Load
FeynCalc and the necessary add-ons or other packages
This example uses a custom QED model created with FeynRules. Please
evaluate the file FeynCalc/Examples/FeynRules/QED/GenerateModelQED.m
before running it for the first time.
= "Renormalization, QED, 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]" ;= { InsertionLevel -> { Particles}, Model -> FileNameJoin [{ "QED" , "QED" }], -> FileNameJoin [{ "QED" , "QED" }], ExcludeParticles -> { F [ 2 , { 2 | 3 }]}} ;top [ i_ , j_ ] := CreateTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs}] ;[ i_ , j_ ] := CreateCTTopologies[ 1 , i -> j , -> { Tadpoles, WFCorrections, WFCorrectionCTs}] ; { diagElectronSE, diagElectronSECT} = InsertFields[ #, { F [ 2 , { 1 }]} -> { F [ 2 , { 1 }]}, Sequence @@ params] & / @ { top [ 1 , 1 ], topCT[ 1 , 1 ]} ;{ diagPhotonSE, diagPhotonSECT} = InsertFields[ #, { V [ 1 ]} -> { V [ 1 ]}, Sequence @@ params] & / @ { top [ 1 , 1 ], topCT[ 1 , 1 ]} ;{ diagVertex, diagVertexCT} = InsertFields[ #, { F [ 2 , { 1 }], V [ 1 ]} -> { F [ 2 , { 1 }]}, Sequence @@ params] & / @ { top [ 2 , 1 ], topCT[ 2 , 1 ]} ;[ 0 ] = diagElectronSE[[ 0 ]][ diagElectronSE[[ 1 ]], diagElectronSECT[[ 1 ]]] ;[ 0 ] = diagPhotonSE[[ 0 ]][ diagPhotonSE[[ 1 ]], diagPhotonSECT[[ 1 ]]] ;[ 0 ] = diagVertex[[ 0 ]][ diagVertex[[ 1 ]], diagVertexCT[[ 1 ]]] ;[ diag1[ 0 ], ColumnsXRows -> { 2 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
[ diag2[ 0 ], ColumnsXRows -> { 2 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
[ diag3[ 0 ], ColumnsXRows -> { 2 , 1 }, SheetHeader -> None , -> Simple, ImageSize -> { 512 , 256 }] ;
Obtain the amplitudes
The 1/(2Pi)^D prefactor is implicit. We need to replace e with -e to
be compatible with the convention D^mu = d^mu + ie A^mu
Electron self-energy including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag1[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p }, OutgoingMomenta -> { p }, -> { mu}, -> { l }, UndoChiralSplittings -> True , -> D , List -> False , SMP -> True , -> { Zm -> SMP[ "Z_m" ], Zpsi -> SMP[ "Z_psi" ], [ "e" ] -> Sqrt [ 4 Pi SMP[ "alpha_fs" ]], GaugeXi[ V [ 1 ]] -> GaugeXi}, -> True ] − 4 π α γ μ . ( m e + γ ⋅ l ) . γ μ ( l 2 − m e 2 ) . ( l − p ) 2 − 4 π α ( 1 − ξ ) ( γ ⋅ ( p − l ) ) . ( m e + γ ⋅ l ) . ( γ ⋅ ( l − p ) ) ( l 2 − m e 2 ) . ( l − p ) 4 − i m e ( Z m Z ψ − 1 ) + i ( Z ψ − 1 ) γ ⋅ p -\frac{4 \pi \alpha \gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).\gamma ^{\mu
}}{\left(l^2-m_e^2\right).(l-p)^2}-\frac{4 \pi \alpha (1-\xi ) (\gamma
\cdot (p-l)).\left(m_e+\gamma \cdot l\right).(\gamma \cdot
(l-p))}{\left(l^2-m_e^2\right).(l-p)^4}-i m_e \left(Z_m Z_{\psi
}-1\right)+i \left(Z_{\psi }-1\right) \gamma \cdot p − ( l 2 − m e 2 ) . ( l − p ) 2 4 π α γ μ . ( m e + γ ⋅ l ) . γ μ − ( l 2 − m e 2 ) . ( l − p ) 4 4 π α ( 1 − ξ ) ( γ ⋅ ( p − l )) . ( m e + γ ⋅ l ) . ( γ ⋅ ( l − p )) − i m e ( Z m Z ψ − 1 ) + i ( Z ψ − 1 ) γ ⋅ p
Photon self-energy including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag2[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p }, OutgoingMomenta -> { p }, -> { mu, nu}, -> { l }, UndoChiralSplittings -> True , -> D , List -> False , SMP -> True , -> { ZA -> SMP[ "Z_A" ], Zxi -> SMP[ "Z_xi" ], [ "e" ] -> Sqrt [ 4 Pi SMP[ "alpha_fs" ]], GaugeXi[ V [ 1 ]] -> GaugeXi}, -> True ] // FCTraceFactor− 4 π α tr ( ( m e − γ ⋅ l ) . γ ν . ( m e + γ ⋅ ( p − l ) ) . γ μ ) ( l 2 − m e 2 ) . ( ( l − p ) 2 − m e 2 ) − i p 2 ( Z A − 1 ) g μ ν − i p μ p ν ( Z A − Z ξ ) ξ Z ξ + i ( Z A − 1 ) p μ p ν -\frac{4
