Load
FeynCalc and the necessary add-ons or other packages
This example uses a custom QCD model created with FeynRules. Please
evaluate the file FeynCalc/Examples/FeynRules/QCD/GenerateModelQCD.m
before running it for the first time.
= "Renormalization, QCD, MS and MSbar, 1-loop" ;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
version, 2023-12-20 22:40:59 +01:00, dff3b835). For help, use the
}\underline{\text{online} \;\text{documentation}}\;\text{, check out the
}\underline{\text{wiki}}\;\text{ or visit the
}\underline{\text{forum}.} 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 .
Please check our FAQ ‾ for answers to some common FeynCalc questions and have a look at the supplied examples . ‾ \text{Please check our
}\underline{\text{FAQ}}\;\text{ for answers to some common FeynCalc
questions and have a look at the supplied
}\underline{\text{examples}.} Please check our FAQ for answers to some common FeynCalc questions and have a look at the supplied examples .
If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software. \text{If you use FeynCalc in your
research, please evaluate FeynCalcHowToCite[] to learn how to cite this
software.} If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.
Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package! \text{Please keep in mind that the proper
academic attribution of our work is crucial to ensure the future
development of this package!} Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!
FeynArts 3.11 (3 Aug 2020) patched for use with FeynCalc, for documentation see the manual ‾ or visit www . feynarts . de . ‾ \text{FeynArts }\;\text{3.11 (3 Aug 2020)
patched for use with FeynCalc, for documentation see the
}\underline{\text{manual}}\;\text{ or visit
}\underline{\text{www}.\text{feynarts}.\text{de}.} FeynArts 3.11 (3 Aug 2020) patched for use with FeynCalc, for documentation see the manual or visit www . feynarts . de .
If you use FeynArts in your research, please cite \text{If you use FeynArts in your
research, please cite} If you use FeynArts in your research, please cite
∙ 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
We keep scaleless B0 functions, since otherwise the UV part would not
come out right.
= True ;[ PatchModelsOnly -> True ] ;(*Successfully patched FeynArts.*) Generate Feynman diagrams
Nicer typesetting
MakeBoxes [ mu, TraditionalForm ] := " \[ Mu]" ;MakeBoxes [ nu, TraditionalForm ] := " \[ Nu]" ;MakeBoxes [ rho, TraditionalForm ] := " \[ Rho]" ;MakeBoxes [ si, TraditionalForm ] := " \[ Sigma]" ;= { InsertionLevel -> { Particles}, Model -> FileNameJoin [{ "QCD" , "QCD" }], -> FileNameJoin [{ "QCD" , "QCD" }], ExcludeParticles -> { F [ 3 | 4 , { 2 | 3 }], F [ 4 , { 1 }]}} ;top [ i_ , j_ ] := CreateTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs}] ;[ i_ , j_ ] := CreateTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs, SelfEnergies}] ; [ i_ , j_ ] := CreateCTTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs}] ;[ i_ , j_ ] := CreateCTTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs, SelfEnergyCTs}] ; { diagQuarkSE, diagQuarkSECT} = InsertFields[ #, { F [ 3 , { 1 }]} -> { F [ 3 , { 1 }]}, Sequence @@ params] & / @ { top [ 1 , 1 ], topCT[ 1 , 1 ]} ; { diagGluonSE, diagGluonSECT} = InsertFields[ #, { V [ 5 ]} -> { V [ 5 ]}, Sequence @@ params] & / @ { top [ 1 , 1 ], topCT[ 1 , 1 ]} ; { diagGhostSE, diagGhostSECT} = InsertFields[ #, { U [ 5 ]} -> { U [ 5 ]}, Sequence @@ params] & / @ { top [ 1 , 1 ], topCT[ 1 , 1 ]} ; { diagQuarkGluonVertex, diagQuarkGluonVertexCT} = InsertFields[ #, { F [ 3 , { 1 }], V [ 5 ]} -> { F [ 3 , { 1 }]}, Sequence @@ params] & / @ { topTriangle[ 2 , 1 ], topTriangleCT[ 2 , 1 ]} ;[ 0 ] = diagQuarkSE[[ 0 ]][ Sequence @@ diagQuarkSE, Sequence @@ diagQuarkSECT] ;[ 0 ] = diagGluonSE[[ 0 ]][ Sequence @@ diagGluonSE, Sequence @@ diagGluonSECT] ;[ 0 ] = diagGhostSE[[ 0 ]][ Sequence @@ diagGhostSE, Sequence @@ diagGhostSECT] ;[ 0 ] = diagQuarkGluonVertex[[ 0 ]][ Sequence @@ diagQuarkGluonVertex, Sequence @@ diagQuarkGluonVertexCT] ;[ diag1[ 0 ], ColumnsXRows -> { 2 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
[ diag2[ 0 ], ColumnsXRows -> { 4 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 128 }] ;
[ diag3[ 0 ], ColumnsXRows -> { 2 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
[ diag4[ 0 ], ColumnsXRows -> { 3 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
Obtain the amplitudes
The 1/(2Pi)^D prefactor is implicit.
Quark self-energy including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag1[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p }, OutgoingMomenta -> { p }, -> { mu, nu}, DropSumOver -> True , -> { l }, UndoChiralSplittings -> True , -> D , List -> False , SMP -> True , -> { Zm -> SMP[ "Z_m" ], Zpsi -> SMP[ "Z_psi" ], [ "m_u" ] -> SMP[ "m_q" ]}] ( ( 1 − ξ G ) ( l − p ) μ ( p − l ) ν ( l 2 − m q 2 ) . ( l − p ) 4 + g μ ν ( l 2 − m q 2 ) . ( l − p ) 2 ) ( − i γ ν g s T Col2 Col3 Glu3 ) . ( γ ⋅ l + m q ) . ( − i γ μ g s T Col3 Col1 Glu3 ) − i m q δ Col1 Col2 ( Z m Z ψ − 1 ) + i ( Z ψ − 1 ) δ Col1 Col2 γ ⋅ p \left(\frac{\left(1-\xi _{\text{G}}\right)
(l-p)^{\mu } (p-l)^{\nu }}{\left(l^2-m_q^2\right).(l-p)^4}+\frac{g^{\mu
\nu }}{\left(l^2-m_q^2\right).(l-p)^2}\right) \left(-i \gamma ^{\nu }
g_s T_{\text{Col2}\;\text{Col3}}^{\text{Glu3}}\right).\left(\gamma \cdot
l+m_q\right).\left(-i \gamma ^{\mu } g_s
T_{\text{Col3}\;\text{Col1}}^{\text{Glu3}}\right)-i m_q \delta
_{\text{Col1}\;\text{Col2}} \left(Z_m Z_{\psi }-1\right)+i \left(Z_{\psi
}-1\right) \delta _{\text{Col1}\;\text{Col2}} \gamma \cdot p ( ( l 2 − m q 2 ) . ( l − p ) 4 ( 1 − ξ G ) ( l − p ) μ ( p − l ) ν + ( l 2 − m q 2 ) . ( l − p ) 2 g μν ) ( − i γ ν g s T Col2 Col3 Glu3 ) . ( γ ⋅ l + m q ) . ( − i γ μ g s T Col3 Col1 Glu3 ) − i m q δ Col1 Col2 ( Z m Z ψ − 1 ) + i ( Z ψ − 1 ) δ Col1 Col2 γ ⋅ p
