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= "Anel El -> Z Z, EW, total cross section, tree" ;If [ $FrontEnd === Null , = False ; Print [ description] ; ] ;If [ $Notebooks === False , = False ] ;= { "FeynArts" } ;= 0 ; [ 9 , 3 , 1 ] ;FeynCalc 10.0.0 (dev version, 2023-12-20 22:40:59 +01:00, dff3b835). For help, use the online documentation ‾ , check out the wiki ‾ or visit the forum . ‾ \text{FeynCalc }\;\text{10.0.0 (dev
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patched for use with FeynCalc, for documentation see the
}\underline{\text{manual}}\;\text{ or visit
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∙ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260 \text{ $\bullet $ T. Hahn, Comput. Phys.
Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260} ∙ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260
Generate Feynman diagrams
Nicer typesetting
MakeBoxes [ p1, TraditionalForm ] := " \!\(\* SubscriptBox[ \( p \) , \( 1 \) ] \) " ;MakeBoxes [ p2, TraditionalForm ] := " \!\(\* SubscriptBox[ \( p \) , \( 2 \) ] \) " ;MakeBoxes [ k1, TraditionalForm ] := " \!\(\* SubscriptBox[ \( k \) , \( 1 \) ] \) " ;MakeBoxes [ k2, TraditionalForm ] := " \!\(\* SubscriptBox[ \( k \) , \( 2 \) ] \) " ;= InsertFields[ CreateTopologies[ 0 , 2 -> 2 ], { F [ 2 , { 1 }], - F [ 2 , { 1 }]} -> { V [ 2 ], V [ 2 ]}, InsertionLevel -> { Classes}] ; [ diags, ColumnsXRows -> { 3 , 1 }, Numbering -> Simple, -> None , ImageSize -> { 512 , 256 }] ;
Obtain the amplitude
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diags], IncomingMomenta -> { p1, p2}, -> { k1, k2}, UndoChiralSplittings -> True , ChangeDimension -> 4 , -> { k1, k2}, List -> True , SMP -> True , -> True , FinalSubstitutions -> { SMP[ "e" ] -> Sqrt [ 4 Pi SMP[ "alpha_fs" ]], [ "m_W" ] -> SMP[ "m_Z" ] SMP[ "cos_W" ]}] ;Let us separately mark the Higgs contribution separately
[ 1 ] = { markHiggs amp[ 0 ][[ 1 ]], amp[ 0 ][[ 2 ]], amp[ 0 ][[ 3 ]]} ;[ 2 ] = Total [ amp[ 1 ]] // DiracSimplify;Fix the kinematics
[] ;[ s , t , u , p1, p2, - k1, - k2, SMP[ "m_e" ], SMP[ "m_e" ], SMP[ "m_Z" ], SMP[ "m_Z" ]] ;Square the amplitude
We need to multiply by 1/2 to account for two identical particles in
the final state
[ 0 ] = 1 / 2 * (amp[ 2 ] (ComplexConjugate[ amp[ 2 ]] )) // // FermionSpinSum[ #, ExtraFactor -> 1 / 2 ^ 2 ] & // // DoPolarizationSums[ #, k1] & // DoPolarizationSums[ #, k2] & // [ #, { s , t , u , 2 SMP[ "m_e" ] ^ 2 + 2 SMP[ "m_Z" ] ^ 2 }] &;[ 0 ] = (ampSquared[ 0 ] /. markHiggs -> 1 /. u -> 2 SMP[ "m_e" ] ^ 2 + 2 SMP[ "m_Z" ] ^ 2 - s - t ) // Simplify ;[ 1 ] = Simplify [ Numerator [ ampSquaredFull[ 0 ]] /. [ "cos_W" ] -> Sqrt [ 1 - SMP[ "sin_W" ] ^ 2 ]] / Denominator [ ampSquaredFull[ 0 ]] − ( ( π 2 α 2 ( 4 ( 12 m Z 4 − 4 s m Z 2 + s 2 ) m e 12 − ( ( s − 4 m Z 2 ) m H 4 + 4 m Z 2 ( 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 2 + s ) m H 2 + 4 ( − 48 m Z 6 + ( − 32 s ( sin ( θ W ) ) 4 + 16 s ( sin ( θ W ) ) 2 + 39 s + 48 t ) m Z 4 − s ( 13 s + 16 t ) m Z 2 + 2 s 2 ( s + 2 t ) ) ) m e 10 + ( 192 m Z 8 + 64 ( 12 s ( sin ( θ W ) ) 4 − 6 s ( sin ( θ W ) ) 2 − 4 s − 9 t ) m Z 6 + 4 ( 32 s 2 ( sin ( θ W ) ) 8 − 32 s 2 ( sin ( θ W ) ) 6 − 8 s ( 9 s + 16 t ) ( sin ( θ W ) ) 4 + 8 s ( 5 s + 8 t ) ( sin ( θ W ) ) 2 + 43 s 2 + 72 t 2 + 116 s t ) m Z 4 − 16 s ( 3 s 2 + 10 t s + 6 t 2 ) m Z 2 + s 2 ( 5 s 2 + 24 t s + 24 t 2 ) + m H 4 ( 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 7 ) m Z 4 − 8 ( s + 2 t ) m Z 2 + s ( s + 4 t ) ) + 4 m H 2 ( − 12 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 m Z 6 − 8 ( 8 s ( sin ( θ W ) ) 8 − 8 s ( sin ( θ W ) ) 6 − 2 ( 3 s + 8 t ) ( sin ( θ W ) ) 4 + 4 ( s + 2 t ) ( sin ( θ W ) ) 2 + s − 2 t ) m Z 4 + s ( 3 s + 4 t ) m Z 2 ) ) m e 8 − ( 192 ( − 6 s ( sin ( θ W ) ) 4 + 3 s ( sin ( θ W ) ) 2 + 2 t ) m Z 8 − 4 ( 128 s 2 ( sin ( θ W ) ) 8 − 128 s 2 ( sin ( θ W ) ) 6 − 32 s ( 9 s + 16 t ) ( sin ( θ W ) ) 4 + 32 s ( 5 s + 8 t ) ( sin ( θ W ) ) 2 + 13 s 2 + 144 t 2 + 124 s t ) m Z 6 − 8 ( − 64 s 2 t ( sin ( θ W ) ) 8 + 64 s 2 t ( sin ( θ W ) ) 6 + 16 s ( s 2 + 5 t s + 6 t 2 ) ( sin ( θ W ) ) 4 − 8 s ( s 2 + 6 t s + 6 t 2 ) ( sin ( θ W ) ) 2 − 3 ( 3 s 3 + 15 t s 2 + 19 t 2 s + 8 t 3 ) ) m Z 4 − 8 s ( 2 s 3 ( sin ( θ W ) ) 4 − s 3 ( sin ( θ W ) ) 2 + 2 s 3 + 8 t 3 + 21 s t 2 + 13 s 2 t ) m Z 2 + s 2 ( s 3 + 10 t s 2 + 24 t 2 s + 16 t 3 ) + 2 m H 4 ( − 2 ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 + 8 ( sin ( θ W ) ) 2 + 5 ) m Z 6 + 2 ( 128 t ( sin ( θ W ) ) 8 − 128 t ( sin ( θ W ) ) 6 + 48 ( s + 2 t ) ( sin ( θ W ) ) 4 − 8 ( 3 s + 4 t ) ( sin ( θ W ) ) 2 + 9 s + 20 t ) m Z 4 − 2 ( 4 s 2 ( sin ( θ W ) ) 4 − 2 s 2 ( sin ( θ W ) ) 2 + s 2 + 6 t 2 + 7 s t ) m Z 2 + s t ( 2 s + 3 t ) ) + 4 m H 2 ( 12 ( 24 ( sin ( θ W ) ) 4 − 12 ( sin ( θ W ) ) 2 + 1 ) m Z 8 + 2 ( 128 s ( sin ( θ W ) ) 8 − 128 s ( sin ( θ W ) ) 6 − 8 ( 17 s + 32 t ) ( sin ( θ W ) ) 4 + 4 ( 21 s + 32 t ) ( sin ( θ W ) ) 2 + s − 20 t ) m Z 6 + ( − 256 s t ( sin ( θ W ) ) 8 + 256 s t ( sin ( θ W ) ) 6 − 16 ( s 2 − 4 t s − 12 t 2 ) ( sin ( θ W ) ) 4 + 8 ( s 2 − 8 t s − 12 t 2 ) ( sin ( θ W ) ) 2 − 19 s 2 + 24 t 2 − 16 s t ) m Z 4 + s ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + 3 s 2 + 6 t 2 + 8 s t ) m Z 2 ) ) m e 6 + ( 32 s ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 10 + 8 ( 240 s 2 ( sin ( θ W ) ) 8 − 240 s 2 ( sin ( θ W ) ) 6 − 4 s ( 11 s + 56 t ) ( sin ( θ W ) ) 4 + 4 s ( 13 s + 28 t ) ( sin ( θ W ) ) 2 − 9 s 2 + 24 t 2 − 4 s t ) m Z 8 − 8 ( 64 s 2 ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 64 s 2 ( 2 s + 3 t ) ( sin ( θ W ) ) 6 − 16 s ( s 2 + 11 t s + 14 t 2 ) ( sin ( θ W ) ) 4 + 8 s ( 3 s 2 + 14 t s + 14 t 2 ) ( sin ( θ W ) ) 2 − 2 s 3 + 24 t 3 + 28 s t 2 + 5 s 2 t ) m Z 6 + ( 64 s 2 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s 2 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 s ( 9 s 3 + 4 t s 2 − 20 t 2 s − 32 t 3 ) ( sin ( θ W ) ) 4 + 16 s ( − 3 s 3 + 16 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 2 + 15 s 4 + 48 t 4 + 144 s t 3 + 208 s 2 t 2 + 104 s 3 t ) m Z 4 − 2 s ( s + 2 t ) ( 8 s 3 ( sin ( θ W ) ) 4 − 4 s 3 ( sin ( θ W ) ) 2 + s 3 + 4 t 3 + 14 s t 2 + 9 s 2 t ) m Z 2 + 2 m H 2 ( − 16 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 8 − 16 ( 120 s ( sin ( θ W ) ) 8 − 120 s ( sin ( θ W ) ) 6 − 2 ( 3 s + 28 t ) ( sin ( θ W ) ) 4 + 2 ( 9 s + 14 t ) ( sin ( θ W ) ) 2 − 3 s − 4 t ) m Z 6 + 4 ( 128 s ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 128 s ( 2 s + 3 t ) ( sin ( θ W ) ) 6 + 8 ( s 2 − 14 t s − 28 t 2 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 2 + 26 t s + 28 t 2 ) ( sin ( θ W ) ) 2 − s 2 − 22 t 2 − 6 s t ) m Z 4 − ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 ( 11 s 3 + 20 t s 2 + 8 t 2 s − 16 t 3 ) ( sin ( θ W ) ) 4 − 32 ( 2 s 3 + 4 t s 2 − t 2 s − 4 t 3 ) ( sin ( θ W ) ) 2 + 17 s 3 − 32 t 3 + 20 s t 2 + 60 s 2 t ) m Z 2 + 2 s ( 8 s 2 ( s + 2 t ) ( sin ( θ W ) ) 4 − 4 s 2 ( s + 2 t ) ( sin ( θ W ) ) 2 + s 3 + 4 t 3 + 7 s t 2 + 5 s 2 t ) ) m Z 2 + s 2 t ( s + t ) ( s + 2 t ) 2 + m H 4 ( 8 ( 240 ( sin ( θ W ) ) 8 − 240 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 + 20 ( sin ( θ W ) ) 2 − 3 ) m Z 8 − 8 ( 64 ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 64 ( 2 s + 3 t ) ( sin ( θ W ) ) 6 + 8 ( 3 s + 8 t ) ( sin ( θ W ) ) 4 + 4 ( s − 2 t ) ( sin ( θ W ) ) 2 + s + 5 t ) m Z 6 + ( 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 ( 13 s 2 + 36 t s + 36 t 2 ) ( sin ( θ W ) ) 4 − 16 ( 5 s 2 + 16 t s + 12 t 2 ) ( sin ( θ W ) ) 2 + 19 s 2 + 80 t 2 + 80 s t ) m Z 4 − 2 ( 8 s 2 ( s + 2 t ) ( sin ( θ W ) ) 4 − 4 s 2 ( s + 2 t ) ( sin ( θ W ) ) 2 + s 3 + 8 t 3 + 16 s t 2 + 7 s 2 t ) m Z 2 + s t ( s 2 + 5 t s + 4 t 2 ) ) ) m e 4 + ( − ( ( − 32 ( sin ( θ W ) ) 2 ( 56 ( sin ( θ W ) ) 6 − 56 ( sin ( θ W ) ) 4 + 20 ( sin ( θ W ) ) 2 − 3 ) m Z 10 + 4 ( 32 ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s + 22 t ) ( sin ( θ W ) ) 4 + 16 ( s − 2 t ) ( sin ( θ W ) ) 2 − 5 s ) m Z 8 − 2 ( 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 2 − 28 t s − 52 t 2 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 2 − 20 t s − 28 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 + 18 t 2 + 10 s t ) m Z 6 + 4 ( 16 ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 16 ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 4 ( s 3 + 20 t s 2 + 36 t 2 s + 24 t 3 ) ( sin ( θ W ) ) 4 − 4 t ( 9 s 2 + 14 t s + 8 t 2 ) ( sin ( θ W ) ) 2 + t ( 6 s 2 + 13 t s + 8 t 2 ) ) m Z 4 − 2 t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + 2 t 2 + 4 s t ) m Z 2 + s t 2 ( s + t ) 2 ) m H 4 ) − 4 m Z 2 ( − 8 ( − 112 s ( sin ( θ W ) ) 8 + 112 s ( sin ( θ W ) ) 6 + 8 ( t − 5 s ) ( sin ( θ W ) ) 4 + ( 6 s − 4 t ) ( sin ( θ W ) ) 2 + t ) m Z 8 − 2 ( 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s 2 + 14 t s − 10 t 2 ) ( sin ( θ W ) ) 4 + 8 ( 2 s 2 + 5 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 − 10 t 2 − 6 s t ) m Z 6 + ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 3 − 18 t s 2 − 30 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 3 − 10 t s 2 − 6 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 2 − 5 s 3 − 16 t 3 − 6 s t 2 + 4 s 2 t ) m Z 4 − ( 32 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 32 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 8 ( s 4 + 18 t s 3 + 30 t 2 s 2 + 16 t 3 s − 4 t 4 ) ( sin ( θ W ) ) 4 − 8 t ( 8 s 3 + 11 t s 2 + 4 t 2 s − 2 t 3 ) ( sin ( θ W ) ) 2 + t ( 11 s 3 + 15 t s 2 + 4 t 2 s − 4 t 3 ) ) m Z 2 + s t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + t 2 + s t ) ) m H 2 + 2 s m Z 2 ( − 16 ( − 56 s ( sin ( θ W ) ) 8 + 56 s ( sin ( θ W ) ) 6 + ( 8 t − 20 s ) ( sin ( θ W ) ) 4 + ( 3 s − 4 t ) ( sin ( θ W ) ) 2 + t ) m Z 8 − 2 ( 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s 2 + 6 t s − 20 t 2 ) ( sin ( θ W ) ) 4 + 16 ( s 2 + 2 t s + 5 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 − 8 t 2 − 12 s t ) m Z 6 + ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 3 − 8 t s 2 − 8 t 2 s + 32 t 3 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 3 + 16 t 2 s + 32 t 3 ) ( sin ( θ W ) ) 2 − 5 s 3 − 8 t 3 + 2 s t 2 − 2 s 2 t ) m Z 4 − 2 ( 16 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 16 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 4 ( s 4 + 16 t s 3 + 24 t 2 s 2 + 8 t 3 s − 8 t 4 ) ( sin ( θ W ) ) 4 + 4 t ( − 7 s 3 − 8 t s 2 + 4 t 3 ) ( sin ( θ W ) ) 2 + t ( 5 s 3 + 10 t s 2 + 6 t 2 s − t 3 ) ) m Z 2 + s t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + 2 t 2 + 2 s t ) ) ) m e 2 + 2 ( s − m H 2 ) 2 m Z 4 ( 4 m Z 8 − 4 ( s + 3 t ) m Z 6 + ( s 2 + 6 t s + 14 t 2 ) m Z 4 − 2 t ( s + 2 t ) 2 m Z 2 + t ( s 3 + 3 t s 2 + 4 t 2 s + 2 t 3 ) ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 4 ( cos ( θ W ) ) 4 ( t − m e 2 ) 2 ( s − m H 2 ) 2 m Z 4 ( − m e 2 − 2 m Z 2 + s + t ) 2 ( sin ( θ W ) ) 4 ) ) -\left(\left(\pi ^2 \alpha ^2 \left(4
\left(12 m_Z^4-4 s m_Z^2+s^2\right) m_e^{12}-\left(\left(s-4
m_Z^2\right) m_H^4+4 m_Z^2 \left(4 \left(8 \left(\left.\sin (\theta
_W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) m_Z^2+s\right) m_H^2+4 \left(-48
m_Z^6+\left(-32 s \left(\left.\sin (\theta _W\right)\right){}^4+16 s
\left(\left.\sin (\theta _W\right)\right){}^2+39 s+48 t\right) m_Z^4-s
(13 s+16 t) m_Z^2+2 s^2 (s+2 t)\right)\right) m_e^{10}+\left(192
m_Z^8+64 \left(12 s \left(\left.\sin (\theta _W\right)\right){}^4-6 s
\left(\left.\sin (\theta _W\right)\right){}^2-4 s-9 t\right) m_Z^6+4
\left(32 s^2 \left(\left.\sin (\theta _W\right)\right){}^8-32 s^2
\left(\left.\sin (\theta _W\right)\right){}^6-8 s (9 s+16 t)
\left(\left.\sin (\theta _W\right)\right){}^4+8 s (5 s+8 t)
\left(\left.\sin (\theta _W\right)\right){}^2+43 s^2+72 t^2+116 s
t\right) m_Z^4-16 s \left(3 s^2+10 t s+6 t^2\right) m_Z^2+s^2 \left(5
s^2+24 t s+24 t^2\right)+m_H^4 \left(4 \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+7\right) m_Z^4-8 (s+2 t) m_Z^2+s (s+4 t)\right)+4
m_H^2 \left(-12 \left(1-4 \left(\left.\sin (\theta
_W\right)\right){}^2\right){}^2 m_Z^6-8 \left(8 s \left(\left.\sin
(\theta _W\right)\right){}^8-8 s \left(\left.\sin (\theta
_W\right)\right){}^6-2 (3 s+8 t) \left(\left.\sin (\theta
_W\right)\right){}^4+4 (s+2 t) \left(\left.\sin (\theta
_W\right)\right){}^2+s-2 t\right) m_Z^4+s (3 s+4 t) m_Z^2\right)\right)
m_e^8-\left(192 \left(-6 s \left(\left.\sin (\theta
_W\right)\right){}^4+3 s \left(\left.\sin (\theta _W\right)\right){}^2+2
t\right) m_Z^8-4 \left(128 s^2 \left(\left.\sin (\theta
_W\right)\right){}^8-128 s^2 \left(\left.\sin (\theta
_W\right)\right){}^6-32 s (9 s+16 t) \left(\left.\sin (\theta
_W\right)\right){}^4+32 s (5 s+8 t) \left(\left.\sin (\theta
_W\right)\right){}^2+13 s^2+144 t^2+124 s t\right) m_Z^6-8 \left(-64 s^2
t \left(\left.\sin (\theta _W\right)\right){}^8+64 s^2 t
\left(\left.\sin (\theta _W\right)\right){}^6+16 s \left(s^2+5 t s+6
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^4-8 s \left(s^2+6
t s+6 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^2-3 \left(3
s^3+15 t s^2+19 t^2 s+8 t^3\right)\right) m_Z^4-8 s \left(2 s^3
\left(\left.\sin (\theta _W\right)\right){}^4-s^3 \left(\left.\sin
(\theta _W\right)\right){}^2+2 s^3+8 t^3+21 s t^2+13 s^2 t\right)
m_Z^2+s^2 \left(s^3+10 t s^2+24 t^2 s+16 t^3\right)+2 m_H^4 \left(-2
\left(128 \left(\left.\sin (\theta _W\right)\right){}^8-128
\left(\left.\sin (\theta _W\right)\right){}^6+16 \left(\left.\sin
(\theta _W\right)\right){}^4+8 \left(\left.\sin (\theta
_W\right)\right){}^2+5\right) m_Z^6+2 \left(128 t \left(\left.\sin
(\theta _W\right)\right){}^8-128 t \left(\left.\sin (\theta
_W\right)\right){}^6+48 (s+2 t) \left(\left.\sin (\theta
_W\right)\right){}^4-8 (3 s+4 t) \left(\left.\sin (\theta
_W\right)\right){}^2+9 s+20 t\right) m_Z^4-2 \left(4 s^2
\left(\left.\sin (\theta _W\right)\right){}^4-2 s^2 \left(\left.\sin
(\theta _W\right)\right){}^2+s^2+6 t^2+7 s t\right) m_Z^2+s t (2 s+3
t)\right)+4 m_H^2 \left(12 \left(24 \left(\left.\sin (\theta
_W\right)\right){}^4-12 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) m_Z^8+2 \left(128 s \left(\left.\sin
(\theta _W\right)\right){}^8-128 s \left(\left.\sin (\theta
_W\right)\right){}^6-8 (17 s+32 t) \left(\left.\sin (\theta
_W\right)\right){}^4+4 (21 s+32 t) \left(\left.\sin (\theta
_W\right)\right){}^2+s-20 t\right) m_Z^6+\left(-256 s t \left(\left.\sin
(\theta _W\right)\right){}^8+256 s t \left(\left.\sin (\theta
_W\right)\right){}^6-16 \left(s^2-4 t s-12 t^2\right) \left(\left.\sin
(\theta _W\right)\right){}^4+8 \left(s^2-8 t s-12 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^2-19 s^2+24 t^2-16 s
t\right) m_Z^4+s \left(8 s^2 \left(\left.\sin (\theta
_W\right)\right){}^4-4 s^2 \left(\left.\sin (\theta
_W\right)\right){}^2+3 s^2+6 t^2+8 s t\right) m_Z^2\right)\right)
m_e^6+\left(32 s \left(8 \left(\left.\sin (\theta _W\right)\right){}^4-4
\left(\left.\sin (\theta _W\right)\right){}^2+1\right) m_Z^{10}+8
\left(240 s^2 \left(\left.\sin (\theta _W\right)\right){}^8-240 s^2
\left(\left.\sin (\theta _W\right)\right){}^6-4 s (11 s+56 t)
\left(\left.\sin (\theta _W\right)\right){}^4+4 s (13 s+28 t)
\left(\left.\sin (\theta _W\right)\right){}^2-9 s^2+24 t^2-4 s t\right)
m_Z^8-8 \left(64 s^2 (2 s+3 t) \left(\left.\sin (\theta
_W\right)\right){}^8-64 s^2 (2 s+3 t) \left(\left.\sin (\theta
_W\right)\right){}^6-16 s \left(s^2+11 t s+14 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^4+8 s \left(3 s^2+14 t s+14
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^2-2 s^3+24 t^3+28
s t^2+5 s^2 t\right) m_Z^6+\left(64 s^2 \left(3 s^2+4 t s+12 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^8-64 s^2 \left(3 s^2+4 t
s+12 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^6+16 s
\left(9 s^3+4 t s^2-20 t^2 s-32 t^3\right) \left(\left.\sin (\theta
_W\right)\right){}^4+16 s \left(-3 s^3+16 t^2 s+16 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^2+15 s^4+48 t^4+144 s