\pi \alpha \;\text{tr}\left(\left(m_e-\gamma \cdot l\right).\gamma
^{\nu }.\left(m_e+\gamma \cdot (p-l)\right).\gamma ^{\mu
}\right)}{\left(l^2-m_e^2\right).\left((l-p)^2-m_e^2\right)}-i p^2
\left(Z_A-1\right) g^{\mu \nu }-\frac{i p^{\mu } p^{\nu }
\left(Z_A-Z_{\xi }\right)}{\xi Z_{\xi }}+i \left(Z_A-1\right) p^{\mu }
p^{\nu } − ( l 2 − m e 2 ) . ( ( l − p ) 2 − m e 2 ) 4 π α tr ( ( m e − γ ⋅ l ) . γ ν . ( m e + γ ⋅ ( p − l ) ) . γ μ ) − i p 2 ( Z A − 1 ) g μν − ξ Z ξ i p μ p ν ( Z A − Z ξ ) + i ( Z A − 1 ) p μ p ν
Electron-photon vertex including the counter-term
[ 0 ] = FCFAConvert[ CreateFeynAmp[ diag3[ 0 ], Truncated -> True , -> {}, PreFactor -> 1 ], -> { p1, k }, OutgoingMomenta -> { p2}, -> { mu}, LoopMomenta -> { l }, -> True , ChangeDimension -> D , List -> False , SMP -> True , FinalSubstitutions -> { ZA -> SMP[ "Z_A" ], Ze -> SMP[ "Z_e" ], Zpsi -> SMP[ "Z_psi" ], [ "e" ] ^ 3 -> 4 Pi SMP[ "alpha_fs" ] SMP[ "e" ], GaugeXi[ V [ 1 ]] -> GaugeXi}, -> True ] /. SMP[ "e" ] -> - SMP[ "e" ] − 4 π α e γ Lor3 . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . γ Lor3 ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 2 − 4 π α e ( 1 − ξ ) ( γ ⋅ ( − k − l + p2 ) ) . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . ( γ ⋅ ( k + l − p2 ) ) ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 4 − i e γ μ ( Z A Z e Z ψ − 1 ) -\frac{4 \pi \alpha \;\text{e} \gamma
^{\text{Lor3}}.\left(m_e+\gamma \cdot (k+l)\right).\gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).\gamma
^{\text{Lor3}}}{\left(l^2-m_e^2\right).\left((k+l)^2-m_e^2\right).(k+l-\text{p2})^2}-\frac{4
\pi \alpha \;\text{e} (1-\xi ) (\gamma \cdot
(-k-l+\text{p2})).\left(m_e+\gamma \cdot (k+l)\right).\gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).(\gamma \cdot
(k+l-\text{p2}))}{\left(l^2-m_e^2\right).\left((k+l)^2-m_e^2\right).(k+l-\text{p2})^4}-i
\;\text{e} \gamma ^{\mu } \left(\sqrt{Z_A} Z_e Z_{\psi
}-1\right) − ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 2 4 π α e γ Lor3 . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . γ Lor3 − ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 4 4 π α e ( 1 − ξ ) ( γ ⋅ ( − k − l + p2 )) . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . ( γ ⋅ ( k + l − p2 )) − i e γ μ ( Z A Z e Z ψ − 1 )
Calculate the amplitudes
Electron self-energy
[ 1 ] = amp1[ 0 ] // ReplaceAll [ #, { SMP[ "Z_psi" ] -> 1 + alpha SMP[ "d_psi" ], [ "Z_m" ] -> 1 + alpha SMP[ "d_m" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] &− 4 π α γ μ . ( m e + γ ⋅ l ) . γ μ ( l 2 − m e 2 ) . ( l − p ) 2 − 4 π α ( 1 − ξ ) ( γ ⋅ ( p − l ) ) . ( m e + γ ⋅ l ) . ( γ ⋅ ( l − p ) ) ( l 2 − m e 2 ) . ( l − p ) 4 − i m e ( δ ψ + δ m ) + i δ ψ γ ⋅ p -\frac{4 \pi \alpha \gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).\gamma ^{\mu
}}{\left(l^2-m_e^2\right).(l-p)^2}-\frac{4 \pi \alpha (1-\xi ) (\gamma
\cdot (p-l)).\left(m_e+\gamma \cdot l\right).(\gamma \cdot
(l-p))}{\left(l^2-m_e^2\right).(l-p)^4}-i m_e \left(\delta _{\psi
}+\delta _m\right)+i \delta _{\psi } \gamma \cdot p − ( l 2 − m e 2 ) . ( l − p ) 2 4 π α γ μ . ( m e + γ ⋅ l ) . γ μ − ( l 2 − m e 2 ) . ( l − p ) 4 4 π α ( 1 − ξ ) ( γ ⋅ ( p − l )) . ( m e + γ ⋅ l ) . ( γ ⋅ ( l − p )) − i m e ( δ ψ + δ m ) + i δ ψ γ ⋅ p
Tensor reduction allows us to express the electron self-energy in
tems of the Passarino-Veltman coefficient functions.
