -
660            6\ u\^2\ s\^2 + 8\ D\ t\ u\ s\^2 - 20\ t\ u\ s\^2 +
661            2\ D\ t\^3\ s - 4\ t\^3\ s + 2\ D\ u\^3\ s - 4\ u\^3\ s +
662            8\ D\ t\ u\^2\ s - 20\ t\ u\^2\ s + 8\ D\ t\^2\ u\ s -
663            20\ t\^2\ u\ s + D\ t\^4 - 2\ t\^4 + D\ u\^4 - 2\ u\^4 +
664            2\ D\ t\ u\^3 - 4\ t\ u\^3 + 3\ D\ t\^2\ u\^2 - 6\ t\^2\ u\^2 +
665            2\ D\ t\^3\ u - 4\ t\^3\ u)\))\)\)\)], "Output"]
666}, Open  ]],
667
668Cell["\<\
669
670Amp2=Amp2 /. D->4-2e//Simplify;\
671\>", "Input"]
672}, Open  ]],
673
674Cell[CellGroupData[{
675
676Cell["Real Amplitude Squared in D dimensions", "Subsection",
677  PageWidth->PaperWidth],
678
679Cell[BoxData[
680    \(\(\(\[IndentingNewLine]\)\(\(Emme =
681        1/s \(\(\((\((mh2^4 + s^4 + t^4 + u^4)\)\ \((1 - 2\ e)\) + \
682                    e/2\ \((mh2^2 + s^2 + t^2 + u^2)\)^2)\)/s\)/t\)/
683            u;\)\[IndentingNewLine]
684    \(RealD = Emme/\((1 - e)\)^2;\)\[IndentingNewLine]
685    \(Real4 = RealD /. \ e \[Rule] 0;\)\)\)\)], "Input",
686  PageWidth->PaperWidth]
687}, Open  ]],
688
689Cell[CellGroupData[{
690
691Cell["Phase space in D dimensions", "Subsection",
692  PageWidth->PaperWidth],
693
694Cell[BoxData[{
695    \(\[IndentingNewLine]\(PS = \(1\/\(8  \[Pi]\)\) \(\((\(\(4\)\(\ \)\(\[Pi]\
696\)\(\ \)\)\/mh2)\)\^e\)
697          1\/Gamma[1 - e]\ \((mh2\/s)\)\^e\ \ \((1 - mh2\/s)\)\^\(1 - 2  e\)\ \
698v\^\(-e\)\ \((omv)\)\^\(-e\);\)\), "\[IndentingNewLine]",
699    \(\(substu = {t \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) \((omv)\), \
700          u \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) v\ ,
701          s \[Rule] \ mh2/z};\)\), "\[IndentingNewLine]",
702    \(\(cGamma = \((1/16)\)/Pi^2\ mh2^\((\(-e\))\) \((4\ Pi)\)^e\ Gamma[
703            1 + e]\ Gamma[1 - e]^2/Gamma[1 - 2\ e];\)\), "\n",
704    \(\(pgg =
705        2 \((z\ PlusDistribution[1/\((1 - z)\)] + \((1 - z)\)/z +
706              z \((1 - z)\) +
707              11/12\ DeltaFunction[1 - z])\);\)\), "\[IndentingNewLine]",
708    \(\(s0 = z;\)\)}], "Input",
709  PageWidth->PaperWidth]
710}, Closed]],
711
712Cell[CellGroupData[{
713
714Cell["CDR", "Subsection",
715  PageWidth->PaperWidth],