t^3+208 s^2 t^2+104 s^3 t\right) m_Z^4-2 s (s+2 t) \left(8 s^3
\left(\left.\sin (\theta _W\right)\right){}^4-4 s^3 \left(\left.\sin
(\theta _W\right)\right){}^2+s^3+4 t^3+14 s t^2+9 s^2 t\right) m_Z^2+2
m_H^2 \left(-16 \left(8 \left(\left.\sin (\theta _W\right)\right){}^4-4
\left(\left.\sin (\theta _W\right)\right){}^2+1\right) m_Z^8-16
\left(120 s \left(\left.\sin (\theta _W\right)\right){}^8-120 s
\left(\left.\sin (\theta _W\right)\right){}^6-2 (3 s+28 t)
\left(\left.\sin (\theta _W\right)\right){}^4+2 (9 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^2-3 s-4 t\right) m_Z^6+4
\left(128 s (2 s+3 t) \left(\left.\sin (\theta _W\right)\right){}^8-128
s (2 s+3 t) \left(\left.\sin (\theta _W\right)\right){}^6+8 \left(s^2-14
t s-28 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^4+4
\left(7 s^2+26 t s+28 t^2\right) \left(\left.\sin (\theta
_W\right)\right){}^2-s^2-22 t^2-6 s t\right) m_Z^4-\left(64 s \left(3
s^2+4 t s+12 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^8-64
s \left(3 s^2+4 t s+12 t^2\right) \left(\left.\sin (\theta
_W\right)\right){}^6+16 \left(11 s^3+20 t s^2+8 t^2 s-16 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^4-32 \left(2 s^3+4 t s^2-t^2
s-4 t^3\right) \left(\left.\sin (\theta _W\right)\right){}^2+17 s^3-32
t^3+20 s t^2+60 s^2 t\right) m_Z^2+2 s \left(8 s^2 (s+2 t)
\left(\left.\sin (\theta _W\right)\right){}^4-4 s^2 (s+2 t)
\left(\left.\sin (\theta _W\right)\right){}^2+s^3+4 t^3+7 s t^2+5 s^2
t\right)\right) m_Z^2+s^2 t (s+t) (s+2 t)^2+m_H^4 \left(8 \left(240
\left(\left.\sin (\theta _W\right)\right){}^8-240 \left(\left.\sin
(\theta _W\right)\right){}^6+20 \left(\left.\sin (\theta
_W\right)\right){}^4+20 \left(\left.\sin (\theta
_W\right)\right){}^2-3\right) m_Z^8-8 \left(64 (2 s+3 t)
\left(\left.\sin (\theta _W\right)\right){}^8-64 (2 s+3 t)
\left(\left.\sin (\theta _W\right)\right){}^6+8 (3 s+8 t)
\left(\left.\sin (\theta _W\right)\right){}^4+4 (s-2 t) \left(\left.\sin
(\theta _W\right)\right){}^2+s+5 t\right) m_Z^6+\left(64 \left(3 s^2+4 t
s+12 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^8-64 \left(3
s^2+4 t s+12 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^6+16
\left(13 s^2+36 t s+36 t^2\right) \left(\left.\sin (\theta
_W\right)\right){}^4-16 \left(5 s^2+16 t s+12 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^2+19 s^2+80 t^2+80 s
t\right) m_Z^4-2 \left(8 s^2 (s+2 t) \left(\left.\sin (\theta
_W\right)\right){}^4-4 s^2 (s+2 t) \left(\left.\sin (\theta
_W\right)\right){}^2+s^3+8 t^3+16 s t^2+7 s^2 t\right) m_Z^2+s t
\left(s^2+5 t s+4 t^2\right)\right)\right) m_e^4+\left(-\left(\left(-32
\left(\left.\sin (\theta _W\right)\right){}^2 \left(56 \left(\left.\sin
(\theta _W\right)\right){}^6-56 \left(\left.\sin (\theta
_W\right)\right){}^4+20 \left(\left.\sin (\theta
_W\right)\right){}^2-3\right) m_Z^{10}+4 \left(32 (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^8-32 (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^6+8 (7 s+22 t)
\left(\left.\sin (\theta _W\right)\right){}^4+16 (s-2 t)
\left(\left.\sin (\theta _W\right)\right){}^2-5 s\right) m_Z^8-2
\left(64 \left(3 s^2+4 t s+12 t^2\right) \left(\left.\sin (\theta
_W\right)\right){}^8-64 \left(3 s^2+4 t s+12 t^2\right) \left(\left.\sin
(\theta _W\right)\right){}^6-8 \left(s^2-28 t s-52 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^4+4 \left(7 s^2-20 t s-28
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^2-5 s^2+18 t^2+10
s t\right) m_Z^6+4 \left(16 \left(s^3+2 t s^2+8 t^2 s+8 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^8-16 \left(s^3+2 t s^2+8 t^2
s+8 t^3\right) \left(\left.\sin (\theta _W\right)\right){}^6+4
\left(s^3+20 t s^2+36 t^2 s+24 t^3\right) \left(\left.\sin (\theta
_W\right)\right){}^4-4 t \left(9 s^2+14 t s+8 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^2+t \left(6 s^2+13 t s+8
t^2\right)\right) m_Z^4-2 t (s+t) \left(8 s^2 \left(\left.\sin (\theta
_W\right)\right){}^4-4 s^2 \left(\left.\sin (\theta
_W\right)\right){}^2+s^2+2 t^2+4 s t\right) m_Z^2+s t^2 (s+t)^2\right)
m_H^4\right)-4 m_Z^2 \left(-8 \left(-112 s \left(\left.\sin (\theta
_W\right)\right){}^8+112 s \left(\left.\sin (\theta
_W\right)\right){}^6+8 (t-5 s) \left(\left.\sin (\theta
_W\right)\right){}^4+(6 s-4 t) \left(\left.\sin (\theta
_W\right)\right){}^2+t\right) m_Z^8-2 \left(32 s (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^8-32 s (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^6+8 \left(7 s^2+14 t s-10
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^4+8 \left(2 s^2+5
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^2-5 s^2-10 t^2-6
s t\right) m_Z^6+\left(64 s \left(3 s^2+4 t s+12 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^8-64 s \left(3 s^2+4 t s+12
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^6-8 \left(s^3-18
t s^2-30 t^2 s+16 t^3\right) \left(\left.\sin (\theta
_W\right)\right){}^4+4 \left(7 s^3-10 t s^2-6 t^2 s+16 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^2-5 s^3-16 t^3-6 s t^2+4 s^2
t\right) m_Z^4-\left(32 s \left(s^3+2 t s^2+8 t^2 s+8 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^8-32 s \left(s^3+2 t s^2+8
t^2 s+8 t^3\right) \left(\left.\sin (\theta _W\right)\right){}^6+8
\left(s^4+18 t s^3+30 t^2 s^2+16 t^3 s-4 t^4\right) \left(\left.\sin
(\theta _W\right)\right){}^4-8 t \left(8 s^3+11 t s^2+4 t^2 s-2
t^3\right) \left(\left.\sin (\theta _W\right)\right){}^2+t \left(11
s^3+15 t s^2+4 t^2 s-4 t^3\right)\right) m_Z^2+s t (s+t) \left(8 s^2
\left(\left.\sin (\theta _W\right)\right){}^4-4 s^2 \left(\left.\sin
(\theta _W\right)\right){}^2+s^2+t^2+s t\right)\right) m_H^2+2 s m_Z^2
\left(-16 \left(-56 s \left(\left.\sin (\theta _W\right)\right){}^8+56 s
\left(\left.\sin (\theta _W\right)\right){}^6+(8 t-20 s)
\left(\left.\sin (\theta _W\right)\right){}^4+(3 s-4 t) \left(\left.\sin
(\theta _W\right)\right){}^2+t\right) m_Z^8-2 \left(32 s (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^8-32 s (11 s+14 t)
\left(\left.\sin (\theta _W\right)\right){}^6+8 \left(7 s^2+6 t s-20
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^4+16 \left(s^2+2
t s+5 t^2\right) \left(\left.\sin (\theta _W\right)\right){}^2-5 s^2-8
t^2-12 s t\right) m_Z^6+\left(64 s \left(3 s^2+4 t s+12 t^2\right)
\left(\left.\sin (\theta _W\right)\right){}^8-64 s \left(3 s^2+4 t s+12
t^2\right) \left(\left.\sin (\theta _W\right)\right){}^6-8 \left(s^3-8 t
s^2-8 t^2 s+32 t^3\right) \left(\left.\sin (\theta
_W\right)\right){}^4+4 \left(7 s^3+16 t^2 s+32 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^2-5 s^3-8 t^3+2 s t^2-2 s^2
t\right) m_Z^4-2 \left(16 s \left(s^3+2 t s^2+8 t^2 s+8 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^8-16 s \left(s^3+2 t s^2+8
t^2 s+8 t^3\right) \left(\left.\sin (\theta _W\right)\right){}^6+4
\left(s^4+16 t s^3+24 t^2 s^2+8 t^3 s-8 t^4\right) \left(\left.\sin
(\theta _W\right)\right){}^4+4 t \left(-7 s^3-8 t s^2+4 t^3\right)
\left(\left.\sin (\theta _W\right)\right){}^2+t \left(5 s^3+10 t s^2+6
t^2 s-t^3\right)\right) m_Z^2+s t (s+t) \left(8 s^2 \left(\left.\sin
(\theta _W\right)\right){}^4-4 s^2 \left(\left.\sin (\theta
_W\right)\right){}^2+s^2+2 t^2+2 s t\right)\right)\right) m_e^2+2
\left(s-m_H^2\right){}^2 m_Z^4 \left(4 m_Z^8-4 (s+3 t) m_Z^6+\left(s^2+6
t s+14 t^2\right) m_Z^4-2 t (s+2 t)^2 m_Z^2+t \left(s^3+3 t s^2+4 t^2
s+2 t^3\right)\right) \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)\right)/\left(4 \left(\left.\cos
(\theta _W\right)\right){}^4 \left(t-m_e^2\right){}^2
\left(s-m_H^2\right){}^2 m_Z^4 \left(-m_e^2-2 m_Z^2+s+t\right){}^2