[ 2 ] = TID[ amp1[ 1 ], l , ToPaVe -> True ] 1 p 2 2 i π 3 α B 0 ( p 2 , 0 , m e 2 ) ( − D ( p 2 − m e 2 ) γ ⋅ p − 2 D p 2 m e + 2 D p 2 γ ⋅ p + ξ m e 2 γ ⋅ p − 2 ξ m e ( γ ⋅ p ) . ( γ ⋅ p ) − m e 2 γ ⋅ p + 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) + 2 ( p 2 − m e 2 ) γ ⋅ p + ξ p 2 γ ⋅ p − 5 p 2 γ ⋅ p ) − 2 i π 3 α ( 1 − ξ ) B 0 ( 0 , 0 , 0 ) ( m e 2 ( − ( γ ⋅ p ) ) + 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) − 2 p 2 m e + p 2 γ ⋅ p ) p 2 + 2 i π 3 α ( 1 − ξ ) ( − m e 2 ( p 2 − m e 2 ) γ ⋅ p − 4 p 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) + 2 m e ( p 2 − m e 2 ) ( γ ⋅ p ) . ( γ ⋅ p ) + p 2 ( p 2 − m e 2 ) γ ⋅ p + 2 p 2 m e 3 + 2 p 4 m e ) C 0 ( 0 , p 2 , p 2 , 0 , 0 , m e 2 ) p 2 + 2 i π 3 α ( 2 − D ) γ ⋅ p A 0 ( m e 2 ) p 2 + i ( − δ ψ m e − m e δ m + δ ψ γ ⋅ p ) \frac{1}{p^2}2 i \pi ^3
\alpha \;\text{B}_0\left(p^2,0,m_e^2\right) \left(-D
\left(p^2-m_e^2\right) \gamma \cdot p-2 D p^2 m_e+2 D p^2 \gamma \cdot
p+\xi m_e^2 \gamma \cdot p-2 \xi m_e (\gamma \cdot p).(\gamma \cdot
p)-m_e^2 \gamma \cdot p+2 m_e (\gamma \cdot p).(\gamma \cdot p)+2
\left(p^2-m_e^2\right) \gamma \cdot p+\xi p^2 \gamma \cdot p-5 p^2
\gamma \cdot p\right)-\frac{2 i \pi ^3 \alpha (1-\xi )
\;\text{B}_0(0,0,0) \left(m_e^2 (-(\gamma \cdot p))+2 m_e (\gamma \cdot
p).(\gamma \cdot p)-2 p^2 m_e+p^2 \gamma \cdot p\right)}{p^2}+\frac{2 i
\pi ^3 \alpha (1-\xi ) \left(-m_e^2 \left(p^2-m_e^2\right) \gamma \cdot
p-4 p^2 m_e (\gamma \cdot p).(\gamma \cdot p)+2 m_e
\left(p^2-m_e^2\right) (\gamma \cdot p).(\gamma \cdot p)+p^2
\left(p^2-m_e^2\right) \gamma \cdot p+2 p^2 m_e^3+2 p^4 m_e\right)
\;\text{C}_0\left(0,p^2,p^2,0,0,m_e^2\right)}{p^2}+\frac{2 i \pi ^3
\alpha (2-D) \gamma \cdot p \;\text{A}_0\left(m_e^2\right)}{p^2}+i
\left(-\delta _{\psi } m_e-m_e \delta _m+\delta _{\psi } \gamma \cdot
p\right) p 2 1 2 i π 3 α B 0 ( p 2 , 0 , m e 2 ) ( − D ( p 2 − m e 2 ) γ ⋅ p − 2 D p 2 m e + 2 D p 2 γ ⋅ p + ξ m e 2 γ ⋅ p − 2 ξ m e ( γ ⋅ p ) . ( γ ⋅ p ) − m e 2 γ ⋅ p + 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) + 2 ( p 2 − m e 2 ) γ ⋅ p + ξ p 2 γ ⋅ p − 5 p 2 γ ⋅ p ) − p 2 2 i π 3 α ( 1 − ξ ) B 0 ( 0 , 0 , 0 ) ( m e 2 ( − ( γ ⋅ p )) + 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) − 2 p 2 m e + p 2 γ ⋅ p ) + p 2 2 i π 3 α ( 1 − ξ ) ( − m e 2 ( p 2 − m e 2 ) γ ⋅ p − 4 p 2 m e ( γ ⋅ p ) . ( γ ⋅ p ) + 2 m e ( p 2 − m e 2 ) ( γ ⋅ p ) . ( γ ⋅ p ) + p 2 ( p 2 − m e 2 ) γ ⋅ p + 2 p 2 m e 3 + 2 p 4 m e ) C 0 ( 0 , p 2 , p 2 , 0 , 0 , m e 2 ) + p 2 2 i π 3 α ( 2 − D ) γ ⋅ p A 0 ( m e 2 ) + i ( − δ ψ m e − m e δ m + δ ψ γ ⋅ p )
Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = PaVeUVPart[ amp1[ 2 ], Prefactor -> 1 / (2 Pi )^ D ] // [ #, D -> 4 - 2 Epsilon] & // Series [ #, { Epsilon, 0 , 0 }] & // Normal // // SelectNotFree2[ #, { SMP[ "Delta" ], SMP[ "d_m" ], [ "d_psi" ]}] & // Simplify i ( γ ⋅ p ( α Δ ξ + 4 π δ ψ ) − m e ( α Δ ( ξ + 3 ) + 4 π ( δ ψ + δ m ) ) ) 4 π \frac{i \left(\gamma \cdot p
\left(\alpha \Delta \xi +4 \pi \delta _{\psi }\right)-m_e
\left(\alpha \Delta (\xi +3)+4 \pi \left(\delta _{\psi }+\delta
_m\right)\right)\right)}{4 \pi } 4 π i ( γ ⋅ p ( α Δ ξ + 4 π δ ψ ) − m e ( α Δ ( ξ + 3 ) + 4 π ( δ ψ + δ m ) ) )
Equating the result to zero and solving for d_psi and d_m we obtain
the renormalization constants in the minimal subtraction schemes.