716
717Cell[CellGroupData[{
718
719Cell[BoxData[{
720    \(intando =
721      FullSimplify[
722          FullSimplify[\(RealD\ PS\ c\[CapitalGamma]\)\/cGamma] \
723//. \[InvisibleSpace]substu] /. \[InvisibleSpace]omv \[Rule]
724          1 - v; \), "\n",
725    \(intando = intando\/\((1 - z)\)\^\(\(-2\)\ e - 1\)\)}], "Input"],
726
727Cell[BoxData[
728    \(TraditionalForm\`\(-\(\((c\[CapitalGamma]\ \((1\/mh2)\)\^e\ mh2\^e\ \
729\[Pi]\ \((1 - v)\)\^\(\(-e\) - 1\)\ v\^\(\(-e\) - 1\)\ z\^e\ \((e\ \((3\ \((1 \
730- v)\)\^4\ \((z - 1)\)\^4 + 3\ v\^4\ \((z - 1)\)\^4 -
731                      2\ v\^2\ \((z\^2 + 1)\)\ \((z - 1)\)\^2 -
732                      2\ \((1 - v)\)\^2\ \((v\^2\ \((z - 1)\)\^2 + z\^2 +
733                            1)\)\ \((z - 1)\)\^2 + 3\ z\^4 - 2\ z\^2 + 3)\) -
734                2\ \((\((1 - v)\)\^4\ \((z - 1)\)\^4 + v\^4\ \((z - 1)\)\^4 +
735                      z\^4 + 1)\))\)\ \(\[CapitalGamma](
736              1 - 2\ e)\))\)/\((\((e - 1)\)\^2\ \(\[CapitalGamma](1 - e)\)\^3\
737\ \(\[CapitalGamma](e + 1)\))\)\)\)\)], "Output"]
738}, Open  ]],
739
740Cell[BoxData[{
741    \(\(\[Sigma]r =
742        Integrate[intando, {v, 0, 1}, \ GenerateConditions \[Rule] False] //
743          PowerExpand;\)\), "\n",
744    \(\(\[Sigma]r = \(Normal[Series[\[Sigma]r, {e, 0, 1}]] // Factor\) //
745          FullSimplify;\)\), "\n",
746    \(\(\[Sigma]r\  = \[Sigma]r\ \ *\((\(\(-1\)\/\(2  e\)\)
747                DeltaFunction[1 - z] + PlusDistribution[1/\((1 - z)\)] -
748              2\ e\ PlusDistribution[Log[1 - z]/\((1 - z)\)])\);\)\), "\n",
749    \(\(\[Sigma]r =
750        Normal[Series[\[Sigma]r/\((1 + e + e^2)\), {e, 0, 0}]] //
751          Expand;\)\)}], "Input",
752  PageWidth->PaperWidth]
753}, Closed]]
754}, Closed]],
755
756Cell[CellGroupData[{
757
758Cell["Results", "Section"],
759
760Cell[CellGroupData[{
761
762Cell["Virtual+Real+Counterterms  in CDR", "Subsection"],
763
764Cell[BoxData[{
765    \(\(\[Sigma]rn = \(\(\[Sigma]r/c\[CapitalGamma]\)/\[Pi]\)/2\ CA -
766          1/e\ 2\ Pgg\ z\ CA + 1/e\ 2\ pgg\ z\ CA;\)\), "\n",
767    \(\(\[Sigma]rn = \((\((\((Coefficient[\[Sigma]rn, DeltaFunction[1 - z]] /.