Gluon self-energy including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag2[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p }, OutgoingMomenta -> { p }, -> { mu, nu, rho, si}, DropSumOver -> True , -> { l }, UndoChiralSplittings -> True , -> D , List -> True , SMP -> True , -> { ZA -> SMP[ "Z_A" ], Zxi -> SMP[ "Z_xi" ], [ "m_u" ] -> SMP[ "m_q" ]}] { − 1 2 i ( g ρ σ l 2 − ( 1 − ξ G ) l ρ l σ ( l 2 ) 2 ) ( i g μ ρ g ν σ f Glu1 Glu3 $AL$14450 f Glu2 Glu3 $AL$14450 g s 2 − 2 i g μ ν g ρ σ f Glu1 Glu3 $AL$14451 f Glu2 Glu3 $AL$14451 g s 2 + i g μ σ g ν ρ f Glu1 Glu3 $AL$14453 f Glu2 Glu3 $AL$14453 g s 2 ) , tr ( ( m q − γ ⋅ l ) . ( − i γ ν g s T Col3 Col4 Glu2 ) . ( γ ⋅ ( p − l ) + m q ) . ( − i γ μ g s T Col4 Col3 Glu1 ) ) ( l 2 − m q 2 ) . ( ( l − p ) 2 − m q 2 ) , − ( l − p ) μ l ν g s 2 f Glu1 Glu3 Glu4 f Glu2 Glu3 Glu4 l 2 . ( l − p ) 2 , − 1 2 ( − ( 1 − ξ G ) 2 ( l − p ) Lor5 ( p − l ) Lor6 l ρ l σ ( l 2 ) 2 . ( l − p ) 4 + ( 1 − ξ G ) ( l − p ) Lor5 ( p − l ) Lor6 g ρ σ l 2 . ( l − p ) 4 − ( 1 − ξ G ) g Lor5 Lor6 l ρ l σ ( l 2 ) 2 . ( l − p ) 2 + g Lor5 Lor6 g ρ σ l 2 . ( l − p ) 2 ) ( − l Lor5 g μ ρ g s f Glu1 Glu3 Glu4 − p Lor5 g μ ρ g s f Glu1 Glu3 Glu4 + g Lor5 ρ l μ g s f Glu1 Glu3 Glu4 + g Lor5 ρ ( l − p ) μ g s f Glu1 Glu3 Glu4 + g Lor5 μ p ρ g s f Glu1 Glu3 Glu4 + g Lor5 μ ( p − l ) ρ g s f Glu1 Glu3 Glu4 ) ( l Lor6 g ν σ g s f Glu2 Glu3 Glu4 + p Lor6 g ν σ g s f Glu2 Glu3 Glu4 − g Lor6 σ l ν g s f Glu2 Glu3 Glu4 + g Lor6 σ ( p − l ) ν g s f Glu2 Glu3 Glu4 + g Lor6 ν ( l − p ) σ g s f Glu2 Glu3 Glu4 − g Lor6 ν p σ g s f Glu2 Glu3 Glu4 ) , i p μ p ν ( Z A − 1 ) δ Glu1 Glu2 − i g μ ν p 2 ( Z A − 1 ) δ Glu1 Glu2 − i p μ p ν ( Z A − Z ξ ) δ Glu1 Glu2 ξ G Z ξ } \left\{-\frac{1}{2} i \left(\frac{g^{\rho
\sigma }}{l^2}-\frac{\left(1-\xi _{\text{G}}\right) l^{\rho } l^{\sigma
}}{\left(l^2\right)^2}\right) \left(i g^{\mu \rho } g^{\nu \sigma }
f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$14450}}
f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$14450}} g_s^2-2 i g^{\mu \nu }
g^{\rho \sigma } f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$14451}}
f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$14451}} g_s^2+i g^{\mu \sigma }
g^{\nu \rho } f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$14453}}
f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$14453}}
g_s^2\right),\frac{\text{tr}\left(\left(m_q-\gamma \cdot
l\right).\left(-i \gamma ^{\nu } g_s
T_{\text{Col3}\;\text{Col4}}^{\text{Glu2}}\right).\left(\gamma \cdot
(p-l)+m_q\right).\left(-i \gamma ^{\mu } g_s
T_{\text{Col4}\;\text{Col3}}^{\text{Glu1}}\right)\right)}{\left(l^2-m_q^2\right).\left((l-p)^2-m_q^2\right)},-\frac{(l-p)^{\mu
} l^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{l^2.(l-p)^2},-\frac{1}{2}
\left(-\frac{\left(1-\xi _{\text{G}}\right){}^2 (l-p)^{\text{Lor5}}
(p-l)^{\text{Lor6}} l^{\rho } l^{\sigma
}}{\left(l^2\right)^2.(l-p)^4}+\frac{\left(1-\xi _{\text{G}}\right)
(l-p)^{\text{Lor5}} (p-l)^{\text{Lor6}} g^{\rho \sigma
}}{l^2.(l-p)^4}-\frac{\left(1-\xi _{\text{G}}\right)
g^{\text{Lor5}\;\text{Lor6}} l^{\rho } l^{\sigma
}}{\left(l^2\right)^2.(l-p)^2}+\frac{g^{\text{Lor5}\;\text{Lor6}}
g^{\rho \sigma }}{l^2.(l-p)^2}\right) \left(-l^{\text{Lor5}} g^{\mu \rho
} g_s f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}-p^{\text{Lor5}} g^{\mu
\rho } g_s f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor5}\rho
} l^{\mu } g_s
f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor5}\rho }
(l-p)^{\mu } g_s
f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor5}\mu } p^{\rho }
g_s f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor5}\mu }
(p-l)^{\rho } g_s f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}\right)
\left(l^{\text{Lor6}} g^{\nu \sigma } g_s
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}+p^{\text{Lor6}} g^{\nu \sigma
} g_s f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}-g^{\text{Lor6}\sigma }
l^{\nu } g_s
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor6}\sigma }
(p-l)^{\nu } g_s
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}+g^{\text{Lor6}\nu }
(l-p)^{\sigma } g_s
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}-g^{\text{Lor6}\nu } p^{\sigma
} g_s f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}\right),i p^{\mu } p^{\nu
} \left(Z_A-1\right) \delta ^{\text{Glu1}\;\text{Glu2}}-i g^{\mu \nu }
p^2 \left(Z_A-1\right) \delta ^{\text{Glu1}\;\text{Glu2}}-\frac{i p^{\mu
} p^{\nu } \left(Z_A-Z_{\xi }\right) \delta
^{\text{Glu1}\;\text{Glu2}}}{\xi _{\text{G}} Z_{\xi
}}\right\} { − 2 1 i ( l 2 g ρ σ − ( l 2 ) 2 ( 1 − ξ G ) l ρ l σ ) ( i g μ ρ g ν σ f Glu1 Glu3 $AL$14450 f Glu2 Glu3 $AL$14450 g s 2 − 2 i g μν g ρ σ f Glu1 Glu3 $AL$14451 f Glu2 Glu3 $AL$14451 g s 2 + i g μ σ g ν ρ f Glu1 Glu3 $AL$14453 f Glu2 Glu3 $AL$14453 g s 2 ) , ( l 2 − m q 2 ) . ( ( l − p ) 2 − m q 2 ) tr ( ( m q − γ ⋅ l ) . ( − i γ ν g s T Col3 Col4 Glu2 ) . ( γ ⋅ ( p − l ) + m q ) . ( − i γ μ g s T Col4 Col3 Glu1 ) ) , − l 2 . ( l − p ) 2 ( l − p ) μ l ν g s 2 f Glu1 Glu3 Glu4 f Glu2 Glu3 Glu4 , − 2 1 ( − ( l 2 ) 2 . ( l − p ) 4 ( 1 − ξ G ) 2 ( l − p ) Lor5 ( p − l ) Lor6 l ρ l σ + l 2 . ( l − p ) 4 ( 1 − ξ G ) ( l − p ) Lor5 ( p − l ) Lor6 g ρ σ − ( l 2 ) 2 . ( l − p ) 2 ( 1 − ξ G ) g Lor5 Lor6 l ρ l σ + l 2 . ( l − p ) 2 g Lor5 Lor6 g ρ σ ) ( − l Lor5 g μ ρ g s f Glu1 Glu3 Glu4 − p Lor5 g μ ρ g s f Glu1 Glu3 Glu4 + g Lor5 ρ l μ g s f Glu1 Glu3 Glu4 + g Lor5 ρ ( l − p ) μ g s f Glu1 Glu3 Glu4 + g Lor5 μ p ρ g s f Glu1 Glu3 Glu4 + g Lor5 μ ( p − l ) ρ g s f Glu1 Glu3 Glu4 ) ( l Lor6 g ν σ g s f Glu2 Glu3 Glu4 + p Lor6 g ν σ g s f Glu2 Glu3 Glu4 − g Lor6 σ l ν g s f Glu2 Glu3 Glu4 + g Lor6 σ ( p − l ) ν g s f Glu2 Glu3 Glu4 + g Lor6 ν ( l − p ) σ g s f Glu2 Glu3 Glu4 − g Lor6 ν p σ g s f Glu2 Glu3 Glu4 ) , i p μ p ν ( Z A − 1 ) δ Glu1 Glu2 − i g μν p 2 ( Z A − 1 ) δ Glu1 Glu2 − ξ G Z ξ i p μ p ν ( Z A − Z ξ ) δ Glu1 Glu2 }