[ 1 ] = Solve [ SelectNotFree2[ amp1Div[ 0 ], DiracGamma] == 0 , [ "d_psi" ]] // Flatten // Simplify ;[ 2 ] = Solve [ (SelectFree2[ amp1Div[ 0 ], DiracGamma] == 0 ) /. sol[ 1 ], [ "d_m" ]] // Flatten // Simplify ;= 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 → − α ξ 4 π ε , δ m MS → − 3 α 4 π ε } \left\{\delta _{\psi }^{\text{MS}}\to
-\frac{\alpha \xi }{4 \pi \varepsilon },\delta _m^{\text{MS}}\to
-\frac{3 \alpha }{4 \pi \varepsilon }\right\} { δ ψ MS → − 4 π ε α ξ , δ m MS → − 4 π ε 3 α }
{ δ ψ MS − − − → − α Δ ξ 4 π , δ m MS − − − → − 3 α Δ 4 π } \left\{\delta _{\psi
}^{\overset{---}{\text{MS}}}\to -\frac{\alpha \Delta \xi }{4 \pi
},\delta _m^{\overset{---}{\text{MS}}}\to -\frac{3 \alpha \Delta }{4
\pi }\right\} { δ ψ MS −−− → − 4 π α Δ ξ , δ m MS −−− → − 4 π 3 α Δ }
Photon self-energy
[ 1 ] = amp2[ 0 ] // ReplaceRepeated [ #, { SMP[ "Z_xi" ] -> SMP[ "Z_A" ], [ "Z_A" ] -> 1 + alpha SMP[ "d_A" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] &− 4 π α tr ( ( m e − γ ⋅ l ) . γ ν . ( m e + γ ⋅ ( p − l ) ) . γ μ ) ( l 2 − m e 2 ) . ( ( l − p ) 2 − m e 2 ) + i ( δ A p μ p ν − p 2 δ A g μ ν ) -\frac{4
\pi \alpha \;\text{tr}\left(\left(m_e-\gamma \cdot l\right).\gamma
^{\nu }.\left(m_e+\gamma \cdot (p-l)\right).\gamma ^{\mu
}\right)}{\left(l^2-m_e^2\right).\left((l-p)^2-m_e^2\right)}+i
\left(\delta _A p^{\mu } p^{\nu }-p^2 \delta _A g^{\mu \nu
}\right) − ( l 2 − m e 2 ) . ( ( l − p ) 2 − m e 2 ) 4 π α tr ( ( m e − γ ⋅ l ) . γ ν . ( m e + γ ⋅ ( p − l ) ) . γ μ ) + i ( δ A p μ p ν − p 2 δ A g μν )
Tensor reduction allows us to express the electron self-energy in
tems of the Passarino-Veltman coefficient functions.