768                      z \[Rule] 1 // FullSimplify)\)*
769                DeltaFunction[
770                  1 - z])\) + \((\((\((Coefficient[\[Sigma]rn,
771                        PlusDistribution[1/\((1 - z)\)]] //
772                      FullSimplify)\))\)
773                PlusDistribution[
774                  1/\((1 - z)\)])\) + \((\((\((Coefficient[\[Sigma]rn,
775                        PlusDistribution[Log[1 - z]/\((1 - z)\)]] //
776                      FullSimplify)\))\)
777                PlusDistribution[
778                  Log[1 - z]/\((1 - z)\)])\) + \((\(\(\[Sigma]rn /.
779                    DeltaFunction[x_] \[Rule] 0\) /.
780                  PlusDistribution[Log[1 - z]/\((1 - z)\)] \[Rule] 0\) /.
781                PlusDistribution[1/\((1 - z)\)] \[Rule] 0)\))\);\)\), "\n",
782    \(\(\[Sigma]rn = \((\((\((Coefficient[\[Sigma]rn,
783                          1/e\ PlusDistribution[1/\((1 - z)\)]]/\((1 - z)\) //
784                      FullSimplify)\))\) 1/e)\) + \((\[Sigma]rn -
785                Coefficient[\[Sigma]rn, 1/e\ PlusDistribution[1/\((1 - z)\)]]
786                  1/e\ PlusDistribution[1/\((1 - z)\)])\) //
787          Expand;\)\)}], "Input",
788  PageWidth->PaperWidth],
789
790Cell[CellGroupData[{
791
792Cell[BoxData[
793    \(divD =
794      Coefficient[\[Sigma]rn,
795          DeltaFunction[1 - z]]*\(as/2\)/\[Pi]\ c\[CapitalGamma]\)], "Input"],
796
797Cell[BoxData[
798    \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(11\ C\_A\)\/\(3\ e\) + \
799\(2\ C\_A\)\/e\^2 - \(\[Pi]\^2\ C\_A\)\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"]
800}, Open  ]],
801
802Cell[CellGroupData[{
803
804Cell[BoxData[
805    \(UVC = \ \(\(1/
806          e\)\(\ \)\(4\)\(*\)\(\(-11\)\/6\) \(CA\)\(\ \)\(as\/\(4  \[Pi]\)\) \
807\(c\[CapitalGamma]\)\(\ \)\)\)], "Input"],
808
809Cell[BoxData[
810    \(TraditionalForm\`\(-\(\(11\ as\ C\_A\ c\[CapitalGamma]\)\/\(6\ e\ \[Pi]\
811\)\)\)\)], "Output"]
812}, Open  ]],
813
814Cell[CellGroupData[{
815
816Cell[BoxData[
817    \(\((\(\((divD + UVC)\)/as\)/c\[CapitalGamma]\ 2\ \[Pi] //
818          Expand)\)\ as\ \(c\[CapitalGamma]/2\)/\[Pi]\)], "Input"],
819
820Cell[BoxData[
821    \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(2\ C\_A\)\/e\^2 - \(\[Pi]\
822\^2\ C\_A\)\/3 + C\_A\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"]
823}, Open  ]],
824
825Cell[BoxData[
826    \(\(sally =
827        as/\[Pi]\ \((\(-11\)/2\ \((1 - z)\)^3 +
828              6\ \((1 + z^4 + \((1 - z)\)^4)\)\ PlusDistribution[
829                  Log[1 - z]/\((1 - z)\)] -
830              6\ \((z^2\ PlusDistribution[1/\((1 - z)\)] + \((1 - z)\) +
831                    z^2 \((1 - z)\))\) Log[z])\);\)\)], "Input"],
832
833Cell[BoxData[
834    \(\((\(\(\((\[Sigma]rn*\(as/2\)/\[Pi] /. DeltaFunction[x_] \[Rule] 0)\) -
835                    sally /. CA \[Rule] 3\) /. Pgg \[Rule] 0 // Expand\) //
836            Simplify)\) /.
837        PlusDistribution[1/\((1 - z)\)] \[Rule] 1/\((1 - z)\) //
838      Factor\)], "Input"]
839}, Open  ]]
840}, Closed]],
841
842Cell[CellGroupData[{
843
844Cell["Gluon initiated process", "Section",
845  PageWidth->PaperWidth],
846
847Cell[CellGroupData[{
848
849Cell["Kinematics for the Born", "Subsection",
850  PageWidth->PaperWidth],
851
852Cell["\<\
853-----I take all momenta outgoing 
854
855     p1 + p2 + p3  = 0     
856     
857     p1^2=0
858     p2^2=0
859     p3^3=Q^2
860     
861
862        \
863\>", "Text",
864  PageWidth->PaperWidth],
865
866Cell[BoxData[{
867    \(\(ScalarProduct[p1, p1] = 0;\)\), "\[IndentingNewLine]",
868    \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]",
869    \(\(ScalarProduct[p3, p3] = Q2;\)\), "\[IndentingNewLine]",