\left(\left.\sin (\theta _W\right)\right){}^4\right)\right) − ( ( π 2 α 2 ( 4 ( 12 m Z 4 − 4 s m Z 2 + s 2 ) m e 12 − ( ( s − 4 m Z 2 ) m H 4 + 4 m Z 2 ( 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 2 + s ) m H 2 + 4 ( − 48 m Z 6 + ( − 32 s ( sin ( θ W ) ) 4 + 16 s ( sin ( θ W ) ) 2 + 39 s + 48 t ) m Z 4 − s ( 13 s + 16 t ) m Z 2 + 2 s 2 ( s + 2 t ) ) ) m e 10 + ( 192 m Z 8 + 64 ( 12 s ( sin ( θ W ) ) 4 − 6 s ( sin ( θ W ) ) 2 − 4 s − 9 t ) m Z 6 + 4 ( 32 s 2 ( sin ( θ W ) ) 8 − 32 s 2 ( sin ( θ W ) ) 6 − 8 s ( 9 s + 16 t ) ( sin ( θ W ) ) 4 + 8 s ( 5 s + 8 t ) ( sin ( θ W ) ) 2 + 43 s 2 + 72 t 2 + 116 s t ) m Z 4 − 16 s ( 3 s 2 + 10 t s + 6 t 2 ) m Z 2 + s 2 ( 5 s 2 + 24 t s + 24 t 2 ) + m H 4 ( 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 7 ) m Z 4 − 8 ( s + 2 t ) m Z 2 + s ( s + 4 t ) ) + 4 m H 2 ( − 12 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 m Z 6 − 8 ( 8 s ( sin ( θ W ) ) 8 − 8 s ( sin ( θ W ) ) 6 − 2 ( 3 s + 8 t ) ( sin ( θ W ) ) 4 + 4 ( s + 2 t ) ( sin ( θ W ) ) 2 + s − 2 t ) m Z 4 + s ( 3 s + 4 t ) m Z 2 ) ) m e 8 − ( 192 ( − 6 s ( sin ( θ W ) ) 4 + 3 s ( sin ( θ W ) ) 2 + 2 t ) m Z 8 − 4 ( 128 s 2 ( sin ( θ W ) ) 8 − 128 s 2 ( sin ( θ W ) ) 6 − 32 s ( 9 s + 16 t ) ( sin ( θ W ) ) 4 + 32 s ( 5 s + 8 t ) ( sin ( θ W ) ) 2 + 13 s 2 + 144 t 2 + 124 s t ) m Z 6 − 8 ( − 64 s 2 t ( sin ( θ W ) ) 8 + 64 s 2 t ( sin ( θ W ) ) 6 + 16 s ( s 2 + 5 t s + 6 t 2 ) ( sin ( θ W ) ) 4 − 8 s ( s 2 + 6 t s + 6 t 2 ) ( sin ( θ W ) ) 2 − 3 ( 3 s 3 + 15 t s 2 + 19 t 2 s + 8 t 3 ) ) m Z 4 − 8 s ( 2 s 3 ( sin ( θ W ) ) 4 − s 3 ( sin ( θ W ) ) 2 + 2 s 3 + 8 t 3 + 21 s t 2 + 13 s 2 t ) m Z 2 + s 2 ( s 3 + 10 t s 2 + 24 t 2 s + 16 t 3 ) + 2 m H 4 ( − 2 ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 + 8 ( sin ( θ W ) ) 2 + 5 ) m Z 6 + 2 ( 128 t ( sin ( θ W ) ) 8 − 128 t ( sin ( θ W ) ) 6 + 48 ( s + 2 t ) ( sin ( θ W ) ) 4 − 8 ( 3 s + 4 t ) ( sin ( θ W ) ) 2 + 9 s + 20 t ) m Z 4 − 2 ( 4 s 2 ( sin ( θ W ) ) 4 − 2 s 2 ( sin ( θ W ) ) 2 + s 2 + 6 t 2 + 7 s t ) m Z 2 + s t ( 2 s + 3 t ) ) + 4 m H 2 ( 12 ( 24 ( sin ( θ W ) ) 4 − 12 ( sin ( θ W ) ) 2 + 1 ) m Z 8 + 2 ( 128 s ( sin ( θ W ) ) 8 − 128 s ( sin ( θ W ) ) 6 − 8 ( 17 s + 32 t ) ( sin ( θ W ) ) 4 + 4 ( 21 s + 32 t ) ( sin ( θ W ) ) 2 + s − 20 t ) m Z 6 + ( − 256 s t ( sin ( θ W ) ) 8 + 256 s t ( sin ( θ W ) ) 6 − 16 ( s 2 − 4 t s − 12 t 2 ) ( sin ( θ W ) ) 4 + 8 ( s 2 − 8 t s − 12 t 2 ) ( sin ( θ W ) ) 2 − 19 s 2 + 24 t 2 − 16 s t ) m Z 4 + s ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + 3 s 2 + 6 t 2 + 8 s t ) m Z 2 ) ) m e 6 + ( 32 s ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 10 + 8 ( 240 s 2 ( sin ( θ W ) ) 8 − 240 s 2 ( sin ( θ W ) ) 6 − 4 s ( 11 s + 56 t ) ( sin ( θ W ) ) 4 + 4 s ( 13 s + 28 t ) ( sin ( θ W ) ) 2 − 9 s 2 + 24 t 2 − 4 s t ) m Z 8 − 8 ( 64 s 2 ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 64 s 2 ( 2 s + 3 t ) ( sin ( θ W ) ) 6 − 16 s ( s 2 + 11 t s + 14 t 2 ) ( sin ( θ W ) ) 4 + 8 s ( 3 s 2 + 14 t s + 14 t 2 ) ( sin ( θ W ) ) 2 − 2 s 3 + 24 t 3 + 28 s t 2 + 5 s 2 t ) m Z 6 + ( 64 s 2 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s 2 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 s ( 9 s 3 + 4 t s 2 − 20 t 2 s − 32 t 3 ) ( sin ( θ W ) ) 4 + 16 s ( − 3 s 3 + 16 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 2 + 15 s 4 + 48 t 4 + 144 s t 3 + 208 s 2 t 2 + 104 s 3 t ) m Z 4 − 2 s ( s + 2 t ) ( 8 s 3 ( sin ( θ W ) ) 4 − 4 s 3 ( sin ( θ W ) ) 2 + s 3 + 4 t 3 + 14 s t 2 + 9 s 2 t ) m Z 2 + 2 m H 2 ( − 16 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 8 − 16 ( 120 s ( sin ( θ W ) ) 8 − 120 s ( sin ( θ W ) ) 6 − 2 ( 3 s + 28 t ) ( sin ( θ W ) ) 4 + 2 ( 9 s + 14 t ) ( sin ( θ W ) ) 2 − 3 s − 4 t ) m Z 6 + 4 ( 128 s ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 128 s ( 2 s + 3 t ) ( sin ( θ W ) ) 6 + 8 ( s 2 − 14 t s − 28 t 2 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 2 + 26 t s + 28 t 2 ) ( sin ( θ W ) ) 2 − s 2 − 22 t 2 − 6 s t ) m Z 4 − ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 ( 11 s 3 + 20 t s 2 + 8 t 2 s − 16 t 3 ) ( sin ( θ W ) ) 4 − 32 ( 2 s 3 + 4 t s 2 − t 2 s − 4 t 3 ) ( sin ( θ W ) ) 2 + 17 s 3 − 32 t 3 + 20 s t 2 + 60 s 2 t ) m Z 2 + 2 s ( 8 s 2 ( s + 2 t ) ( sin ( θ W ) ) 4 − 4 s 2 ( s + 2 t ) ( sin ( θ W ) ) 2 + s 3 + 4 t 3 + 7 s t 2 + 5 s 2 t ) ) m Z 2 + s 2 t ( s + t ) ( s + 2 t ) 2 + m H 4 ( 8 ( 240 ( sin ( θ W ) ) 8 − 240 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 + 20 ( sin ( θ W ) ) 2 − 3 ) m Z 8 − 8 ( 64 ( 2 s + 3 t ) ( sin ( θ W ) ) 8 − 64 ( 2 s + 3 t ) ( sin ( θ W ) ) 6 + 8 ( 3 s + 8 t ) ( sin ( θ W ) ) 4 + 4 ( s − 2 t ) ( sin ( θ W ) ) 2 + s + 5 t ) m Z 6 + ( 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 + 16 ( 13 s 2 + 36 t s + 36 t 2 ) ( sin ( θ W ) ) 4 − 16 ( 5 s 2 + 16 t s + 12 t 2 ) ( sin ( θ W ) ) 2 + 19 s 2 + 80 t 2 + 80 s t ) m Z 4 − 2 ( 8 s 2 ( s + 2 t ) ( sin ( θ W ) ) 4 − 4 s 2 ( s + 2 t ) ( sin ( θ W ) ) 2 + s 3 + 8 t 3 + 16 s t 2 + 7 s 2 t ) m Z 2 + s t ( s 2 + 5 t s + 4 t 2 ) ) ) m e 4 + ( − ( ( − 32 ( sin ( θ W ) ) 2 ( 56 ( sin ( θ W ) ) 6 − 56 ( sin ( θ W ) ) 4 + 20 ( sin ( θ W ) ) 2 − 3 ) m Z 10 + 4 ( 32 ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s + 22 t ) ( sin ( θ W ) ) 4 + 16 ( s − 2 t ) ( sin ( θ W ) ) 2 − 5 s ) m Z 8 − 2 ( 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 2 − 28 t s − 52 t 2 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 2 − 20 t s − 28 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 + 18 t 2 + 10 s t ) m Z 6 + 4 ( 16 ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 16 ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 4 ( s 3 + 20 t s 2 + 36 t 2 s + 24 t 3 ) ( sin ( θ W ) ) 4 − 4 t ( 9 s 2 + 14 t s + 8 t 2 ) ( sin ( θ W ) ) 2 + t ( 6 s 2 + 13 t s + 8 t 2 ) ) m Z 4 − 2 t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + 2 t 2 + 4 s t ) m Z 2 + s t 2 ( s + t ) 2 ) m H 4 ) − 4 m Z 2 ( − 8 ( − 112 s ( sin ( θ W ) ) 8 + 112 s ( sin ( θ W ) ) 6 + 8 ( t − 5 s ) ( sin ( θ W ) ) 4 + ( 6 s − 4 t ) ( sin ( θ W ) ) 2 + t ) m Z 8 − 2 ( 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s 2 + 14 t s − 10 t 2 ) ( sin ( θ W ) ) 4 + 8 ( 2 s 2 + 5 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 − 10 t 2 − 6 s t ) m Z 6 + ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 3 − 18 t s 2 − 30 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 3 − 10 t s 2 − 6 t 2 s + 16 t 3 ) ( sin ( θ W ) ) 2 − 5 s 3 − 16 t 3 − 6 s t 2 + 4 s 2 t ) m Z 4 − ( 32 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 32 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 8 ( s 4 + 18 t s 3 + 30 t 2 s 2 + 16 t 3 s − 4 t 4 ) ( sin ( θ W ) ) 4 − 8 t ( 8 s 3 + 11 t s 2 + 4 t 2 s − 2 t 3 ) ( sin ( θ W ) ) 2 + t ( 11 s 3 + 15 t s 2 + 4 t 2 s − 4 t 3 ) ) m Z 2 + s t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + t 2 + s t ) ) m H 2 + 2 s m Z 2 ( − 16 ( − 56 s ( sin ( θ W ) ) 8 + 56 s ( sin ( θ W ) ) 6 + ( 8 t − 20 s ) ( sin ( θ W ) ) 4 + ( 3 s − 4 t ) ( sin ( θ W ) ) 2 + t ) m Z 8 − 2 ( 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 8 − 32 s ( 11 s + 14 t ) ( sin ( θ W ) ) 6 + 8 ( 7 s 2 + 6 t s − 20 t 2 ) ( sin ( θ W ) ) 4 + 16 ( s 2 + 2 t s + 5 t 2 ) ( sin ( θ W ) ) 2 − 5 s 2 − 8 t 2 − 12 s t ) m Z 6 + ( 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 8 − 64 s ( 3 s 2 + 4 t s + 12 t 2 ) ( sin ( θ W ) ) 6 − 8 ( s 3 − 8 t s 2 − 8 t 2 s + 32 t 3 ) ( sin ( θ W ) ) 4 + 4 ( 7 s 3 + 16 t 2 s + 32 t 3 ) ( sin ( θ W ) ) 2 − 5 s 3 − 8 t 3 + 2 s t 2 − 2 s 2 t ) m Z 4 − 2 ( 16 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 8 − 16 s ( s 3 + 2 t s 2 + 8 t 2 s + 8 t 3 ) ( sin ( θ W ) ) 6 + 4 ( s 4 + 16 t s 3 + 24 t 2 s 2 + 8 t 3 s − 8 t 4 ) ( sin ( θ W ) ) 4 + 4 t ( − 7 s 3 − 8 t s 2 + 4 t 3 ) ( sin ( θ W ) ) 2 + t ( 5 s 3 + 10 t s 2 + 6 t 2 s − t 3 ) ) m Z 2 + s t ( s + t ) ( 8 s 2 ( sin ( θ W ) ) 4 − 4 s 2 ( sin ( θ W ) ) 2 + s 2 + 2 t 2 + 2 s t ) ) ) m e 2 + 2 ( s − m H 2 ) 2 m Z 4 ( 4 m Z 8 − 4 ( s + 3 t ) m Z 6 + ( s 2 + 6 t s + 14 t 2 ) m Z 4 − 2 t ( s + 2 t ) 2 m Z 2 + t ( s 3 + 3 t s 2 + 4 t 2 s + 2 t 3 ) ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 4 ( cos ( θ W ) ) 4 ( t − m e 2 ) 2 ( s − m H 2 ) 2 m Z 4 ( − m e 2 − 2 m Z 2 + s + t ) 2 ( sin ( θ W ) ) 4 ) )
If we neglect the electron mass, then the Higgs diagram does not
contribute.