[ 2 ] = TID[ amp2[ 1 ], l , ToPaVe -> True ] 8 i π 3 α B 0 ( p 2 , m e 2 , m e 2 ) ( − ( ( 1 − D ) p 4 g μ ν ) + 2 ( 1 − D ) p 2 p μ p ν + D p 2 p μ p ν + 4 p 2 m e 2 g μ ν − 4 m e 2 p μ p ν − p 4 g μ ν ) ( 1 − D ) p 2 − 16 i π 3 α A 0 ( m e 2 ) ( − ( 1 − D ) p 2 g μ ν − D p μ p ν − p 2 g μ ν + 2 p μ p ν ) ( 1 − D ) p 2 + i δ A ( p μ p ν − p 2 g μ ν ) \frac{8 i \pi ^3
\alpha \;\text{B}_0\left(p^2,m_e^2,m_e^2\right) \left(-\left((1-D) p^4
g^{\mu \nu }\right)+2 (1-D) p^2 p^{\mu } p^{\nu }+D p^2 p^{\mu } p^{\nu
}+4 p^2 m_e^2 g^{\mu \nu }-4 m_e^2 p^{\mu } p^{\nu }-p^4 g^{\mu \nu
}\right)}{(1-D) p^2}-\frac{16 i \pi ^3
\alpha \;\text{A}_0\left(m_e^2\right) \left(-(1-D) p^2 g^{\mu \nu }-D
p^{\mu } p^{\nu }-p^2 g^{\mu \nu }+2 p^{\mu } p^{\nu }\right)}{(1-D)
p^2}+i \delta _A \left(p^{\mu } p^{\nu }-p^2 g^{\mu \nu
}\right) ( 1 − D ) p 2 8 i π 3 α B 0 ( p 2 , m e 2 , m e 2 ) ( − ( ( 1 − D ) p 4 g μν ) + 2 ( 1 − D ) p 2 p μ p ν + D p 2 p μ p ν + 4 p 2 m e 2 g μν − 4 m e 2 p μ p ν − p 4 g μν ) − ( 1 − D ) p 2 16 i π 3 α A 0 ( m e 2 ) ( − ( 1 − D ) p 2 g μν − D p μ p ν − p 2 g μν + 2 p μ p ν ) + i δ A ( p μ p ν − p 2 g μν )
Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = PaVeUVPart[ amp2[ 2 ], Prefactor -> 1 / (2 Pi )^ D ] // [ #, D -> 4 - 2 Epsilon] & // Series [ #, { Epsilon, 0 , 0 }] & // Normal // // SelectNotFree2[ #, { SMP[ "Delta" ], SMP[ "d_A" ]}] & // Simplify i ( α Δ + 3 π δ A ) ( p μ p ν − p 2 g μ ν ) 3 π \frac{i \left(\alpha \Delta +3
\pi \delta _A\right) \left(p^{\mu } p^{\nu }-p^2 g^{\mu \nu }\right)}{3
\pi } 3 π i ( α Δ + 3 π δ A ) ( p μ p ν − p 2 g μν )
Equating this to zero and solving for d_A we obtain the wave-function
renormalization constant for the photon in the minimal subtraction
schemes.
[ 3 ] = Solve [ amp2Div[ 0 ] == 0 , SMP[ "d_A" ]] // Flatten ;= sol[ 3 ] /. { SMP[ "d_A" ] -> SMP[ "d_A^MS" ], SMP[ "Delta" ] -> 1 / Epsilon} = sol[ 3 ] /. { SMP[ "d_A" ] -> SMP[ "d_A^MSbar" ]} { δ A MS → − α 3 π ε } \left\{\delta _A^{\text{MS}}\to
-\frac{\alpha }{3 \pi \varepsilon }\right\} { δ A MS → − 3 π ε α }
{ δ A MS − − − → − α Δ 3 π } \left\{\delta
_A^{\overset{---}{\text{MS}}}\to -\frac{\alpha \Delta }{3 \pi
}\right\} { δ A MS −−− → − 3 π α Δ }
Electron-photon vertex
[ 1 ] = amp3[ 0 ] // ReplaceRepeated [ #, { SMP[ "Z_psi" ] -> 1 + alpha SMP[ "d_psi" ], [ "Z_A" ] -> 1 + alpha SMP[ "d_A" ], SMP[ "Z_e" ] -> 1 + alpha SMP[ "d_e" ]}] & // Series [ #, { alpha, 0 , 1 }] & // Normal // ReplaceAll [ #, alpha -> 1 ] &− 4 π α e γ Lor3 . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . γ Lor3 ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 2 − 4 π α e ( 1 − ξ ) ( γ ⋅ ( − k − l + p2 ) ) . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . ( γ ⋅ ( k + l − p2 ) ) ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 4 − i e γ μ ( δ A 2 + δ ψ + δ e ) -\frac{4 \pi \alpha \;\text{e} \gamma
^{\text{Lor3}}.\left(m_e+\gamma \cdot (k+l)\right).\gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).\gamma
^{\text{Lor3}}}{\left(l^2-m_e^2\right).\left((k+l)^2-m_e^2\right).(k+l-\text{p2})^2}-\frac{4
\pi \alpha \;\text{e} (1-\xi ) (\gamma \cdot
(-k-l+\text{p2})).\left(m_e+\gamma \cdot (k+l)\right).\gamma ^{\mu
}.\left(m_e+\gamma \cdot l\right).(\gamma \cdot
(k+l-\text{p2}))}{\left(l^2-m_e^2\right).\left((k+l)^2-m_e^2\right).(k+l-\text{p2})^4}-i
\;\text{e} \gamma ^{\mu } \left(\frac{\delta _A}{2}+\delta _{\psi
}+\delta _e\right) − ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 2 4 π α e γ Lor3 . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . γ Lor3 − ( l 2 − m e 2 ) . ( ( k + l ) 2 − m e 2 ) . ( k + l − p2 ) 4 4 π α e ( 1 − ξ ) ( γ ⋅ ( − k − l + p2 )) . ( m e + γ ⋅ ( k + l ) ) . γ μ . ( m e + γ ⋅ l ) . ( γ ⋅ ( k + l − p2 )) − i e γ μ ( 2 δ A + δ ψ + δ e )
The result of the tensor reduction is quite large, since we keep the
full gauge dependence and do not specify the kinematics