Ghost self-energy including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag3[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p }, OutgoingMomenta -> { p }, -> { mu, nu}, DropSumOver -> True , -> { l }, UndoChiralSplittings -> True , -> D , List -> False , SMP -> True , -> { Zu -> SMP[ "Z_u" ]}] − g s 2 l μ p ν f Glu1 Glu3 Glu4 f Glu2 Glu3 Glu4 ( ( 1 − ξ G ) ( l − p ) μ ( p − l ) ν l 2 . ( l − p ) 4 + g μ ν l 2 . ( l − p ) 2 ) + i p 2 ( Z u − 1 ) δ Glu1 Glu2 -g_s^2 l^{\mu } p^{\nu }
f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}}
f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}} \left(\frac{\left(1-\xi
_{\text{G}}\right) (l-p)^{\mu } (p-l)^{\nu }}{l^2.(l-p)^4}+\frac{g^{\mu
\nu }}{l^2.(l-p)^2}\right)+i p^2 \left(Z_u-1\right) \delta
^{\text{Glu1}\;\text{Glu2}} − g s 2 l μ p ν f Glu1 Glu3 Glu4 f Glu2 Glu3 Glu4 ( l 2 . ( l − p ) 4 ( 1 − ξ G ) ( l − p ) μ ( p − l ) ν + l 2 . ( l − p ) 2 g μν ) + i p 2 ( Z u − 1 ) δ Glu1 Glu2
Quark-gluon vertex including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag4[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p1, k }, OutgoingMomenta -> { p2}, -> { mu, nu, rho}, DropSumOver -> True , LoopMomenta -> { l }, -> True , ChangeDimension -> D , List -> False , SMP -> True , FinalSubstitutions -> { ZA -> SMP[ "Z_A" ], Zg -> SMP[ "Z_g" ], Zpsi -> SMP[ "Z_psi" ], [ "m_u" ] -> SMP[ "m_q" ]}] i ( ( 1 − ξ G ) ( k + l − p2 ) ν ( − k − l + p2 ) ρ ( l 2 − m q 2 ) . ( ( k + l ) 2 − m q 2 ) . ( k + l − p2 ) 4 + g ν ρ ( l 2 − m q 2 ) . ( ( k + l ) 2 − m q 2 ) . ( k + l − p2 ) 2 ) ( − i γ ρ g s T Col3 Col5 Glu4 ) . ( γ ⋅ ( k + l ) + m q ) . ( − i γ μ g s T Col5 Col4 Glu2 ) . ( γ ⋅ l + m q ) . ( − i γ ν g s T Col4 Col1 Glu4 ) − i ( g s g Lor4 ρ ( − k − l ) μ f Glu2 Glu4 Glu5 + g s g Lor4 μ ( k + l ) ρ f Glu2 Glu4 Glu5 + g s ( − k Lor4 ) g μ ρ f Glu2 Glu4 Glu5 + g s k ρ g Lor4 μ f Glu2 Glu4 Glu5 + g s l Lor4 g μ ρ f Glu2 Glu4 Glu5 − g s l μ g Lor4 ρ f Glu2 Glu4 Glu5 ) ( ( 1 − ξ G ) g ν ρ ( − k − l ) Lor4 ( k + l ) Lor5 l 2 . ( k + l ) 4 . ( ( k + l − p2 ) 2 − m q 2 ) − ( 1 − ξ G ) l ν l ρ g Lor4 Lor5 ( l 2 ) 2 . ( k + l ) 2 . ( ( k + l − p2 ) 2 − m q 2 ) + − ( 1 − ξ G ) 2 l ν l ρ ( − k − l ) Lor4 ( k + l ) Lor5 ( l 2 ) 2 . ( k + l ) 4 . ( ( k + l − p2 ) 2 − m q 2 ) + g Lor4 Lor5 g ν ρ l 2 . ( k + l ) 2 . ( ( k + l − p2 ) 2 − m q 2 ) ) ( − i g s γ Lor5 T Col3 Col4 Glu5 ) . ( γ ⋅ ( − k − l + p2 ) + m q ) . ( − i γ ν g s T Col4 Col1 Glu4 ) − i γ μ g s ( Z A Z g Z ψ − 1 ) T Col3 Col1 Glu2 i \left(\frac{\left(1-\xi
_{\text{G}}\right) (k+l-\text{p2})^{\nu } (-k-l+\text{p2})^{\rho
}}{\left(l^2-m_q^2\right).\left((k+l)^2-m_q^2\right).(k+l-\text{p2})^4}+\frac{g^{\nu
\rho
}}{\left(l^2-m_q^2\right).\left((k+l)^2-m_q^2\right).(k+l-\text{p2})^2}\right)
\left(-i \gamma ^{\rho } g_s
T_{\text{Col3}\;\text{Col5}}^{\text{Glu4}}\right).\left(\gamma \cdot
(k+l)+m_q\right).\left(-i \gamma ^{\mu } g_s
T_{\text{Col5}\;\text{Col4}}^{\text{Glu2}}\right).\left(\gamma \cdot
l+m_q\right).\left(-i \gamma ^{\nu } g_s
T_{\text{Col4}\;\text{Col1}}^{\text{Glu4}}\right)-i \left(g_s
g^{\text{Lor4}\rho } (-k-l)^{\mu }
f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}+g_s g^{\text{Lor4}\mu }
(k+l)^{\rho } f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}+g_s
\left(-k^{\text{Lor4}}\right) g^{\mu \rho }
f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}+g_s k^{\rho }
g^{\text{Lor4}\mu } f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}+g_s
l^{\text{Lor4}} g^{\mu \rho }
f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}-g_s l^{\mu }
g^{\text{Lor4}\rho } f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}}\right)
\left(\frac{\left(1-\xi _{\text{G}}\right) g^{\nu \rho }
(-k-l)^{\text{Lor4}}
(k+l)^{\text{Lor5}}}{l^2.(k+l)^4.\left((k+l-\text{p2})^2-m_q^2\right)}-\frac{\left(1-\xi
_{\text{G}}\right) l^{\nu } l^{\rho }
g^{\text{Lor4}\;\text{Lor5}}}{\left(l^2\right)^2.(k+l)^2.\left((k+l-\text{p2})^2-m_q^2\right)}+-\frac{\left(1-\xi
_{\text{G}}\right){}^2 l^{\nu } l^{\rho } (-k-l)^{\text{Lor4}}
(k+l)^{\text{Lor5}}}{\left(l^2\right)^2.(k+l)^4.\left((k+l-\text{p2})^2-m_q^2\right)}+\frac{g^{\text{Lor4}\;\text{Lor5}}
g^{\nu \rho }}{l^2.(k+l)^2.\left((k+l-\text{p2})^2-m_q^2\right)}\right)
\left(-i g_s \gamma ^{\text{Lor5}}
T_{\text{Col3}\;\text{Col4}}^{\text{Glu5}}\right).\left(\gamma \cdot
(-k-l+\text{p2})+m_q\right).\left(-i \gamma ^{\nu } g_s
T_{\text{Col4}\;\text{Col1}}^{\text{Glu4}}\right)-i \gamma ^{\mu } g_s
\left(\sqrt{Z_A} Z_g Z_{\psi }-1\right)
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}} i ( ( l 2 − m q 2 ) . ( ( k + l ) 2 − m q 2 ) . ( k + l − p2 ) 4 ( 1 − ξ G ) ( k + l − p2 ) ν ( − k − l + p2 ) ρ + ( l 2 − m q 2 ) . ( ( k + l ) 2 − m q 2 ) . ( k + l − p2 ) 2 g ν ρ ) ( − i γ ρ g s T Col3 Col5 Glu4 ) . ( γ ⋅ ( k + l ) + m q ) . ( − i γ μ g s T Col5 Col4 Glu2 ) . ( γ ⋅ l + m q ) . ( − i γ ν g s T Col4 Col1 Glu4 ) − i ( g s g Lor4 ρ ( − k − l ) μ f Glu2 Glu4 Glu5 + g s g Lor4 μ ( k + l ) ρ f Glu2 Glu4 Glu5 + g s ( − k Lor4 ) g μ ρ f Glu2 Glu4 Glu5 + g s k ρ g Lor4 μ f Glu2 Glu4 Glu5 + g s l Lor4 g μ ρ f Glu2 Glu4 Glu5 − g s l μ g Lor4 ρ f Glu2 Glu4 Glu5 ) ( l 2 . ( k + l ) 4 . ( ( k + l − p2 ) 2 − m q 2 ) ( 1 − ξ G ) g ν ρ ( − k − l ) Lor4 ( k + l ) Lor5 − ( l 2 ) 2 . ( k + l ) 2 . ( ( k + l − p2 ) 2 − m q 2 ) ( 1 − ξ G ) l ν l ρ g Lor4 Lor5 + − ( l 2 ) 2 . ( k + l ) 4 . ( ( k + l − p2 ) 2 − m q 2 ) ( 1 − ξ G ) 2 l ν l ρ ( − k − l ) Lor4 ( k + l ) Lor5 + l 2 . ( k + l ) 2 . ( ( k + l − p2 ) 2 − m q 2 ) g Lor4 Lor5 g ν ρ ) ( − i g s γ Lor5 T Col3 Col4 Glu5 ) . ( γ ⋅ ( − k − l + p2 ) + m q ) . ( − i γ ν g s T Col4 Col1 Glu4 ) − i γ μ g s ( Z A Z g Z ψ − 1 ) T Col3 Col1 Glu2
Calculate the amplitudes
Quark self-energy
Tensor reduction allows us to express the quark self-energy in tems
of the Passarino-Veltman coefficient functions.
[ 1 ] = ampQuarkSE[ 0 ] // SUNSimplify // DiracSimplify // [ #, l , UsePaVeBasis -> True , ToPaVe -> True ] &;Discard all the finite pieces of the 1-loop amplitude.