870    \(\(ScalarProduct[p1, p3] = \(-Q2\)/2;\)\), "\[IndentingNewLine]",
871    \(\(ScalarProduct[p1, p2] = Q2/2;\)\), "\[IndentingNewLine]",
872    \(\(ScalarProduct[p2, p3] = \(-\ Q2\)/2;\)\), "\[IndentingNewLine]",
873    \(\(ScalarProduct[p1, e1] = 0;\)\), "\[IndentingNewLine]",
874    \(\(ScalarProduct[p2, e2] = 0;\)\), "\[IndentingNewLine]",
875    \(\(ScalarProduct[p1, e2] = 0;\)\), "\[IndentingNewLine]",
876    \(\(\(ScalarProduct[p2, e1] = 0;\)\(\[IndentingNewLine]\)
877    \)\), "\[IndentingNewLine]",
878    \(\(ScalarProduct[p1, p1, Dimension \[Rule] D] =
879        0;\)\), "\[IndentingNewLine]",
880    \(\(ScalarProduct[p2, p2, Dimension \[Rule] D] =
881        0;\)\), "\[IndentingNewLine]",
882    \(\(ScalarProduct[p3, p3, Dimension \[Rule] D] =
883        Q2;\)\), "\[IndentingNewLine]",
884    \(\(ScalarProduct[p1, p3, Dimension \[Rule] D] = \(-Q2\)/
885          2;\)\), "\[IndentingNewLine]",
886    \(\(ScalarProduct[p1, p2, Dimension \[Rule] D] =
887        Q2/2;\)\), "\[IndentingNewLine]",
888    \(\(ScalarProduct[p2, p3, Dimension \[Rule] D] = \(-\ Q2\)/
889          2;\)\), "\[IndentingNewLine]",
890    \(\(ScalarProduct[p1, e1, Dimension \[Rule] D] =
891        0;\)\), "\[IndentingNewLine]",
892    \(\(ScalarProduct[p2, e2, Dimension \[Rule] D] =
893        0;\)\), "\[IndentingNewLine]",
894    \(\(ScalarProduct[p1, e2, Dimension \[Rule] D] =
895        0;\)\), "\[IndentingNewLine]",
896    \(\(ScalarProduct[p2, e1, Dimension \[Rule] D] =
897        0;\)\), "\[IndentingNewLine]",
898    \(\)}], "Input",
899  PageWidth->PaperWidth]
900}, Open  ]],
901
902Cell[CellGroupData[{
903
904Cell["Real Amplitude", "Subsection",
905  PageWidth->PaperWidth],
906
907Cell[BoxData[{
908    \(\(Emme =
909        CF\ 1/s \((\((s^2 + u^2)\)\  - e\ \((s + u)\)^2)\)/
910            t;\)\), "\[IndentingNewLine]",
911    \(\(RealD = Emme/\((1 - e)\);\)\), "\[IndentingNewLine]",
912    \(\(Real4 = RealD /. \ e \[Rule] 0;\)\)}], "Input"]
913}, Open  ]],
914
915Cell[CellGroupData[{
916
917Cell["Integration over the phase space in D dimensions", "Subsection",
918  PageWidth->PaperWidth],
919
920Cell[BoxData[{
921    \(\(PS = \(1\/\(8  \[Pi]\)\) \(\((\(\(4\)\(\ \)\(\[Pi]\)\(\ \
922\)\)\/mh2)\)\^e\)
923          1\/Gamma[1 - e]\ \((mh2\/s)\)\^e\ \ \((1 - mh2\/s)\)\^\(1 - 2  e\)\ \
924v\^\(-e\)\ \((1 - v)\)\^\(-e\);\)\), "\[IndentingNewLine]",
925    \(\(substu = {t \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) \((1 - v)\), \
926          u \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) v\ ,
927          s \[Rule] \ mh2/z};\)\), "\[IndentingNewLine]",
928    \(\(cGamma = \((1/16)\)/Pi^2\ mh2^\((\(-e\))\) \((4\ Pi)\)^e\ Gamma[
929            1 + e]\ Gamma[1 - e]^2/Gamma[1 - 2\ e];\)\), "\n",
930    \(\(pgq =
931        CF \((\((1 - z)\)\^2 + 1)\)/z\  //
932          Factor;\)\), "\[IndentingNewLine]",
933    \(\)}], "Input",
934  PageWidth->PaperWidth],
935
936Cell[CellGroupData[{
937
938Cell["dim-reg", "Subsubsection",
939  PageWidth->PaperWidth],
940
941Cell[CellGroupData[{
942
943Cell[BoxData[{
944    \(\(\[Sigma]r =
945        Integrate[\((RealD\ PS\ c\[CapitalGamma]/cGamma\  //
946                    Simplify)\) //. \ substu // Expand, {v, 0, 1}, \
947            GenerateConditions \[Rule] False] // Simplify;\)\), "\n",
948    \(\(\[Sigma]r =
949        Normal[Series[\[Sigma]r/\((1 + \ e)\), {e, 0, 0}]] /. \
950            gs^2\  \[Rule] \ as\ 4\ \[Pi]\  // FullSimplify;\)\), "\n",
951    \(\(\[Sigma]r = \(\(\(-\[Sigma]r\)/c\[CapitalGamma]\)/\[Pi]\)/2 -
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960      2\ C\_F\ \(log(z)\)\ z - \(3\ C\_F\)\/2 + 4\ C\_F\ \(log(1 - z)\) -
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