[ 2 ] = TID[ amp3[ 1 ], l , ToPaVe -> True , UsePaVeBasis -> True ] 8 i ( 1 − ξ ) π 3 B 0 ( 0 , 0 , 0 ) γ μ α e + 4 i ( 2 − D ) π 3 B 0 ( p2 2 , 0 , m e 2 ) γ μ α e − 4 i D ( 1 − ξ ) π 3 γ μ C 00 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 ) α e − 8 i ( 2 − D ) π 3 γ μ C 00 ( k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 γ ⋅ k k μ C 11 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 γ ⋅ p2 p2 μ C 11 ( p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , m e 2 , 0 , m e 2 ) α e − 4 i ( 1 − ξ ) π 3 γ μ ( k 2 − 2 ( k ⋅ p2 ) + p2 2 ) C 11 ( k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 ( γ ⋅ p2 k μ + γ ⋅ k p2 μ ) C 12 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α e − 1 2 i γ μ ( δ A + 2 δ e + 2 δ ψ ) e + 4 i ( 1 − ξ ) π 3 C 1 ( k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , 0 ) α ( γ μ . ( − ( γ ⋅ ( k − p2 ) ) ) . ( γ ⋅ k ) − γ μ . ( − ( γ ⋅ ( k − p2 ) ) ) . ( γ ⋅ p2 ) − 2 γ ⋅ p2 k μ + 2 γ ⋅ p2 p2 μ − 2 γ μ k 2 + 4 γ μ ( k ⋅ p2 ) − 2 γ μ p2 2 + γ μ . ( − ( γ ⋅ ( k − p2 ) ) ) m e ) e + 4 i ( 1 − ξ ) π 3 C 1 ( p2 2 , p2 2 , 0 , 0 , m e 2 , 0 ) α ( ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ + ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ − ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ − 2 γ μ ( k ⋅ p2 ) + ( γ ⋅ p2 ) . γ μ m e ) e + 4 i π 3 C 0 ( k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + ξ ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) + 2 D γ ⋅ k k μ − 4 γ ⋅ k k μ − 2 ξ γ ⋅ p2 k μ + 2 γ ⋅ p2 k μ + D γ μ . ( γ ⋅ k ) m e − 4 γ μ . ( γ ⋅ k ) m e + 4 k μ m e ) e + 4 i π 3 C 1 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 4 D γ ⋅ k k μ − 8 γ ⋅ k k μ − 2 ξ γ ⋅ p2 k μ + 2 γ ⋅ p2 k μ + D γ μ . ( γ ⋅ k ) m e − 4 γ μ . ( γ ⋅ k ) m e + D ( γ ⋅ k ) . γ μ m e − 4 ( γ ⋅ k ) . γ μ m e + 8 k μ m e ) e + 4 i π 3 C 1 ( p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , m e 2 , 0 , m e 2 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) + 4 ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − 2 ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + 2 D γ ⋅ p2 k μ − 4 γ ⋅ p2 k μ + 2 D γ ⋅ k p2 μ − 4 γ ⋅ k p2 μ − 2 ξ γ ⋅ p2 p2 μ + 2 γ ⋅ p2 p2 μ + D γ μ . ( γ ⋅ p2 ) m e − 4 γ μ . ( γ ⋅ p2 ) m e + D ( γ ⋅ p2 ) . γ μ m e − 4 ( γ ⋅ p2 ) . γ μ m e + 8 p2 μ m e ) e − 4 i ( 1 − ξ ) π 3 C 0 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 ) α ( − γ μ m e 2 + γ μ . ( γ ⋅ k ) m e − γ μ . ( γ ⋅ p2 ) m e + γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) − γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) − γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + 2 γ ⋅ p2 k μ − 2 γ ⋅ p2 p2 μ + γ μ k 2 − 2 γ μ ( k ⋅ p2 ) + γ μ p2 2 ) e − 4 i ( 1 − ξ ) π 3 C 0 ( 0 , p2 2 , p2 2 , 0 , 0 , m e 2 ) α ( − γ μ m e 2 − ( γ ⋅ p2 ) . γ μ m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ − ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ + 2 γ ⋅ p2 p2 μ − 2 γ μ k 2 + 2 γ μ ( k ⋅ p2 ) − γ μ p2 2 ) e − 8 i ( 1 − ξ ) π 3 D 00 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ m e 2 − γ μ . ( γ ⋅ k ) m e + γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + γ μ p2 2 ) e − 8 i ( 1 − ξ ) π 3 k μ D 11 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 , 0 ) α ( − ( γ ⋅ k ) m e 2 − ( γ ⋅ k ) . ( γ ⋅ k ) m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) + γ ⋅ k p2 2 ) e − 16 i ( 1 − ξ ) π 3 p2 μ D 11 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ ⋅ p2 p2 2 ) e − 16 i ( 1 − ξ ) π 3 p2 μ D 12 ( p2 2 , 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , 0 , 0 , m e 2 ) α ( − ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ ⋅ p2 p2 2 ) e − 16 i ( 1 − ξ ) π 3 D 12 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − ( γ ⋅ p2 ) k μ m e 2 − γ ⋅ k p2 μ m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) k μ m e − ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) p2 