[ 0 ] = ampQuarkSE[ 1 ] // PaVeUVPart[ #, Prefactor -> 1 / (2 Pi )^ D ] &;[ 1 ] = FCReplaceD[ ampQuarkSEDiv[ 0 ], D -> 4 - 2 Epsilon] // Series [ #, { Epsilon, 0 , 0 }] & // Normal // FCHideEpsilon // Simplify 1 32 π 2 i δ Col1 Col2 ( m q ( − ( ( C A 2 − 1 ) g s 2 ( 3 Δ ( C A − 2 C F ) + ( Δ + γ + log ( π ) ) ξ G ( C A − 2 C F ) + 3 γ C A + 3 log ( π ) C A + log ( 16384 ) C A − log ( 256 ) C A − log ( 64 ) C A − 6 γ C F − 6 log ( π ) C F ) ) − 32 π 2 C A ( C A − 2 C F ) ( Z m Z ψ − 1 ) ) + ( C A − 2 C F ) γ ⋅ p ( ( C A 2 − 1 ) ( Δ + γ + log ( π ) ) ξ G g s 2 + 32 π 2 C A ( Z ψ − 1 ) ) ) \frac{1}{32 \pi ^2}i \delta
_{\text{Col1}\;\text{Col2}} \left(m_q \left(-\left(\left(C_A^2-1\right)
g_s^2 \left(3 \Delta \left(C_A-2 C_F\right)+(\Delta +\gamma +\log (\pi
)) \xi _{\text{G}} \left(C_A-2 C_F\right)+3 \gamma C_A+3 \log (\pi )
C_A+\log (16384) C_A-\log (256) C_A-\log (64) C_A-6 \gamma C_F-6 \log
(\pi ) C_F\right)\right)-32 \pi ^2 C_A \left(C_A-2 C_F\right) \left(Z_m
Z_{\psi }-1\right)\right)+\left(C_A-2 C_F\right) \gamma \cdot p
\left(\left(C_A^2-1\right) (\Delta +\gamma +\log (\pi )) \xi _{\text{G}}
g_s^2+32 \pi ^2 C_A \left(Z_{\psi }-1\right)\right)\right) 32 π 2 1 i δ Col1 Col2 ( m q ( − ( ( C A 2 − 1 ) g s 2 ( 3Δ ( C A − 2 C F ) + ( Δ + γ + log ( π )) ξ G ( C A − 2 C F ) + 3 γ C A + 3 log ( π ) C A + log ( 16384 ) C A − log ( 256 ) C A − log ( 64 ) C A − 6 γ C F − 6 log ( π ) C F ) ) − 32 π 2 C A ( C A − 2 C F ) ( Z m Z ψ − 1 ) ) + ( C A − 2 C F ) γ ⋅ p ( ( C A 2 − 1 ) ( Δ + γ + log ( π )) ξ G g s 2 + 32 π 2 C A ( Z ψ − 1 ) ) )
[ 2 ] = ampQuarkSEDiv[ 1 ] // ReplaceRepeated [ #, { [ "Z_m" ] -> 1 + alpha SMP[ "d_m" ], [ "Z_psi" ] -> 1 + alpha SMP[ "d_psi" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] & // SelectNotFree2[ #, SMP[ "Delta" ], SMP[ "d_m" ], [ "d_psi" ]] &− i Δ ( C A 2 − 1 ) ξ G g s 2 m q ( C A − 2 C F ) δ Col1 Col2 32 π 2 + i Δ ( C A 2 − 1 ) ξ G g s 2 ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p 32 π 2 − 3 i Δ ( C A 2 − 1 ) g s 2 m q ( C A − 2 C F ) δ Col1 Col2 32 π 2 − i C A δ ψ m q ( C A − 2 C F ) δ Col1 Col2 − i C A δ m m q ( C A − 2 C F ) δ Col1 Col2 + i C A δ ψ ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p -\frac{i \Delta \left(C_A^2-1\right) \xi
_{\text{G}} g_s^2 m_q \left(C_A-2 C_F\right) \delta
_{\text{Col1}\;\text{Col2}}}{32 \pi ^2}+\frac{i
\Delta \left(C_A^2-1\right) \xi _{\text{G}} g_s^2 \left(C_A-2
C_F\right) \delta _{\text{Col1}\;\text{Col2}} \gamma \cdot p}{32 \pi
^2}-\frac{3 i \Delta \left(C_A^2-1\right) g_s^2 m_q \left(C_A-2
C_F\right) \delta _{\text{Col1}\;\text{Col2}}}{32 \pi ^2}-i C_A \delta
_{\psi } m_q \left(C_A-2 C_F\right) \delta _{\text{Col1}\;\text{Col2}}-i
C_A \delta _m m_q \left(C_A-2 C_F\right) \delta
_{\text{Col1}\;\text{Col2}}+i C_A \delta _{\psi } \left(C_A-2 C_F\right)
\delta _{\text{Col1}\;\text{Col2}} \gamma \cdot p − 32 π 2 i Δ ( C A 2 − 1 ) ξ G g s 2 m q ( C A − 2 C F ) δ Col1 Col2 + 32 π 2 i Δ ( C A 2 − 1 ) ξ G g s 2 ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p − 32 π 2 3 i Δ ( C A 2 − 1 ) g s 2 m q ( C A − 2 C F ) δ Col1 Col2 − i C A δ ψ m q ( C A − 2 C F ) δ Col1 Col2 − i C A δ m m q ( C A − 2 C F ) δ Col1 Col2 + i C A δ ψ ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p
[ 3 ] = ampQuarkSEDiv[ 2 ] // SUNSimplify // [ #, DiracGamma, Factoring -> Simplify ] &i ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p ( 32 π 2 C A δ ψ + Δ ( C A 2 − 1 ) ξ G g s 2 ) 32 π 2 − i m q ( C A − 2 C F ) δ Col1 Col2 ( 32 π 2 C A δ ψ + Δ ( C A 2 − 1 ) ( ξ G + 3 ) g s 2 + 32 π 2 C A δ m ) 32 π 2 \frac{i \left(C_A-2 C_F\right) \delta
_{\text{Col1}\;\text{Col2}} \gamma \cdot p \left(32 \pi ^2 C_A \delta
_{\psi }+\Delta \left(C_A^2-1\right) \xi _{\text{G}} g_s^2\right)}{32
\pi ^2}-\frac{i m_q \left(C_A-2 C_F\right) \delta
_{\text{Col1}\;\text{Col2}} \left(32 \pi ^2 C_A \delta _{\psi
}+\Delta \left(C_A^2-1\right) \left(\xi _{\text{G}}+3\right) g_s^2+32
\pi ^2 C_A \delta _m\right)}{32 \pi ^2} 32 π 2 i ( C A − 2 C F ) δ Col1 Col2 γ ⋅ p ( 32 π 2 C A δ ψ + Δ ( C A 2 − 1 ) ξ G g s 2 ) − 32 π 2 i m q ( C A − 2 C F ) δ Col1 Col2 ( 32 π 2 C A δ ψ + Δ ( C A 2 − 1 ) ( ξ G + 3 ) g s 2 + 32 π 2 C A δ m )
[ 1 ] = Solve [ SelectNotFree2[ ampQuarkSEDiv[ 3 ], DiracGamma] == 0 , SMP[ "d_psi" ]] // Flatten // ReplaceAll [ #, Rule [ a_ , b_ ] :> Rule [ a , SUNSimplify[ b ]]] & // ReplaceAll [ #, SMP[ "g_s" ] ^ 2 -> 4 Pi SMP[ "alpha_s" ]] &; [ 2 ] = Solve [ (SelectFree2[ ampQuarkSEDiv[ 3 ], DiracGamma] == 0 ) /. sol[ 1 ], SMP[ "d_m" ]] // Flatten // ReplaceAll [ #, Rule [ a_ , b_ ] :> Rule [ a , SUNSimplify[ b ]]] & // ReplaceAll [ #, SMP[ "g_s" ] ^ 2 -> 4 Pi SMP[ "alpha_s" ]] &; = Join [ sol[ 1 ], sol[ 2 ]] /. { [ "d_psi" ] -> SMP[ "d_psi^MS" ], [ "d_m" ] -> SMP[ "d_m^MS" ], SMP[ "Delta" ] -> 1 / Epsilon } = Join [ sol[ 1 ], sol[ 2 ]] /. { [ "d_psi" ] -> SMP[ "d_psi^MSbar" ], [ "d_m" ] -> SMP[ "d_m^MSbar" ] } { δ ψ MS → − C F ξ G α s 4 π ε , δ m MS → − 3 C F α s 4 π ε } \left\{\delta _{\psi }^{\text{MS}}\to
-\frac{C_F \xi _{\text{G}} \alpha _s}{4 \pi \varepsilon },\delta
_m^{\text{MS}}\to -\frac{3 C_F \alpha _s}{4 \pi \varepsilon
}\right\} { δ ψ MS → − 4 π ε C F ξ G α s , δ m MS → − 4 π ε 3 C F α s }
{ δ ψ MS − − − → − Δ C F ξ G α s 4 π , δ m MS − − − → − 3 Δ C F α s 4 π } \left\{\delta _{\psi
}^{\overset{---}{\text{MS}}}\to -\frac{\Delta C_F \xi _{\text{G}}
\alpha _s}{4 \pi },\delta _m^{\overset{---}{\text{MS}}}\to -\frac{3
\Delta C_F \alpha _s}{4 \pi }\right\} { δ ψ MS −−− → − 4 π Δ C F ξ G α s , δ m MS −−− → − 4 π 3Δ C F α s }
Gluon self-energy
Tensor reduction allows us to express the gluon self-energy in tems
of the Passarino-Veltman coefficient functions.