[ 0 ] = (ampSquared[ 0 ] /. SMP[ "m_e" ] -> 0 /. u -> 2 SMP[ "m_Z" ] ^ 2 - s - t ) // Simplify − π 2 α 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( m Z 4 ( s 2 + 6 s t + 14 t 2 ) − 4 m Z 6 ( s + 3 t ) − 2 t m Z 2 ( s + 2 t ) 2 + 4 m Z 8 + t ( s 3 + 3 s 2 t + 4 s t 2 + 2 t 3 ) ) 2 t 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 ( − 2 m Z 2 + s + t ) 2 -\frac{\pi ^2 \alpha ^2 \left(32
\left(\left.\sin (\theta _W\right)\right){}^8-32 \left(\left.\sin
(\theta _W\right)\right){}^6+24 \left(\left.\sin (\theta
_W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) \left(m_Z^4 \left(s^2+6 s t+14
t^2\right)-4 m_Z^6 (s+3 t)-2 t m_Z^2 (s+2 t)^2+4 m_Z^8+t \left(s^3+3 s^2
t+4 s t^2+2 t^3\right)\right)}{2 t^2 \left(\left.\cos (\theta
_W\right)\right){}^4 \left(\left.\sin (\theta _W\right)\right){}^4
\left(-2 m_Z^2+s+t\right){}^2} − 2 t 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 ( − 2 m Z 2 + s + t ) 2 π 2 α 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( m Z 4 ( s 2 + 6 s t + 14 t 2 ) − 4 m Z 6 ( s + 3 t ) − 2 t m Z 2 ( s + 2 t ) 2 + 4 m Z 8 + t ( s 3 + 3 s 2 t + 4 s t 2 + 2 t 3 ) )
[ 1 ] = Simplify [ Numerator [ ampSquaredMassless[ 0 ]] /. [ "cos_W" ] -> Sqrt [ 1 - SMP[ "sin_W" ] ^ 2 ]] / Denominator [ ampSquaredMassless[ 0 ]] − π 2 α 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( m Z 4 ( s 2 + 6 s t + 14 t 2 ) − 4 m Z 6 ( s + 3 t ) − 2 t m Z 2 ( s + 2 t ) 2 + 4 m Z 8 + t ( s 3 + 3 s 2 t + 4 s t 2 + 2 t 3 ) ) 2 t 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 ( − 2 m Z 2 + s + t ) 2 -\frac{\pi ^2 \alpha ^2 \left(32
\left(\left.\sin (\theta _W\right)\right){}^8-32 \left(\left.\sin
(\theta _W\right)\right){}^6+24 \left(\left.\sin (\theta
_W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) \left(m_Z^4 \left(s^2+6 s t+14
t^2\right)-4 m_Z^6 (s+3 t)-2 t m_Z^2 (s+2 t)^2+4 m_Z^8+t \left(s^3+3 s^2
t+4 s t^2+2 t^3\right)\right)}{2 t^2 \left(\left.\cos (\theta
_W\right)\right){}^4 \left(\left.\sin (\theta _W\right)\right){}^4
\left(-2 m_Z^2+s+t\right){}^2} − 2 t 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 ( − 2 m Z 2 + s + t ) 2 π 2 α 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( m Z 4 ( s 2 + 6 s t + 14 t 2 ) − 4 m Z 6 ( s + 3 t ) − 2 t m Z 2 ( s + 2 t ) 2 + 4 m Z 8 + t ( s 3 + 3 s 2 t + 4 s t 2 + 2 t 3 ) )
Total cross section
= 1 / (16 Pi s ^ 2 );= prefac* Integrate [ ampSquaredFull[ 1 ], t ] ;= SelectFree2[ integral, Log ] ;= SelectNotFree2[ integral, Log ] // Simplify ;= - 1 / 2 (s - 2 SMP[ "m_Z" ] ^ 2 - 2 SMP[ "m_e" ] ^ 2 - Sqrt [ (s - 4 SMP[ "m_e" ] ^ 2 ) (s - 4 SMP[ "m_Z" ] ^ 2 )] );= - 1 / 2 (s - 2 SMP[ "m_Z" ] ^ 2 - 2 SMP[ "m_e" ] ^ 2 + Sqrt [ (s - 4 SMP[ "m_e" ] ^ 2 ) (s - 4 SMP[ "m_Z" ] ^ 2 )] );= ((logFreePart /. { t -> tUpper} ) - (logFreePart /. { t -> tLower} )) // Simplify − ( ( π α 2 ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ( m e 2 m Z 4 ( − 4 m H 2 ( s m Z 2 ( − 176 ( sin ( θ W ) ) 4 + 88 ( sin ( θ W ) ) 2 − 13 ) + 4 m Z 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + 4 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) + m H 4 ( s ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 160 ( sin ( θ W ) ) 4 − 64 ( sin ( θ W ) ) 2 + 7 ) − 8 m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) ) + 4 s ( − 2 s m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) + m Z 4 ( 32 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 1 ) + 2 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) + m e 4 ( − 4 m H 2 m Z 2 ( s m Z 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 64 ( sin ( θ W ) ) 4 + 1 ) − 16 m Z 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + s 2 ) + m H 4 ( 8 s m Z 2 + 2 m Z 4 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 32 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 − 7 ) − s 2 ) + 4 s 3 m Z 2 + 2 s 2 m Z 4 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 96 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 − 3 ) − 32 s m Z 6 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 + 48 m Z 8 ) + 4 m e 6 ( − 8 s 2 m Z 2 + 28 s m Z 4 − 48 m Z 6 + s 3 ) + 8 m Z 8 ( s − m H 2 ) 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 64 s 2 m Z 4 ( cos ( θ W ) ) 4 ( s − m H 2 ) 2 ( sin ( θ W ) ) 4 ( m e 2 ( s − 4 m Z 2 ) + m Z 4 ) ) ) -\left(\left(\pi \alpha ^2
\sqrt{\left(s-4 m_e^2\right) \left(s-4 m_Z^2\right)} \left(m_e^2 m_Z^4
\left(-4 m_H^2 \left(s m_Z^2 \left(-176 \left(\left.\sin (\theta
_W\right)\right){}^4+88 \left(\left.\sin (\theta
_W\right)\right){}^2-13\right)+4 m_Z^4 \left(8 \left(\left.\sin (\theta
_W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)+4 s^2 \left(16 \left(\left.\sin (\theta
_W\right)\right){}^8-16 \left(\left.\sin (\theta _W\right)\right){}^6+20
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)+m_H^4 \left(s \left(128
\left(\left.\sin (\theta _W\right)\right){}^8-128 \left(\left.\sin
(\theta _W\right)\right){}^6+160 \left(\left.\sin (\theta
_W\right)\right){}^4-64 \left(\left.\sin (\theta
_W\right)\right){}^2+7\right)-8 m_Z^2 \left(44 \left(\left.\sin (\theta
_W\right)\right){}^4-22 \left(\left.\sin (\theta
_W\right)\right){}^2+3\right)\right)+4 s \left(-2 s m_Z^2 \left(44
\left(\left.\sin (\theta _W\right)\right){}^4-22 \left(\left.\sin
(\theta _W\right)\right){}^2+3\right)+m_Z^4 \left(32 \left(\left.\sin
(\theta _W\right)\right){}^4-16 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)+2 s^2 \left(16 \left(\left.\sin (\theta
_W\right)\right){}^8-16 \left(\left.\sin (\theta _W\right)\right){}^6+20
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)\right)+m_e^4 \left(-4 m_H^2 m_Z^2
\left(s m_Z^2 \left(256 \left(\left.\sin (\theta
_W\right)\right){}^8-256 \left(\left.\sin (\theta
_W\right)\right){}^6+64 \left(\left.\sin (\theta
_W\right)\right){}^4+1\right)-16 m_Z^4 \left(8 \left(\left.\sin (\theta
_W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)+s^2\right)+m_H^4 \left(8 s m_Z^2+2 m_Z^4
\left(256 \left(\left.\sin (\theta _W\right)\right){}^8-256
\left(\left.\sin (\theta _W\right)\right){}^6+32 \left(\left.\sin
(\theta _W\right)\right){}^4+16 \left(\left.\sin (\theta
_W\right)\right){}^2-7\right)-s^2\right)+4 s^3 m_Z^2+2 s^2 m_Z^4
\left(256 \left(\left.\sin (\theta _W\right)\right){}^8-256
\left(\left.\sin (\theta _W\right)\right){}^6+96 \left(\left.\sin
(\theta _W\right)\right){}^4-16 \left(\left.\sin (\theta
_W\right)\right){}^2-3\right)-32 s m_Z^6 \left(1-4 \left(\left.\sin
(\theta _W\right)\right){}^2\right){}^2+48 m_Z^8\right)+4 m_e^6 \left(-8
s^2 m_Z^2+28 s m_Z^4-48 m_Z^6+s^3\right)+8 m_Z^8
\left(s-m_H^2\right){}^2 \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)\right)/\left(64 s^2 m_Z^4
\left(\left.\cos (\theta _W\right)\right){}^4 \left(s-m_H^2\right){}^2
\left(\left.\sin (\theta _W\right)\right){}^4 \left(m_e^2 \left(s-4
m_Z^2\right)+m_Z^4\right)\right)\right) − ( ( π α 2 ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ( m e 2 m Z 4 ( − 4 m H 2 ( s m Z 2 ( − 176 ( sin ( θ W ) ) 4 + 88 ( sin ( θ W ) ) 2 − 13 ) + 4 m Z 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + 4 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) + m H 4 ( s ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 160 ( sin ( θ W ) ) 4 − 64 ( sin ( θ W ) ) 2 + 7 ) − 8 m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) ) + 4 s ( − 2 s m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) + m Z 4 ( 32 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 1 ) + 2 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) + m e 4 ( − 4 m H 2 m Z 2 ( s m Z 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 64 ( sin ( θ W ) ) 4 + 1 ) − 16 m Z 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + s 2 ) + m H 4 ( 8 s m Z 2 + 2 m Z 4 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 32 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 − 7 ) − s 2 ) + 4 s 3 m Z 2 + 2 s 2 m Z 4 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 96 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 − 3 ) − 32 s m Z 6 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 + 48 m Z 8 ) + 4 m e 6 ( − 8 s 2 m Z 2 + 28 s m Z 4 − 48 m Z 6 + s 3 ) + 8 m Z 8 ( s − m H 2 ) 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 64 s 2 m Z 4 ( cos ( θ W ) ) 4 ( s − m H 2 ) 2 ( sin ( θ W ) ) 4 ( m e 2 ( s − 4 m Z 2 ) + m Z 4 ) ) )