μ m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ + ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ + γ ⋅ p2 k μ p2 2 + γ ⋅ k p2 μ p2 2 ) e + 4 i ( 1 − ξ ) π 3 D 0 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ . ( γ ⋅ k ) m e 3 − γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e 2 + γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e 2 − 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ m e 2 + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 + 4 γ ⋅ k k μ m e 2 − 2 γ ⋅ p2 k μ m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) k μ m e − 4 ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e − γ μ . ( γ ⋅ p2 ) k 2 m e + ( γ ⋅ p2 ) . γ μ k 2 m e + 2 γ μ . ( γ ⋅ p2 ) ( k ⋅ p2 ) m e − γ μ . ( γ ⋅ k ) p2 2 m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ − 2 γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) − 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) ( k ⋅ p2 ) + 4 γ ⋅ p2 k μ ( k ⋅ p2 ) + ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) p2 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) p2 2 − 2 γ ⋅ k k μ p2 2 − 2 γ ⋅ p2 k μ p2 2 ) e − 4 i ( 1 − ξ ) π 3 D 1 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 , 0 ) α ( γ μ . ( γ ⋅ k ) m e 3 + ( γ ⋅ k ) . γ μ m e 3 + γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e 2 − γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e 2 + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ m e 2 − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ m e 2 − 4 γ ⋅ k k μ m e 2 + 2 γ ⋅ p2 k μ m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ k ) k μ m e + 4 ( γ ⋅ k ) . ( γ ⋅ p2 ) k μ m e + 4 ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e + γ μ . ( γ ⋅ k ) k 2 m e + γ μ . ( γ ⋅ k ) p2 2 m e + ( γ ⋅ k ) . γ μ p2 2 m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) k μ − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) k 2 + 2 γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) − 4 γ ⋅ p2 k μ ( k ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) p2 2 + 4 γ ⋅ k k μ p2 2 + 2 γ ⋅ p2 k μ p2 2 ) e + 8 i ( 1 − ξ ) π 3 D 1 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ . ( γ ⋅ p2 ) m e 3 − ( γ ⋅ p2 ) . γ μ m e 3 − γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) m e 2 + γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ m e 2 − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 + 2 ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ m e 2 + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ m e 2 + 2 γ ⋅ p2 k μ m e 2 + 2 γ ⋅ k p2 μ m e 2 − 2 γ ⋅ p2 p2 μ m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e − 4 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) k μ m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) p2 μ m e − γ μ . ( γ ⋅ p2 ) k 2 m e − γ μ . ( γ ⋅ p2 ) p2 2 m e − ( γ ⋅ p2 ) . γ μ p2 2 m e + 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) p2 μ − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) k 2 − 2 γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) + 4 γ ⋅ p2 p2 μ ( k ⋅ p2 ) + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) p2 2 − 2 γ ⋅ p2 k μ p2 2 − 2 γ ⋅ k p2 μ p2 2 − 2 γ ⋅ p2 p2 μ p2 2 ) e 8 i (1-\xi ) \pi ^3 \;\text{B}_0(0,0,0)
\gamma ^{\mu } \alpha \;\text{e}+4 i (2-D) \pi ^3
\;\text{B}_0\left(\text{p2}^2,0,m_e^2\right) \gamma ^{\mu }
\alpha \;\text{e}-4 i D (1-\xi ) \pi ^3 \gamma ^{\mu }
\;\text{C}_{00}\left(0,k^2-2 (k\cdot \;\text{p2})+\text{p2}^2,k^2-2
(k\cdot \;\text{p2})+\text{p2}^2,0,0,m_e^2\right) \alpha \;\text{e}-8 i
(2-D) \pi ^3 \gamma ^{\mu } \;\text{C}_{00}\left(k^2,\text{p2}^2,k^2-2
(k\cdot \;\text{p2})+\text{p2}^2,m_e^2,m_e^2,0\right)
\alpha \;\text{e}-8 i (2-D) \pi ^3 \gamma \cdot k k^{\mu }
\;\text{C}_{11}\left(k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,m_e^2,m_e^2,0\right)
\alpha \;\text{e}-8 i (2-D) \pi ^3 \gamma \cdot \;\text{p2}