[ 1 ] = (ampGluonSE[ 0 ][[ 1 ]] + Nf ampGluonSE[ 0 ][[ 2 ]] + Total [ ampGluonSE[ 0 ][[ 3 ;;]]] ) // SUNSimplify // DiracSimplify;[ 2 ] = TID[ ampGluonSE[ 1 ], l , UsePaVeBasis -> True , ToPaVe -> True ] ;Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = ampGluonSE[ 2 ] // PaVeUVPart[ #, Prefactor -> 1 / (2 Pi )^ D ] &1 ( D − 4 ) ( D − 1 ) ξ G Z ξ i 2 − 2 D − 1 π − 2 D ( − 2 D C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 3 + C A ( 2 π ) D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 3 − 7 2 D + 1 C A π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 2 D + 1 C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 2 − 2 D + 1 C A π D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 2 D + 1 C A D π D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 15 2 D + 1 C A π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 3 N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 4 N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 D N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 9 2 D + 1 C A π D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 1 C A D π D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 N f π D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G + D N f ( 2 π ) D + 2 g μ ν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 5 N f π D + 2 g μ ν g s 2 m q 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 D N f π D + 2 g μ ν g s 2 m q 2 Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 1 D 2 π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G + 5 2 2 D + 1 D π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 3 π 2 D g μ ν p 2 Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 1 D 2 π 2 D g μ ν p 2 Z ξ δ Glu1 Glu2 ξ G − 5 2 2 D + 1 D π 2 D g μ ν p 2 Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 3 π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 1 D 2 π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G − 5 2 2 D + 1 D π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D g μ ν p 2 Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 1 D 2 π 2 D g μ ν p 2 Z A Z ξ δ Glu1 Glu2 ξ G + 5 2 2 D + 1 D π 2 D g μ ν p 2 Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D p μ p ν Z A δ Glu1 Glu2 − 2 2 D + 1 D 2 π 2 D p μ p ν Z A δ Glu1 Glu2 + 5 2 2 D + 1 D π 2 D p μ p ν Z A δ Glu1 Glu2 + 2 2 D + 3 π 2 D p μ p ν Z ξ δ Glu1 Glu2 + 2 2 D + 1 D 2 π 2 D p μ p ν Z ξ δ Glu1 Glu2 − 5 2 2 D + 1 D π 2 D p μ p ν Z ξ δ Glu1 Glu2 ) \frac{1}{(D-4) (D-1) \xi _{\text{G}}
Z_{\xi }}i 2^{-2 D-1} \pi ^{-2 D} \left(-2^D C_A D \pi ^{D+2} p^{\mu }
p^{\nu } g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}^3+C_A (2 \pi )^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}^3-7\ 2^{D+1} C_A \pi ^{D+2}
p^{\mu } p^{\nu } g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}^2+2^{D+1} C_A D \pi ^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}^2-2^{D+1} C_A \pi
^{D+2} g^{\mu \nu } p^2 g_s^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}^2+2^{D+1} C_A D \pi ^{D+2}
g^{\mu \nu } p^2 g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}^2+15\ 2^{D+1} C_A \pi ^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi
} \delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^D C_A D \pi
^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}+2^{D+3} N_f \pi ^{D+2}
p^{\mu } p^{\nu } g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}+2^{D+4} N_f \pi ^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{D+3} D N_f \pi
^{D+2} p^{\mu } p^{\nu } g_s^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-9\ 2^{D+1} C_A \pi ^{D+2}
g^{\mu \nu } p^2 g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}-2^{D+1} C_A D \pi ^{D+2} g^{\mu \nu } p^2 g_s^2 Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{D+3} N_f \pi
^{D+2} g^{\mu \nu } p^2 g_s^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}+D N_f (2 \pi )^{D+2} g^{\mu
\nu } p^2 g_s^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}+2^{D+5} N_f \pi ^{D+2} g^{\mu \nu } g_s^2 m_q^2 Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{D+3} D N_f \pi
^{D+2} g^{\mu \nu } g_s^2 m_q^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{2 D+3} \pi ^{2 D} p^{\mu
} p^{\nu } Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}-2^{2 D+1} D^2 \pi ^{2 D} p^{\mu } p^{\nu } Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}+5\ 2^{2 D+1} D \pi ^{2 D}
p^{\mu } p^{\nu } Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}+2^{2 D+3} \pi ^{2 D} g^{\mu \nu } p^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}+2^{2 D+1} D^2 \pi ^{2 D}
g^{\mu \nu } p^2 Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}-5\ 2^{2 D+1} D \pi ^{2 D} g^{\mu \nu } p^2 Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}+2^{2 D+3} \pi ^{2 D} p^{\mu
} p^{\nu } Z_A Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}+2^{2 D+1} D^2 \pi ^{2 D} p^{\mu } p^{\nu } Z_A Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-5\ 2^{2 D+1} D \pi
^{2 D} p^{\mu } p^{\nu } Z_A Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}}
\xi _{\text{G}}-2^{2 D+3} \pi ^{2 D} g^{\mu \nu } p^2 Z_A Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{2 D+1} D^2 \pi ^{2
D} g^{\mu \nu } p^2 Z_A Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}} \xi
_{\text{G}}+5\ 2^{2 D+1} D \pi ^{2 D} g^{\mu \nu } p^2 Z_A Z_{\xi }
\delta ^{\text{Glu1}\;\text{Glu2}} \xi _{\text{G}}-2^{2 D+3} \pi ^{2 D}
p^{\mu } p^{\nu } Z_A \delta ^{\text{Glu1}\;\text{Glu2}}-2^{2 D+1} D^2
\pi ^{2 D} p^{\mu } p^{\nu } Z_A \delta ^{\text{Glu1}\;\text{Glu2}}+5\
2^{2 D+1} D \pi ^{2 D} p^{\mu } p^{\nu } Z_A \delta
^{\text{Glu1}\;\text{Glu2}}+2^{2 D+3} \pi ^{2 D} p^{\mu } p^{\nu }
Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}}+2^{2 D+1} D^2 \pi ^{2 D}
p^{\mu } p^{\nu } Z_{\xi } \delta ^{\text{Glu1}\;\text{Glu2}}-5\ 2^{2
D+1} D \pi ^{2 D} p^{\mu } p^{\nu } Z_{\xi } \delta
^{\text{Glu1}\;\text{Glu2}}\right) ( D − 4 ) ( D − 1 ) ξ G Z ξ 1 i 2 − 2 D − 1 π − 2 D ( − 2 D C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 3 + C A ( 2 π ) D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 3 − 7 2 D + 1 C A π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 2 D + 1 C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G 2 − 2 D + 1 C A π D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 2 D + 1 C A D π D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G 2 + 15 2 D + 1 C A π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D C A D π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 3 N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 4 N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 D N f π D + 2 p μ p ν g s 2 Z ξ δ Glu1 Glu2 ξ G − 9 2 D + 1 C A π D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 1 C A D π D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 N f π D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G + D N f ( 2 π ) D + 2 g μν p 2 g s 2 Z ξ δ Glu1 Glu2 ξ G + 2 D + 5 N f π D + 2 g μν g s 2 m q 2 Z ξ δ Glu1 Glu2 ξ G − 2 D + 3 D N f π D + 2 g μν g s 2 m q 2 Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 1 D 2 π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G + 5 2 2 D + 1 D π 2 D p μ p ν Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 3 π 2 D g μν p 2 Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 1 D 2 π 2 D g μν p 2 Z ξ δ Glu1 Glu2 ξ G − 5 2 2 D + 1 D π 2 D g μν p 2 Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 3 π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G + 2 2 D + 1 D 2 π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G − 5 2 2 D + 1 D π 2 D p μ p ν Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D g μν p 2 Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 1 D 2 π 2 D g μν p 2 Z A Z ξ δ Glu1 Glu2 ξ G + 5 2 2 D + 1 D π 2 D g μν p 2 Z A Z ξ δ Glu1 Glu2 ξ G − 2 2 D + 3 π 2 D p μ p ν Z A δ Glu1 Glu2 − 2 2 D + 1 D 2 π 2 D p μ p ν Z A δ Glu1 Glu2 + 5 2 2 D + 1 D π 2 D p μ p ν Z A δ Glu1 Glu2 + 2 2 D + 3 π 2 D p μ p ν Z ξ δ Glu1 Glu2 + 2 2 D + 1 D 2 π 2 D p μ p ν Z ξ δ Glu1 Glu2 − 5 2 2 D + 1 D π 2 D p μ p ν Z ξ δ Glu1 Glu2 )