= logPart /. Log [ t + a_ : 0 ] :> Log [ (tUpper + a )/ (tLower + a )] // Simplify − ( ( π α 2 ( log ( − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ) ) ( m e 4 ( m H 2 ( − 4 s m Z 2 + 2 m Z 4 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 3 ) + s 2 ) − 8 s 2 m Z 2 + 2 s m Z 4 ( − 256 ( sin ( θ W ) ) 8 + 256 ( sin ( θ W ) ) 6 − 96 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 + 15 ) + 16 m Z 6 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 1 ) + s 3 ) + 2 m e 2 m Z 2 ( m H 2 ( 2 s m Z 2 ( − 64 ( sin ( θ W ) ) 8 + 64 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 3 ) + 8 m Z 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 22 ( sin ( θ W ) ) 4 − 7 ( sin ( θ W ) ) 2 + 1 ) + s 2 ( − 8 ( sin ( θ W ) ) 4 + 4 ( sin ( θ W ) ) 2 − 1 ) ) + 4 s 2 m Z 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 4 ( sin ( θ W ) ) 2 − 1 ) − 8 s m Z 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 34 ( sin ( θ W ) ) 4 − 13 ( sin ( θ W ) ) 2 + 2 ) + 8 m Z 6 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + s 3 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) ) + 2 m Z 4 ( s − m H 2 ) ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 64 s 2 m Z 4 ( cos ( θ W ) ) 4 ( s − m H 2 ) ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 ) ) -\left(\left(\pi \alpha ^2 \left(\log
\left(\frac{-\sqrt{\left(s-4 m_e^2\right) \left(s-4 m_Z^2\right)}-2
m_Z^2+s}{\sqrt{\left(s-4 m_e^2\right) \left(s-4 m_Z^2\right)}-2
m_Z^2+s}\right)-\log \left(\frac{\sqrt{\left(s-4 m_e^2\right) \left(s-4
m_Z^2\right)}-2 m_Z^2+s}{-\sqrt{\left(s-4 m_e^2\right) \left(s-4
m_Z^2\right)}-2 m_Z^2+s}\right)\right) \left(m_e^4 \left(m_H^2 \left(-4
s m_Z^2+2 m_Z^4 \left(1-4 \left(\left.\sin (\theta
_W\right)\right){}^2\right){}^2 \left(16 \left(\left.\sin (\theta
_W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2-3\right)+s^2\right)-8 s^2 m_Z^2+2 s m_Z^4
\left(-256 \left(\left.\sin (\theta _W\right)\right){}^8+256
\left(\left.\sin (\theta _W\right)\right){}^6-96 \left(\left.\sin
(\theta _W\right)\right){}^4+16 \left(\left.\sin (\theta
_W\right)\right){}^2+15\right)+16 m_Z^6 \left(16 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2-1\right)+s^3\right)+2 m_e^2 m_Z^2 \left(m_H^2
\left(2 s m_Z^2 \left(-64 \left(\left.\sin (\theta
_W\right)\right){}^8+64 \left(\left.\sin (\theta _W\right)\right){}^6+16
\left(\left.\sin (\theta _W\right)\right){}^4-16 \left(\left.\sin
(\theta _W\right)\right){}^2+3\right)+8 m_Z^4 \left(32 \left(\left.\sin
(\theta _W\right)\right){}^8-32 \left(\left.\sin (\theta
_W\right)\right){}^6+22 \left(\left.\sin (\theta _W\right)\right){}^4-7
\left(\left.\sin (\theta _W\right)\right){}^2+1\right)+s^2 \left(-8
\left(\left.\sin (\theta _W\right)\right){}^4+4 \left(\left.\sin (\theta
_W\right)\right){}^2-1\right)\right)+4 s^2 m_Z^2 \left(32
\left(\left.\sin (\theta _W\right)\right){}^8-32 \left(\left.\sin
(\theta _W\right)\right){}^6+4 \left(\left.\sin (\theta
_W\right)\right){}^2-1\right)-8 s m_Z^4 \left(32 \left(\left.\sin
(\theta _W\right)\right){}^8-32 \left(\left.\sin (\theta
_W\right)\right){}^6+34 \left(\left.\sin (\theta _W\right)\right){}^4-13
\left(\left.\sin (\theta _W\right)\right){}^2+2\right)+8 m_Z^6 \left(8
\left(\left.\sin (\theta _W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)+s^3 \left(8 \left(\left.\sin (\theta
_W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)+2 m_Z^4 \left(s-m_H^2\right)
\left(4 m_Z^4+s^2\right) \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)\right)/\left(64 s^2 m_Z^4
\left(\left.\cos (\theta _W\right)\right){}^4 \left(s-m_H^2\right)
\left(s-2 m_Z^2\right) \left(\left.\sin (\theta
_W\right)\right){}^4\right)\right) − ( ( π α 2 ( log ( ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s ) ) ( m e 4 ( m H 2 ( − 4 s m Z 2 + 2 m Z 4 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 3 ) + s 2 ) − 8 s 2 m Z 2 + 2 s m Z 4 ( − 256 ( sin ( θ W ) ) 8 + 256 ( sin ( θ W ) ) 6 − 96 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 + 15 ) + 16 m Z 6 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 1 ) + s 3 ) + 2 m e 2 m Z 2 ( m H 2 ( 2 s m Z 2 ( − 64 ( sin ( θ W ) ) 8 + 64 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 3 ) + 8 m Z 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 22 ( sin ( θ W ) ) 4 − 7 ( sin ( θ W ) ) 2 + 1 ) + s 2 ( − 8 ( sin ( θ W ) ) 4 + 4 ( sin ( θ W ) ) 2 − 1 ) ) + 4 s 2 m Z 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 4 ( sin ( θ W ) ) 2 − 1 ) − 8 s m Z 4 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 34 ( sin ( θ W ) ) 4 − 13 ( sin ( θ W ) ) 2 + 2 ) + 8 m Z 6 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + s 3 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) ) + 2 m Z 4 ( s − m H 2 ) ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) / ( 64 s 2 m Z 4 ( cos ( θ W ) ) 4 ( s − m H 2 ) ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 ) )
= (xsectionPart1 + xsectionPart2)− ( ( π ( log ( − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s + ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ) − log ( − 2 m Z 2 + s + ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ) ) ( ( 16 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 1 ) m Z 6 + 2 s ( − 256 ( sin ( θ W ) ) 8 + 256 ( sin ( θ W ) ) 6 − 96 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 + 15 ) m Z 4 − 8 s 2 m Z 2 + s 3 + m H 2 ( 2 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 3 ) m Z 4 − 4 s m Z 2 + s 2 ) ) m e 4 + 2 m Z 2 ( 8 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 6 − 8 s ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 34 ( sin ( θ W ) ) 4 − 13 ( sin ( θ W ) ) 2 + 2 ) m Z 4 + 4 s 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 4 ( sin ( θ W ) ) 2 − 1 ) m Z 2 + s 3 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + m H 2 ( 8 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 22 ( sin ( θ W ) ) 4 − 7 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + 2 s ( − 64 ( sin ( θ W ) ) 8 + 64 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 3 ) m Z 2 + s 2 ( − 8 ( sin ( θ W ) ) 4 + 4 ( sin ( θ W ) ) 2 − 1 ) ) ) m e 2 + 2 ( s − m H 2 ) m Z 4 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) α 2 ) / ( 64 s 2 ( cos ( θ W ) ) 4 ( s − m H 2 ) m Z 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 ) ) − ( π ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ( 8 ( s − m H 2 ) 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) m Z 8 + m e 2 ( ( s ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 160 ( sin ( θ W ) ) 4 − 64 ( sin ( θ W ) ) 2 + 7 ) − 8 m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) ) m H 4 − 4 ( 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + s ( − 176 ( sin ( θ W ) ) 4 + 88 ( sin ( θ W ) ) 2 − 13 ) m Z 2 + 4 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) m H 2 + 4 s ( ( 32 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 1 ) m Z 4 − 2 s ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) m Z 2 + 2 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) m Z 4 + 4 m e 6 ( − 48 m Z 6 + 28 s m Z 4 − 8 s 2 m Z 2 + s 3 ) + m e 4 ( 48 m Z 8 − 32 s ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 m Z 6 + 2 s 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 96 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 − 3 ) m Z 4 + 4 s 3 m Z 2 − 4 m H 2 ( − 16 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + s ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 64 ( sin ( θ W ) ) 4 + 1 ) m Z 2 + s 2 ) m Z 2 + m H 4 ( 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 32 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 − 7 ) m Z 4 + 8 s m Z 2 − s 2 ) ) ) α 2 ) / ( 64 s 2 ( cos ( θ W ) ) 4 ( s − m H 2 ) 2 m Z 4 ( m Z 4 + m e 2 ( s − 4 m Z 2 ) ) ( sin ( θ W ) ) 4 ) -\left(\left(\pi \left(\log