[ 1 ] = FCReplaceD[ ampGluonSEDiv[ 0 ], D -> 4 - 2 Epsilon] // Series [ #, { Epsilon, 0 , 0 }] & // Normal // FCHideEpsilon // SUNSimplify;[ 2 ] = ampGluonSEDiv[ 1 ] // ReplaceRepeated [ #, { [ "Z_A" ] -> 1 + alpha SMP[ "d_A" ], [ "Z_xi" ] -> 1 + alpha SMP[ "d_A" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] & // SelectNotFree2[ #, SMP[ "Delta" ], SMP[ "d_A" ], [ "d_xi" ]] &i Δ C A ξ G g s 2 p μ p ν δ Glu1 Glu2 32 π 2 − i Δ p 2 C A ξ G g s 2 g μ ν δ Glu1 Glu2 32 π 2 − 13 i Δ C A g s 2 p μ p ν δ Glu1 Glu2 96 π 2 + 13 i Δ p 2 C A g s 2 g μ ν δ Glu1 Glu2 96 π 2 − i p 2 δ A g μ ν δ Glu1 Glu2 + i δ A p μ p ν δ Glu1 Glu2 + i Δ N f g s 2 p μ p ν δ Glu1 Glu2 24 π 2 − i Δ p 2 N f g s 2 g μ ν δ Glu1 Glu2 24 π 2 \frac{i \Delta C_A \xi _{\text{G}} g_s^2
p^{\mu } p^{\nu } \delta ^{\text{Glu1}\;\text{Glu2}}}{32 \pi ^2}-\frac{i
\Delta p^2 C_A \xi _{\text{G}} g_s^2 g^{\mu \nu } \delta
^{\text{Glu1}\;\text{Glu2}}}{32 \pi ^2}-\frac{13 i \Delta C_A g_s^2
p^{\mu } p^{\nu } \delta ^{\text{Glu1}\;\text{Glu2}}}{96 \pi
^2}+\frac{13 i \Delta p^2 C_A g_s^2 g^{\mu \nu } \delta
^{\text{Glu1}\;\text{Glu2}}}{96 \pi ^2}-i p^2 \delta _A g^{\mu \nu }
\delta ^{\text{Glu1}\;\text{Glu2}}+i \delta _A p^{\mu } p^{\nu } \delta
^{\text{Glu1}\;\text{Glu2}}+\frac{i \Delta N_f g_s^2 p^{\mu } p^{\nu }
\delta ^{\text{Glu1}\;\text{Glu2}}}{24 \pi ^2}-\frac{i \Delta p^2 N_f
g_s^2 g^{\mu \nu } \delta ^{\text{Glu1}\;\text{Glu2}}}{24 \pi
^2} 32 π 2 i Δ C A ξ G g s 2 p μ p ν δ Glu1 Glu2 − 32 π 2 i Δ p 2 C A ξ G g s 2 g μν δ Glu1 Glu2 − 96 π 2 13 i Δ C A g s 2 p μ p ν δ Glu1 Glu2 + 96 π 2 13 i Δ p 2 C A g s 2 g μν δ Glu1 Glu2 − i p 2 δ A g μν δ Glu1 Glu2 + i δ A p μ p ν δ Glu1 Glu2 + 24 π 2 i Δ N f g s 2 p μ p ν δ Glu1 Glu2 − 24 π 2 i Δ p 2 N f g s 2 g μν δ Glu1 Glu2
[ 3 ] = Solve [ ampGluonSEDiv[ 2 ] == 0 , SMP[ "d_A" ]] // Flatten // ReplaceAll [ #, SMP[ "g_s" ] ^ 2 -> 4 Pi SMP[ "alpha_s" ]] & // Simplify = sol[ 3 ] /. { SMP[ "d_A" ] -> SMP[ "d_A^MS" ], SMP[ "Delta" ] -> 1 / Epsilon} = sol[ 3 ] /. { SMP[ "d_A" ] -> SMP[ "d_A^MSbar" ]} { δ A → − Δ α s ( 3 C A ξ G − 13 C A + 4 N f ) 24 π } \left\{\delta _A\to -\frac{\Delta \alpha
_s \left(3 C_A \xi _{\text{G}}-13 C_A+4 N_f\right)}{24 \pi
}\right\} { δ A → − 24 π Δ α s ( 3 C A ξ G − 13 C A + 4 N f ) }
{ δ A MS → − α s ( 3 C A ξ G − 13 C A + 4 N f ) 24 π ε } \left\{\delta _A^{\text{MS}}\to
-\frac{\alpha _s \left(3 C_A \xi _{\text{G}}-13 C_A+4 N_f\right)}{24
\pi \varepsilon }\right\} { δ A MS → − 24 π ε α s ( 3 C A ξ G − 13 C A + 4 N f ) }
{ δ A MS − − − → − Δ α s ( 3 C A ξ G − 13 C A + 4 N f ) 24 π } \left\{\delta
_A^{\overset{---}{\text{MS}}}\to -\frac{\Delta \alpha _s \left(3 C_A
\xi _{\text{G}}-13 C_A+4 N_f\right)}{24 \pi }\right\} { δ A MS −−− → − 24 π Δ α s ( 3 C A ξ G − 13 C A + 4 N f ) }
Ghost self-energy
Tensor reduction allows us to express the ghost self-energy in tems
of the Passarino-Veltman coefficient functions.
[ 1 ] = ampGhostSE[ 0 ] // SUNSimplify // DiracSimplify;[ 2 ] = TID[ ampGhostSE[ 1 ], l , UsePaVeBasis -> True , ToPaVe -> True ] ;Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = ampGhostSE[ 2 ] // PaVeUVPart[ #, Prefactor -> 1 / (2 Pi )^ D ] &i 2 − D − 1 π − D p 2 δ Glu1 Glu2 ( π 2 ( − C A ) ξ G g s 2 + 3 π 2 C A g s 2 − 2 D + 3 π D Z u + 2 D + 1 D π D Z u + 2 D + 3 π D − 2 D + 1 D π D ) D − 4 \frac{i 2^{-D-1} \pi ^{-D} p^2 \delta
^{\text{Glu1}\;\text{Glu2}} \left(\pi ^2 \left(-C_A\right) \xi
_{\text{G}} g_s^2+3 \pi ^2 C_A g_s^2-2^{D+3} \pi ^D Z_u+2^{D+1} D \pi ^D
Z_u+2^{D+3} \pi ^D-2^{D+1} D \pi ^D\right)}{D-4} D − 4 i 2 − D − 1 π − D p 2 δ Glu1 Glu2 ( π 2 ( − C A ) ξ G g s 2 + 3 π 2 C A g s 2 − 2 D + 3 π D Z u + 2 D + 1 D π D Z u + 2 D + 3 π D − 2 D + 1 D π D )
[ 1 ] = FCReplaceD[ ampGhostSEDiv[ 0 ], D -> 4 - 2 Epsilon] // Series [ #, { Epsilon, 0 , 0 }] & // Normal // FCHideEpsilon // SUNSimplify− 1 64 π 2 i p 2 δ Glu1 Glu2 ( − Δ C A ξ G g s 2 + γ ( − C A ) ξ G g s 2 + log ( 4 π ) C A ξ G g s 2 − 2 log ( π ) C A ξ G g s 2 − 2 log ( 2 ) C A ξ G g s 2 + 3 Δ C A g s 2 + 3 γ C A g s 2 − 3 log ( 4 π ) C A g s 2 + 6 log ( π ) C A g s 2 + 6 log ( 2 ) C A g s 2 − 64 π 2 Z u + 64 π 2 ) -\frac{1}{64 \pi ^2}i p^2 \delta
^{\text{Glu1}\;\text{Glu2}} \left(-\Delta C_A \xi _{\text{G}}
g_s^2+\gamma \left(-C_A\right) \xi _{\text{G}} g_s^2+\log (4 \pi ) C_A
\xi _{\text{G}} g_s^2-2 \log (\pi ) C_A \xi _{\text{G}} g_s^2-2 \log (2)
C_A \xi _{\text{G}} g_s^2+3 \Delta C_A g_s^2+3 \gamma C_A g_s^2-3 \log
(4 \pi ) C_A g_s^2+6 \log (\pi ) C_A g_s^2+6 \log (2) C_A g_s^2-64 \pi
^2 Z_u+64 \pi ^2\right) − 64 π 2 1 i p 2 δ Glu1 Glu2 ( − Δ C A ξ G g s 2 + γ ( − C A ) ξ G g s 2 + log ( 4 π ) C A ξ G g s 2 − 2 log ( π ) C A ξ G g s 2 − 2 log ( 2 ) C A ξ G g s 2 + 3Δ C A g s 2 + 3 γ C A g s 2 − 3 log ( 4 π ) C A g s 2 + 6 log ( π ) C A g s 2 + 6 log ( 2 ) C A g s 2 − 64 π 2 Z u + 64 π 2 )
[ 2 ] = ampGhostSEDiv[ 1 ] // ReplaceRepeated [ #, { [ "Z_u" ] -> 1 + alpha SMP[ "d_u" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] & // [ #, SMP[ "Delta" ], SMP[ "d_u" ]] & // Simplify i p 2 δ Glu1 Glu2 ( Δ C A ( ξ G − 3 ) g s 2 + 64 π 2 δ u ) 64 π 2 \frac{i p^2 \delta
^{\text{Glu1}\;\text{Glu2}} \left(\Delta C_A \left(\xi
_{\text{G}}-3\right) g_s^2+64 \pi ^2 \delta _u\right)}{64 \pi
^2} 64 π 2 i p 2 δ Glu1 Glu2 ( Δ C A ( ξ G − 3 ) g s 2 + 64 π 2 δ u )
[ 4 ] = Solve [ ampGhostSEDiv[ 2 ] == 0 , SMP[ "d_u" ]] // Flatten // ReplaceAll [ #, SMP[ "g_s" ] ^ 2 -> 4 Pi SMP[ "alpha_s" ]] & // Simplify = sol[ 4 ] /. { SMP[ "d_u" ] -> SMP[ "d_u^MS" ], SMP[ "Delta" ] -> 1 / Epsilon} = sol[ 4 ] /. { SMP[ "d_u" ] -> SMP[ "d_u^MSbar" ]} { δ u → − Δ C A ( ξ G − 3 ) α s 16 π } \left\{\delta _u\to -\frac{\Delta C_A
\left(\xi _{\text{G}}-3\right) \alpha _s}{16 \pi }\right\} { δ u → − 16 π Δ C A ( ξ G − 3 ) α s }
{ δ u MS → − C A ( ξ G − 3 ) α s 16 π ε } \left\{\delta _u^{\text{MS}}\to -\frac{C_A
\left(\xi _{\text{G}}-3\right) \alpha _s}{16 \pi \varepsilon
}\right\} { δ u MS → − 16 π ε C A ( ξ G − 3 ) α s }
{ δ u MS − − − → − Δ C A ( ξ G − 3 ) α s 16 π } \left\{\delta
_u^{\overset{---}{\text{MS}}}\to -\frac{\Delta C_A \left(\xi
_{\text{G}}-3\right) \alpha _s}{16 \pi }\right\} { δ u MS −−− → − 16 π Δ C A ( ξ G − 3 ) α s }
Quark-gluon vertex
Tensor reduction allows us to express the quark-gluon vertex in tems
of the Passarino-Veltman coefficient functions.