\left(\frac{-2 m_Z^2+s-\sqrt{\left(s-4 m_e^2\right) \left(s-4
m_Z^2\right)}}{-2 m_Z^2+s+\sqrt{\left(s-4 m_e^2\right) \left(s-4
m_Z^2\right)}}\right)-\log \left(\frac{-2 m_Z^2+s+\sqrt{\left(s-4
m_e^2\right) \left(s-4 m_Z^2\right)}}{-2 m_Z^2+s-\sqrt{\left(s-4
m_e^2\right) \left(s-4 m_Z^2\right)}}\right)\right) \left(\left(16
\left(16 \left(\left.\sin (\theta _W\right)\right){}^4-8
\left(\left.\sin (\theta _W\right)\right){}^2-1\right) m_Z^6+2 s
\left(-256 \left(\left.\sin (\theta _W\right)\right){}^8+256
\left(\left.\sin (\theta _W\right)\right){}^6-96 \left(\left.\sin
(\theta _W\right)\right){}^4+16 \left(\left.\sin (\theta
_W\right)\right){}^2+15\right) m_Z^4-8 s^2 m_Z^2+s^3+m_H^2 \left(2
\left(1-4 \left(\left.\sin (\theta _W\right)\right){}^2\right){}^2
\left(16 \left(\left.\sin (\theta _W\right)\right){}^4-8
\left(\left.\sin (\theta _W\right)\right){}^2-3\right) m_Z^4-4 s
m_Z^2+s^2\right)\right) m_e^4+2 m_Z^2 \left(8 \left(8 \left(\left.\sin
(\theta _W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) m_Z^6-8 s \left(32 \left(\left.\sin
(\theta _W\right)\right){}^8-32 \left(\left.\sin (\theta
_W\right)\right){}^6+34 \left(\left.\sin (\theta _W\right)\right){}^4-13
\left(\left.\sin (\theta _W\right)\right){}^2+2\right) m_Z^4+4 s^2
\left(32 \left(\left.\sin (\theta _W\right)\right){}^8-32
\left(\left.\sin (\theta _W\right)\right){}^6+4 \left(\left.\sin (\theta
_W\right)\right){}^2-1\right) m_Z^2+s^3 \left(8 \left(\left.\sin (\theta
_W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)+m_H^2 \left(8 \left(32 \left(\left.\sin
(\theta _W\right)\right){}^8-32 \left(\left.\sin (\theta
_W\right)\right){}^6+22 \left(\left.\sin (\theta _W\right)\right){}^4-7
\left(\left.\sin (\theta _W\right)\right){}^2+1\right) m_Z^4+2 s
\left(-64 \left(\left.\sin (\theta _W\right)\right){}^8+64
\left(\left.\sin (\theta _W\right)\right){}^6+16 \left(\left.\sin
(\theta _W\right)\right){}^4-16 \left(\left.\sin (\theta
_W\right)\right){}^2+3\right) m_Z^2+s^2 \left(-8 \left(\left.\sin
(\theta _W\right)\right){}^4+4 \left(\left.\sin (\theta
_W\right)\right){}^2-1\right)\right)\right) m_e^2+2 \left(s-m_H^2\right)
m_Z^4 \left(4 m_Z^4+s^2\right) \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right) \alpha ^2\right)/\left(64 s^2
\left(\left.\cos (\theta _W\right)\right){}^4 \left(s-m_H^2\right) m_Z^4
\left(s-2 m_Z^2\right) \left(\left.\sin (\theta
_W\right)\right){}^4\right)\right)-\left(\pi \sqrt{\left(s-4
m_e^2\right) \left(s-4 m_Z^2\right)} \left(8 \left(s-m_H^2\right){}^2
\left(32 \left(\left.\sin (\theta _W\right)\right){}^8-32
\left(\left.\sin (\theta _W\right)\right){}^6+24 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) m_Z^8+m_e^2 \left(\left(s \left(128
\left(\left.\sin (\theta _W\right)\right){}^8-128 \left(\left.\sin
(\theta _W\right)\right){}^6+160 \left(\left.\sin (\theta
_W\right)\right){}^4-64 \left(\left.\sin (\theta
_W\right)\right){}^2+7\right)-8 m_Z^2 \left(44 \left(\left.\sin (\theta
_W\right)\right){}^4-22 \left(\left.\sin (\theta
_W\right)\right){}^2+3\right)\right) m_H^4-4 \left(4 \left(8
\left(\left.\sin (\theta _W\right)\right){}^4-4 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) m_Z^4+s \left(-176 \left(\left.\sin
(\theta _W\right)\right){}^4+88 \left(\left.\sin (\theta
_W\right)\right){}^2-13\right) m_Z^2+4 s^2 \left(16 \left(\left.\sin
(\theta _W\right)\right){}^8-16 \left(\left.\sin (\theta
_W\right)\right){}^6+20 \left(\left.\sin (\theta _W\right)\right){}^4-8
\left(\left.\sin (\theta _W\right)\right){}^2+1\right)\right) m_H^2+4 s
\left(\left(32 \left(\left.\sin (\theta _W\right)\right){}^4-16
\left(\left.\sin (\theta _W\right)\right){}^2+1\right) m_Z^4-2 s
\left(44 \left(\left.\sin (\theta _W\right)\right){}^4-22
\left(\left.\sin (\theta _W\right)\right){}^2+3\right) m_Z^2+2 s^2
\left(16 \left(\left.\sin (\theta _W\right)\right){}^8-16
\left(\left.\sin (\theta _W\right)\right){}^6+20 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)\right)\right) m_Z^4+4 m_e^6 \left(-48
m_Z^6+28 s m_Z^4-8 s^2 m_Z^2+s^3\right)+m_e^4 \left(48 m_Z^8-32 s
\left(1-4 \left(\left.\sin (\theta _W\right)\right){}^2\right){}^2
m_Z^6+2 s^2 \left(256 \left(\left.\sin (\theta _W\right)\right){}^8-256
\left(\left.\sin (\theta _W\right)\right){}^6+96 \left(\left.\sin
(\theta _W\right)\right){}^4-16 \left(\left.\sin (\theta
_W\right)\right){}^2-3\right) m_Z^4+4 s^3 m_Z^2-4 m_H^2 \left(-16
\left(8 \left(\left.\sin (\theta _W\right)\right){}^4-4 \left(\left.\sin
(\theta _W\right)\right){}^2+1\right) m_Z^4+s \left(256 \left(\left.\sin
(\theta _W\right)\right){}^8-256 \left(\left.\sin (\theta
_W\right)\right){}^6+64 \left(\left.\sin (\theta
_W\right)\right){}^4+1\right) m_Z^2+s^2\right) m_Z^2+m_H^4 \left(2
\left(256 \left(\left.\sin (\theta _W\right)\right){}^8-256
\left(\left.\sin (\theta _W\right)\right){}^6+32 \left(\left.\sin
(\theta _W\right)\right){}^4+16 \left(\left.\sin (\theta
_W\right)\right){}^2-7\right) m_Z^4+8 s m_Z^2-s^2\right)\right)\right)
\alpha ^2\right)/\left(64 s^2 \left(\left.\cos (\theta
_W\right)\right){}^4 \left(s-m_H^2\right){}^2 m_Z^4 \left(m_Z^4+m_e^2
\left(s-4 m_Z^2\right)\right) \left(\left.\sin (\theta
_W\right)\right){}^4\right) − ( ( π ( log ( − 2 m Z 2 + s + ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ) − log ( − 2 m Z 2 + s − ( s − 4 m e 2 ) ( s − 4 m Z 2 ) − 2 m Z 2 + s + ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ) ) ( ( 16 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 1 ) m Z 6 + 2 s ( − 256 ( sin ( θ W ) ) 8 + 256 ( sin ( θ W ) ) 6 − 96 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 + 15 ) m Z 4 − 8 s 2 m Z 2 + s 3 + m H 2 ( 2 ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 ( 16 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 − 3 ) m Z 4 − 4 s m Z 2 + s 2 ) ) m e 4 + 2 m Z 2 ( 8 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 6 − 8 s ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 34 ( sin ( θ W ) ) 4 − 13 ( sin ( θ W ) ) 2 + 2 ) m Z 4 + 4 s 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 4 ( sin ( θ W ) ) 2 − 1 ) m Z 2 + s 3 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) + m H 2 ( 8 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 22 ( sin ( θ W ) ) 4 − 7 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + 2 s ( − 64 ( sin ( θ W ) ) 8 + 64 ( sin ( θ W ) ) 6 + 16 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 3 ) m Z 2 + s 2 ( − 8 ( sin ( θ W ) ) 4 + 4 ( sin ( θ W ) ) 2 − 1 ) ) ) m e 2 + 2 ( s − m H 2 ) m Z 4 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) α 2 ) / ( 64 s 2 ( cos ( θ W ) ) 4 ( s − m H 2 ) m Z 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 ) ) − ( π ( s − 4 m e 2 ) ( s − 4 m Z 2 ) ( 8 ( s − m H 2 ) 2 ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) m Z 8 + m e 2 ( ( s ( 128 ( sin ( θ W ) ) 8 − 128 ( sin ( θ W ) ) 6 + 160 ( sin ( θ W ) ) 4 − 64 ( sin ( θ W ) ) 2 + 7 ) − 8 m Z 2 ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) ) m H 4 − 4 ( 4 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + s ( − 176 ( sin ( θ W ) ) 4 + 88 ( sin ( θ W ) ) 2 − 13 ) m Z 2 + 4 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) m H 2 + 4 s ( ( 32 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 + 1 ) m Z 4 − 2 s ( 44 ( sin ( θ W ) ) 4 − 22 ( sin ( θ W ) ) 2 + 3 ) m Z 2 + 2 s 2 ( 16 ( sin ( θ W ) ) 8 − 16 ( sin ( θ W ) ) 6 + 20 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ) ) m Z 4 + 4 m e 6 ( − 48 m Z 6 + 28 s m Z 4 − 8 s 2 m Z 2 + s 3 ) + m e 4 ( 48 m Z 8 − 32 s ( 1 − 4 ( sin ( θ W ) ) 2 ) 2 m Z 6 + 2 s 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 96 ( sin ( θ W ) ) 4 − 16 ( sin ( θ W ) ) 2 − 3 ) m Z 4 + 4 s 3 m Z 2 − 4 m H 2 ( − 16 ( 8 ( sin ( θ W ) ) 4 − 4 ( sin ( θ W ) ) 2 + 1 ) m Z 4 + s ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 64 ( sin ( θ W ) ) 4 + 1 ) m Z 2 + s 2 ) m Z 2 + m H 4 ( 2 ( 256 ( sin ( θ W ) ) 8 − 256 ( sin ( θ W ) ) 6 + 32 ( sin ( θ W ) ) 4 + 16 ( sin ( θ W ) ) 2 − 7 ) m Z 4 + 8 s m Z 2 − s 2 ) ) ) α 2 ) / ( 64 s 2 ( cos ( θ W ) ) 4 ( s − m H 2 ) 2 m Z 4 ( m Z 4 + m e 2 ( s − 4 m Z 2 ) ) ( sin ( θ W ) ) 4 )