[ 1 ] = ampQGlVertex[ 0 ] // SUNSimplify // DiracSimplify;[ 2 ] = TID[ ampQGlVertex[ 1 ], l , UsePaVeBasis -> True , ToPaVe -> True ] ;Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = ampQGlVertex[ 2 ] // PaVeUVPart[ #, Prefactor -> 1 / (2 Pi )^ D ] &− 1 D − 4 i ( 2 π ) − D ( − 2 D + 3 π D C A γ μ C F g s T Col3 Col1 Glu2 + 2 D + 1 D π D C A γ μ C F g s T Col3 Col1 Glu2 + 2 D + 3 π D C A Z A γ μ C F g s Z g Z ψ T Col3 Col1 Glu2 − 2 D + 1 D π D C A Z A γ μ C F g s Z g Z ψ T Col3 Col1 Glu2 + 2 D + 2 π D C A 2 γ μ g s T Col3 Col1 Glu2 − D ( 2 π ) D C A 2 γ μ g s T Col3 Col1 Glu2 − 2 D + 2 π D C A 2 Z A γ μ g s Z g Z ψ T Col3 Col1 Glu2 + D ( 2 π ) D C A 2 Z A γ μ g s Z g Z ψ T Col3 Col1 Glu2 + π 2 C A γ μ ξ G g s 3 T Col3 Col1 Glu2 − 2 π 2 γ μ C F ξ G g s 3 T Col3 Col1 Glu2 − 3 i π 2 γ μ ξ G g s 3 f Glu2 FCGV ( sun15311 ) FCGV ( sun15312 ) ( T FCGV ( sun15312 ) T FCGV ( sun15311 ) ) Col3 Col1 − 3 i π 2 γ μ g s 3 f Glu2 FCGV ( sun15311 ) FCGV ( sun15312 ) ( T FCGV ( sun15312 ) T FCGV ( sun15311 ) ) Col3 Col1 ) -\frac{1}{D-4}i (2 \pi )^{-D}
\left(-2^{D+3} \pi ^D C_A \gamma ^{\mu } C_F g_s
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+2^{D+1} D \pi ^D C_A \gamma
^{\mu } C_F g_s T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+2^{D+3} \pi
^D C_A \sqrt{Z_A} \gamma ^{\mu } C_F g_s Z_g Z_{\psi }
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-2^{D+1} D \pi ^D C_A
\sqrt{Z_A} \gamma ^{\mu } C_F g_s Z_g Z_{\psi }
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+2^{D+2} \pi ^D C_A^2 \gamma
^{\mu } g_s T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-D (2 \pi )^D
C_A^2 \gamma ^{\mu } g_s
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-2^{D+2} \pi ^D C_A^2
\sqrt{Z_A} \gamma ^{\mu } g_s Z_g Z_{\psi }
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+D (2 \pi )^D C_A^2 \sqrt{Z_A}
\gamma ^{\mu } g_s Z_g Z_{\psi }
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+\pi ^2 C_A \gamma ^{\mu } \xi
_{\text{G}} g_s^3 T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-2 \pi ^2
\gamma ^{\mu } C_F \xi _{\text{G}} g_s^3
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-3 i \pi ^2 \gamma ^{\mu } \xi
_{\text{G}} g_s^3
f^{\text{Glu2}\;\text{FCGV}(\text{sun15311})\text{FCGV}(\text{sun15312})}
\left(T^{\text{FCGV}(\text{sun15312})}T^{\text{FCGV}(\text{sun15311})}\right){}_{\text{Col3}\;\text{Col1}}-3
i \pi ^2 \gamma ^{\mu } g_s^3
f^{\text{Glu2}\;\text{FCGV}(\text{sun15311})\text{FCGV}(\text{sun15312})}
\left(T^{\text{FCGV}(\text{sun15312})}T^{\text{FCGV}(\text{sun15311})}\right){}_{\text{Col3}\;\text{Col1}}\right) − D − 4 1 i ( 2 π ) − D ( − 2 D + 3 π D C A γ μ C F g s T Col3 Col1 Glu2 + 2 D + 1 D π D C A γ μ C F g s T Col3 Col1 Glu2 + 2 D + 3 π D C A Z A γ μ C F g s Z g Z ψ T Col3 Col1 Glu2 − 2 D + 1 D π D C A Z A γ μ C F g s Z g Z ψ T Col3 Col1 Glu2 + 2 D + 2 π D C A 2 γ μ g s T Col3 Col1 Glu2 − D ( 2 π ) D C A 2 γ μ g s T Col3 Col1 Glu2 − 2 D + 2 π D C A 2 Z A γ μ g s Z g Z ψ T Col3 Col1 Glu2 + D ( 2 π ) D C A 2 Z A γ μ g s Z g Z ψ T Col3 Col1 Glu2 + π 2 C A γ μ ξ G g s 3 T Col3 Col1 Glu2 − 2 π 2 γ μ C F ξ G g s 3 T Col3 Col1 Glu2 − 3 i π 2 γ μ ξ G g s 3 f Glu2 FCGV ( sun15311 ) FCGV ( sun15312 ) ( T FCGV ( sun15312 ) T FCGV ( sun15311 ) ) Col3 Col1 − 3 i π 2 γ μ g s 3 f Glu2 FCGV ( sun15311 ) FCGV ( sun15312 ) ( T FCGV ( sun15312 ) T FCGV ( sun15311 ) ) Col3 Col1 )
[ 1 ] = FCReplaceD[ ampQGlVertexDiv[ 0 ], D -> 4 - 2 Epsilon] // Series [ #, { Epsilon, 0 , 0 }] & // Normal // FCHideEpsilon // SUNSimplify;[ 2 ] = ampQGlVertexDiv[ 1 ] // ReplaceRepeated [ #, { [ "Z_g" ] -> 1 + alpha SMP[ "d_g" ], [ "Z_A" ] -> 1 + alpha SMP[ "d_A" ], [ "Z_psi" ] -> 1 + alpha SMP[ "d_psi" ] }] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] & // [ #, SMP[ "Delta" ], SMP[ "d_g" ], SMP[ "d_A" ], SMP[ "d_psi" ]] & // Simplify − 1 32 π 2 i γ μ g s ( 32 π 2 C A δ g ( C A − 2 C F ) T Col3 Col1 Glu2 − 64 π 2 C A C F δ ψ T Col3 Col1 Glu2 + 16 π 2 C A δ A ( C A − 2 C F ) T Col3 Col1 Glu2 − Δ C A ξ G g s 2 T Col3 Col1 Glu2 + 32 π 2 C A 2 δ ψ T Col3 Col1 Glu2 + 2 Δ C F ξ G g s 2 T Col3 Col1 Glu2 + 3 i Δ ξ G g s 2 f Glu2 FCGV ( sun69301 ) FCGV ( sun69302 ) ( T FCGV ( sun69302 ) T FCGV ( sun69301 ) ) Col3 Col1 + 3 i Δ g s 2 f Glu2 FCGV ( sun69301 ) FCGV ( sun69302 ) ( T FCGV ( sun69302 ) T FCGV ( sun69301 ) ) Col3 Col1 ) -\frac{1}{32 \pi ^2}i \gamma ^{\mu } g_s
\left(32 \pi ^2 C_A \delta _g \left(C_A-2 C_F\right)
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-64 \pi ^2 C_A C_F \delta
_{\psi } T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+16 \pi ^2 C_A \delta
_A \left(C_A-2 C_F\right)
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}-\Delta C_A \xi _{\text{G}}
g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+32 \pi ^2 C_A^2 \delta
_{\psi } T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+2 \Delta C_F \xi
_{\text{G}} g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}}+3 i
\Delta \xi _{\text{G}} g_s^2
f^{\text{Glu2}\;\text{FCGV}(\text{sun69301})\text{FCGV}(\text{sun69302})}
\left(T^{\text{FCGV}(\text{sun69302})}T^{\text{FCGV}(\text{sun69301})}\right){}_{\text{Col3}\;\text{Col1}}+3
i \Delta g_s^2
f^{\text{Glu2}\;\text{FCGV}(\text{sun69301})\text{FCGV}(\text{sun69302})}
\left(T^{\text{FCGV}(\text{sun69302})}T^{\text{FCGV}(\text{sun69301})}\right){}_{\text{Col3}\;\text{Col1}}\right) − 32 π 2 1 i γ μ g s ( 32 π 2 C A δ g ( C A − 2 C F ) T Col3 Col1 Glu2 − 64 π 2 C A C F δ ψ T Col3 Col1 Glu2 + 16 π 2 C A δ A ( C A − 2 C F ) T Col3 Col1 Glu2 − Δ C A ξ G g s 2 T Col3 Col1 Glu2 + 32 π 2 C A 2 δ ψ T Col3 Col1 Glu2 + 2Δ C F ξ G g s 2 T Col3 Col1 Glu2 + 3 i Δ ξ G g s 2 f Glu2 FCGV ( sun69301 ) FCGV ( sun69302 ) ( T FCGV ( sun69302 ) T FCGV ( sun69301 ) ) Col3 Col1 + 3 i Δ g s 2 f Glu2 FCGV ( sun69301 ) FCGV ( sun69302 ) ( T FCGV ( sun69302 ) T FCGV ( sun69301 ) ) Col3 Col1 )