Neglecting the electron mass produces a much simpler formula for the
total cross section
= (xsectionPart1 /. SMP[ "m_e" ] -> 0 ) // Simplify − π α 2 s ( s − 4 m Z 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) 8 s 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 -\frac{\pi \alpha ^2 \sqrt{s \left(s-4
m_Z^2\right)} \left(32 \left(\left.\sin (\theta _W\right)\right){}^8-32
\left(\left.\sin (\theta _W\right)\right){}^6+24 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)}{8 s^2 \left(\left.\cos (\theta
_W\right)\right){}^4 \left(\left.\sin (\theta
_W\right)\right){}^4} − 8 s 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 π α 2 s ( s − 4 m Z 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 )
= (xsectionPart2 /. SMP[ "m_e" ] -> 0 ) // Simplify − π α 2 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( log ( − s ( s − 4 m Z 2 ) − 2 m Z 2 + s s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( s ( s − 4 m Z 2 ) − 2 m Z 2 + s − s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) ) 32 s 2 ( cos ( θ W ) ) 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 -\frac{\pi \alpha ^2 \left(4
m_Z^4+s^2\right) \left(32 \left(\left.\sin (\theta
_W\right)\right){}^8-32 \left(\left.\sin (\theta _W\right)\right){}^6+24
\left(\left.\sin (\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) \left(\log \left(\frac{-\sqrt{s \left(s-4
m_Z^2\right)}-2 m_Z^2+s}{\sqrt{s \left(s-4 m_Z^2\right)}-2
m_Z^2+s}\right)-\log \left(\frac{\sqrt{s \left(s-4 m_Z^2\right)}-2
m_Z^2+s}{-\sqrt{s \left(s-4 m_Z^2\right)}-2 m_Z^2+s}\right)\right)}{32
s^2 \left(\left.\cos (\theta _W\right)\right){}^4 \left(s-2 m_Z^2\right)
\left(\left.\sin (\theta _W\right)\right){}^4} − 32 s 2 ( cos ( θ W ) ) 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 π α 2 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( log ( s ( s − 4 m Z 2 ) − 2 m Z 2 + s − s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( − s ( s − 4 m Z 2 ) − 2 m Z 2 + s s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) )
= xsectionMasslessPart1 + xsectionMasslessPart2− π α 2 s ( s − 4 m Z 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) 8 s 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 − π α 2 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( log ( − s ( s − 4 m Z 2 ) − 2 m Z 2 + s s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( s ( s − 4 m Z 2 ) − 2 m Z 2 + s − s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) ) 32 s 2 ( cos ( θ W ) ) 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 -\frac{\pi \alpha ^2 \sqrt{s \left(s-4
m_Z^2\right)} \left(32 \left(\left.\sin (\theta _W\right)\right){}^8-32
\left(\left.\sin (\theta _W\right)\right){}^6+24 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right)}{8 s^2 \left(\left.\cos (\theta
_W\right)\right){}^4 \left(\left.\sin (\theta
_W\right)\right){}^4}-\frac{\pi \alpha ^2 \left(4 m_Z^4+s^2\right)
\left(32 \left(\left.\sin (\theta _W\right)\right){}^8-32
\left(\left.\sin (\theta _W\right)\right){}^6+24 \left(\left.\sin
(\theta _W\right)\right){}^4-8 \left(\left.\sin (\theta
_W\right)\right){}^2+1\right) \left(\log \left(\frac{-\sqrt{s \left(s-4
m_Z^2\right)}-2 m_Z^2+s}{\sqrt{s \left(s-4 m_Z^2\right)}-2
m_Z^2+s}\right)-\log \left(\frac{\sqrt{s \left(s-4 m_Z^2\right)}-2
m_Z^2+s}{-\sqrt{s \left(s-4 m_Z^2\right)}-2 m_Z^2+s}\right)\right)}{32
s^2 \left(\left.\cos (\theta _W\right)\right){}^4 \left(s-2 m_Z^2\right)
\left(\left.\sin (\theta _W\right)\right){}^4} − 8 s 2 ( cos ( θ W ) ) 4 ( sin ( θ W ) ) 4 π α 2 s ( s − 4 m Z 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) − 32 s 2 ( cos ( θ W ) ) 4 ( s − 2 m Z 2 ) ( sin ( θ W ) ) 4 π α 2 ( 4 m Z 4 + s 2 ) ( 32 ( sin ( θ W ) ) 8 − 32 ( sin ( θ W ) ) 6 + 24 ( sin ( θ W ) ) 4 − 8 ( sin ( θ W ) ) 2 + 1 ) ( log ( s ( s − 4 m Z 2 ) − 2 m Z 2 + s − s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) − log ( − s ( s − 4 m Z 2 ) − 2 m Z 2 + s s ( s − 4 m Z 2 ) − 2 m Z 2 + s ) )
We can also plot the full cross-section (in pb) as a function of
Sqrt[s] (in GeV)
= 3.89 * 10 ^ 8 * crossSectionTotal /. { SMP[ "m_e" ] -> 0.51 * 10 ^ (- 3 ), SMP[ "m_H" ] -> 125.0 , SMP[ "m_Z" ] -> 91.2 , SMP[ "sin_W" ] -> Sqrt [ 0.231 ], [ "cos_W" ] -> Sqrt [ 1.0 - 0.231 ], SMP[ "alpha_fs" ] -> 1 / 137 , s -> sqrtS^ 2 } // Simplify 1 sqrtS 4 ( sqrtS 2 − 15625. ) 2 ( − 17424.3 sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 ( 1. sqrtS 6 + 5.09207 × 1 0 14 sqrtS 4 − 1.59127 × 1 0 19 sqrtS 2 + 1.24318 × 1 0 23 ) 1. sqrtS 2 + 2.65974 × 1 0 14 − 1 sqrtS 2 − 16634.9 8339.72 ( sqrtS 2 − 15625. ) ( 1. sqrtS 6 − 15625. sqrtS 4 + 2.76719 × 1 0 8 sqrtS 2 − 4.32374 × 1 0 12 ) ( log ( sqrtS 2 − 1. sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 sqrtS 2 + sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 ) − 1. log ( sqrtS 2 + sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 sqrtS 2 − 1. sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 ) ) ) \frac{1}{\text{sqrtS}^4
\left(\text{sqrtS}^2-15625.\right)^2}\left(-\frac{17424.3
\sqrt{\text{sqrtS}^4-33269.8 \;\text{sqrtS}^2+0.0346139} \left(1.
\;\text{sqrtS}^6+5.09207\times 10^{14} \;\text{sqrtS}^4-1.59127\times
10^{19} \;\text{sqrtS}^2+1.24318\times 10^{23}\right)}{1.
\;\text{sqrtS}^2+2.65974\times
10^{14}}-\frac{1}{\text{sqrtS}^2-16634.9}8339.72
\left(\text{sqrtS}^2-15625.\right) \left(1. \;\text{sqrtS}^6-15625.
\;\text{sqrtS}^4+2.76719\times 10^8 \;\text{sqrtS}^2-4.32374\times
10^{12}\right) \left(\log \left(\frac{\text{sqrtS}^2-1.
\sqrt{\text{sqrtS}^4-33269.8
\;\text{sqrtS}^2+0.0346139}-16634.9}{\text{sqrtS}^2+\sqrt{\text{sqrtS}^4-33269.8
\;\text{sqrtS}^2+0.0346139}-16634.9}\right)-1. \log
\left(\frac{\text{sqrtS}^2+\sqrt{\text{sqrtS}^4-33269.8
\;\text{sqrtS}^2+0.0346139}-16634.9}{\text{sqrtS}^2-1.
\sqrt{\text{sqrtS}^4-33269.8
\;\text{sqrtS}^2+0.0346139}-16634.9}\right)\right)\right) sqrtS 4 ( sqrtS 2 − 15625. ) 2 1 ( − 1. sqrtS 2 + 2.65974 × 1 0 14 17424.3 sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 ( 1. sqrtS 6 + 5.09207 × 1 0 14 sqrtS 4 − 1.59127 × 1 0 19 sqrtS 2 + 1.24318 × 1 0 23 ) − sqrtS 2 − 16634.9 1 8339.72 ( sqrtS 2 − 15625. ) ( 1. sqrtS 6 − 15625. sqrtS 4 + 2.76719 × 1 0 8 sqrtS 2 − 4.32374 × 1 0 12 ) ( log ( sqrtS 2 + sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 sqrtS 2 − 1. sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 ) − 1. log ( sqrtS 2 − 1. sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 sqrtS 2 + sqrtS 4 − 33269.8 sqrtS 2 + 0.0346139 − 16634.9 ) ) )
The plot can be compared to the one in arXiv:hep-ex/9911003, Fig.
8
If [ $FrontEnd = != Null , Plot [ crossSectionTotalPlot, { sqrtS, 183 , 200 }] ]
Check the final results
= { - (Pi * (4 * (- 2 + s )* Sqrt [ (- 4 + s )* s ] + 4 + s ^ 2 )* Log [ (- 2 + s - Sqrt [ (- 4 + s )* s ] )/ (- 2 + s + Sqrt [ (- 4 + s )* s ] )] - 4 + s ^ 2 )* Log [ (- 2 + s + Sqrt [ (- 4 + s )* s ] )/ (- 2 + s - Sqrt [ (- 4 + s )* s ] )] )* [ "alpha_fs" ] ^ 2 * (1 - 8 * SMP[ "sin_W" ] ^ 2 + 24 * SMP[ "sin_W" ] ^ 4 - 32 * SMP[ "sin_W" ] ^ 6 + 32 * SMP[ "sin_W" ] ^ 8 ))/ (32 * (- 2 + s )* s ^ 2 * SMP[ "cos_W" ] ^ 4 * SMP[ "sin_W" ] ^ 4 ) } ;[{ crossSectionTotalMassless /. SMP[ "m_Z" ] -> 1 }, , Text -> { " \t Compare to the known result: " , "CORRECT." , "WRONG!" }, Interrupt -> { Hold [ Quit [ 1 ]], Automatic }] ;Print [ " \t CPU Time used: " , Round [ N [ TimeUsed [], 3 ], 0.001 ], " s." ] ;\ tCompare to the known result: CORRECT. \text{$\backslash $tCompare to the known
result: } \;\text{CORRECT.} \tCompare to the known result: CORRECT.
\ tCPU Time used: 56.255 s. \text{$\backslash $tCPU Time used:
}56.255\text{ s.} \tCPU Time used: 56.255 s.