[ 3 ] = ampQGlVertexDiv[ 2 ] // SUNSimplify[ #, Explicit -> True ] & // ReplaceAll [ #, SUNTrace[ x__ ] :> SUNTrace[ x , Explicit -> True ]] & // [ #, Epsilon, SUNIndex] &3 Δ γ μ ( ξ G + 1 ) g s 3 f Glu2 FCGV ( sun69481 ) FCGV ( sun69482 ) ( T FCGV ( sun69482 ) T FCGV ( sun69481 ) ) Col3 Col1 32 π 2 − i γ μ g s ( C A − 2 C F ) T Col3 Col1 Glu2 ( 32 π 2 C A δ ψ + 16 π 2 C A δ A + 32 π 2 C A δ g + Δ ( − ξ G ) g s 2 ) 32 π 2 \frac{3 \Delta \gamma ^{\mu } \left(\xi
_{\text{G}}+1\right) g_s^3
f^{\text{Glu2}\;\text{FCGV}(\text{sun69481})\text{FCGV}(\text{sun69482})}
\left(T^{\text{FCGV}(\text{sun69482})}T^{\text{FCGV}(\text{sun69481})}\right){}_{\text{Col3}\;\text{Col1}}}{32
\pi ^2}-\frac{i \gamma ^{\mu } g_s \left(C_A-2 C_F\right)
T_{\text{Col3}\;\text{Col1}}^{\text{Glu2}} \left(32 \pi ^2 C_A \delta
_{\psi }+16 \pi ^2 C_A \delta _A+32 \pi ^2 C_A \delta
_g+\Delta \left(-\xi _{\text{G}}\right) g_s^2\right)}{32 \pi
^2} 32 π 2 3Δ γ μ ( ξ G + 1 ) g s 3 f Glu2 FCGV ( sun69481 ) FCGV ( sun69482 ) ( T FCGV ( sun69482 ) T FCGV ( sun69481 ) ) Col3 Col1 − 32 π 2 i γ μ g s ( C A − 2 C F ) T Col3 Col1 Glu2 ( 32 π 2 C A δ ψ + 16 π 2 C A δ A + 32 π 2 C A δ g + Δ ( − ξ G ) g s 2 )
[ 4 ] = ampQGlVertexDiv[ 3 ] // ReplaceAll [ #, suntf[ xx_ , _SUNFIndex, _SUNFIndex] :> SUNT @@ xx] & // ReplaceAll [ #, SUNTF -> suntf] & // ReplaceAll [ #, [ xx_ , _SUNFIndex, _SUNFIndex] :> SUNT @@ xx] & // // Collect2[ #, SMP] &− i Δ γ μ g s 3 T Glu2 ( C A − 2 C F ) ( 3 C A 2 ξ G + 3 C A 2 − 2 ξ G ) 64 π 2 − i C A γ μ δ ψ g s T Glu2 ( C A − 2 C F ) − 1 2 i C A δ A γ μ g s T Glu2 ( C A − 2 C F ) − i C A γ μ δ g g s T Glu2 ( C A − 2 C F ) -\frac{i \Delta \gamma ^{\mu } g_s^3
T^{\text{Glu2}} \left(C_A-2 C_F\right) \left(3 C_A^2 \xi _{\text{G}}+3
C_A^2-2 \xi _{\text{G}}\right)}{64 \pi ^2}-i C_A \gamma ^{\mu } \delta
_{\psi } g_s T^{\text{Glu2}} \left(C_A-2 C_F\right)-\frac{1}{2} i C_A
\delta _A \gamma ^{\mu } g_s T^{\text{Glu2}} \left(C_A-2 C_F\right)-i
C_A \gamma ^{\mu } \delta _g g_s T^{\text{Glu2}} \left(C_A-2
C_F\right) − 64 π 2 i Δ γ μ g s 3 T Glu2 ( C A − 2 C F ) ( 3 C A 2 ξ G + 3 C A 2 − 2 ξ G ) − i C A γ μ δ ψ g s T Glu2 ( C A − 2 C F ) − 2 1 i C A δ A γ μ g s T Glu2 ( C A − 2 C F ) − i C A γ μ δ g g s T Glu2 ( C A − 2 C F )
[ 5 ] = (ampQGlVertexDiv[ 4 ] /. { [ "d_A" ] -> SMP[ "d_A^MS" ], [ "d_psi" ] -> SMP[ "d_psi^MS" ], [ "d_g" ] -> SMP[ "d_g" ], [ "Delta" ] -> 1 / Epsilon } /. solMS1 /. solMS2) // ReplaceAll [ #, SMP[ "g_s" ] ^ 3 -> 4 Pi SMP[ "alpha_s" ] SMP[ "g_s" ]] & // [ #, Epsilon] & // SUNSimplifyi γ μ g s T Glu2 ( − 11 C A α s + 2 N f α s − 24 π ε δ g ) 24 π ε \frac{i \gamma ^{\mu } g_s T^{\text{Glu2}}
\left(-11 C_A \alpha _s+2 N_f \alpha _s-24 \pi \varepsilon \delta
_g\right)}{24 \pi \varepsilon } 24 π ε i γ μ g s T Glu2 ( − 11 C A α s + 2 N f α s − 24 π ε δ g )
[ 5 ] = Solve [ ampQGlVertexDiv[ 5 ] == 0 , SMP[ "d_g" ]] // Flatten // Simplify = sol[ 5 ] /. { SMP[ "d_g" ] -> SMP[ "d_g^MS" ]} = sol[ 5 ] /. { SMP[ "d_g" ] -> SMP[ "d_g^MSbar" ], 1 / Epsilon -> SMP[ "Delta" ]} { δ g → α s ( 2 N f − 11 C A ) 24 π ε } \left\{\delta _g\to \frac{\alpha _s
\left(2 N_f-11 C_A\right)}{24 \pi \varepsilon }\right\} { δ g → 24 π ε α s ( 2 N f − 11 C A ) }
{ δ g MS → α s ( 2 N f − 11 C A ) 24 π ε } \left\{\delta _g^{\text{MS}}\to
\frac{\alpha _s \left(2 N_f-11 C_A\right)}{24 \pi \varepsilon
}\right\} { δ g MS → 24 π ε α s ( 2 N f − 11 C A ) }
{ δ g MS − − − → Δ α s ( 2 N f − 11 C A ) 24 π } \left\{\delta
_g^{\overset{---}{\text{MS}}}\to \frac{\Delta \alpha _s \left(2 N_f-11
C_A\right)}{24 \pi }\right\} { δ g MS −−− → 24 π Δ α s ( 2 N f − 11 C A ) }
Check the final results
= { [ "d_psi^MS" ] -> - SMP[ "alpha_s" ] / (4 Pi ) 1 / Epsilon CF GaugeXi[ "G" ], [ "d_m^MS" ] -> - SMP[ "alpha_s" ] / (4 Pi ) 1 / Epsilon 3 CF, [ "d_psi^MSbar" ] -> - SMP[ "alpha_s" ] / (4 Pi ) SMP[ "Delta" ] CF GaugeXi[ "G" ], [ "d_m^MSbar" ] -> - SMP[ "alpha_s" ] / (4 Pi ) SMP[ "Delta" ] 3 CF, [ "d_A^MS" ] -> SMP[ "alpha_s" ] / (4 Pi ) 1 / Epsilon (1 / 2 CA (13 / 3 - GaugeXi[ "G" ] ) - 2 / 3 Nf), [ "d_A^MSbar" ] -> SMP[ "alpha_s" ] / (4 Pi ) SMP[ "Delta" ] (1 / 2 CA (13 / 3 - GaugeXi[ "G" ] ) - 2 / 3 Nf), [ "d_u^MS" ] -> SMP[ "alpha_s" ] / (4 Pi ) CA/ Epsilon (3 - GaugeXi[ "G" ] )/ 4 , [ "d_u^MSbar" ] -> SMP[ "alpha_s" ] / (4 Pi ) CA SMP[ "Delta" ] (3 - GaugeXi[ "G" ] )/ 4 , [ "d_g^MS" ] -> ((- 11 * CA* SMP[ "alpha_s" ] )/ (24 * Epsilon* Pi ) + (Nf* SMP[ "alpha_s" ] )/ (12 * Epsilon* Pi )), [ "d_g^MSbar" ] -> ((- 11 * CA* SMP[ "alpha_s" ] )/ (24 Pi ) SMP[ "Delta" ] + (Nf* SMP[ "alpha_s" ] )/ (12 * Pi ) SMP[ "Delta" ] ) } // Factor2;[ Join [ solMS1, solMSbar1, solMS2, solMSbar2, solMS3, solMSbar3, solMS4, solMSbar4] // Factor2, knownResult, Text -> { " \t Compare to Muta, Foundations of QCD, Eqs 2.5.131-2.5.147:" , "CORRECT." , "WRONG!" }, Interrupt -> { Hold [ Quit [ 1 ]], Automatic }] ;Print [ " \t CPU Time used: " , Round [ N [ TimeUsed [], 4 ], 0.001 ], " s." ] ;\ tCompare to Muta, Foundations of QCD, Eqs 2.5.131-2.5.147: CORRECT. \text{$\backslash $tCompare to Muta,
Foundations of QCD, Eqs 2.5.131-2.5.147:} \;\text{CORRECT.} \tCompare to Muta, Foundations of QCD, Eqs 2.5.131-2.5.147: CORRECT.
\ tCPU Time used: 47.537 s. \text{$\backslash $tCPU Time used:
}47.537\text{ s.} \tCPU Time used: 47.537 s.