\;\text{p2}^{\mu } \;\text{C}_{11}\left(\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,k^2,m_e^2,0,m_e^2\right) \alpha \;\text{e}-4 i
(1-\xi ) \pi ^3 \gamma ^{\mu } \left(k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2\right) \;\text{C}_{11}\left(k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,0\right) \alpha \;\text{e}-8 i (2-D)
\pi ^3 \left(\gamma \cdot \;\text{p2} k^{\mu }+\gamma \cdot k
\;\text{p2}^{\mu }\right) \;\text{C}_{12}\left(k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,m_e^2,m_e^2,0\right)
\alpha \;\text{e}-\frac{1}{2} i \gamma ^{\mu } \left(\delta _A+2 \delta
_e+2 \delta _{\psi }\right) \;\text{e}+4 i (1-\xi ) \pi ^3
\;\text{C}_1\left(k^2-2 (k\cdot \;\text{p2})+\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,0\right) \alpha \left(\gamma ^{\mu
}.(-(\gamma \cdot (k-\text{p2}))).(\gamma \cdot k)-\gamma ^{\mu
}.(-(\gamma \cdot (k-\text{p2}))).(\gamma \cdot \;\text{p2})-2 \gamma
\cdot \;\text{p2} k^{\mu }+2 \gamma \cdot \;\text{p2} \;\text{p2}^{\mu
}-2 \gamma ^{\mu } k^2+4 \gamma ^{\mu } (k\cdot \;\text{p2})-2 \gamma
^{\mu } \;\text{p2}^2+\gamma ^{\mu }.(-(\gamma \cdot (k-\text{p2})))
m_e\right) \;\text{e}+4 i (1-\xi ) \pi ^3
\;\text{C}_1\left(\text{p2}^2,\text{p2}^2,0,0,m_e^2,0\right)
\alpha \left((\gamma \cdot k).(\gamma \cdot \;\text{p2}).\gamma ^{\mu
}+(\gamma \cdot \;\text{p2}).(\gamma \cdot k).\gamma ^{\mu }-(\gamma
\cdot \;\text{p2}).(\gamma \cdot \;\text{p2}).\gamma ^{\mu }-2 \gamma
^{\mu } (k\cdot \;\text{p2})+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }
m_e\right) \;\text{e}+4 i \pi ^3 \;\text{C}_0\left(k^2,\text{p2}^2,k^2-2
(k\cdot \;\text{p2})+\text{p2}^2,m_e^2,m_e^2,0\right) \alpha \left(-D
(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)+2 (\gamma \cdot
k).\gamma ^{\mu }.(\gamma \cdot k)+\xi (\gamma \cdot k).\gamma ^{\mu
}.(\gamma \cdot \;\text{p2})-(\gamma \cdot k).\gamma ^{\mu }.(\gamma
\cdot \;\text{p2})+2 D \gamma \cdot k k^{\mu }-4 \gamma \cdot k k^{\mu
}-2 \xi \gamma \cdot \;\text{p2} k^{\mu }+2 \gamma \cdot \;\text{p2}
k^{\mu }+D \gamma ^{\mu }.(\gamma \cdot k) m_e-4 \gamma ^{\mu }.(\gamma
\cdot k) m_e+4 k^{\mu } m_e\right) \;\text{e}+4 i \pi ^3
\;\text{C}_1\left(k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,m_e^2,m_e^2,0\right)
\alpha \left(-D (\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)+2
(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)+4 D \gamma \cdot k
k^{\mu }-8 \gamma \cdot k k^{\mu }-2 \xi \gamma \cdot \;\text{p2}
k^{\mu }+2 \gamma \cdot \;\text{p2} k^{\mu }+D \gamma ^{\mu }.(\gamma
\cdot k) m_e-4 \gamma ^{\mu }.(\gamma \cdot k) m_e+D (\gamma \cdot
k).\gamma ^{\mu } m_e-4 (\gamma \cdot k).\gamma ^{\mu } m_e+8 k^{\mu }
m_e\right) \;\text{e}+4 i \pi ^3 \;\text{C}_1\left(\text{p2}^2,k^2-2
(k\cdot \;\text{p2})+\text{p2}^2,k^2,m_e^2,0,m_e^2\right)
\alpha \left(-D (\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot
\;\text{p2})+4 (\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot
\;\text{p2})-2 (\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot
k)+2 D \gamma \cdot \;\text{p2} k^{\mu }-4 \gamma \cdot \;\text{p2}
k^{\mu }+2 D \gamma \cdot k \;\text{p2}^{\mu }-4 \gamma \cdot k
\;\text{p2}^{\mu }-2 \xi \gamma \cdot \;\text{p2} \;\text{p2}^{\mu }+2
\gamma \cdot \;\text{p2} \;\text{p2}^{\mu }+D \gamma ^{\mu }.(\gamma
\cdot \;\text{p2}) m_e-4 \gamma ^{\mu }.(\gamma \cdot \;\text{p2}) m_e+D
(\gamma \cdot \;\text{p2}).\gamma ^{\mu } m_e-4 (\gamma \cdot
\;\text{p2}).\gamma ^{\mu } m_e+8 \;\text{p2}^{\mu } m_e\right)
\;\text{e}-4 i (1-\xi ) \pi ^3 \;\text{C}_0\left(0,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2\right) \alpha \left(-\gamma ^{\mu }
m_e^2+\gamma ^{\mu }.(\gamma \cdot k) m_e-\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}) m_e+\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot k)-\gamma
^{\mu }.(\gamma \cdot k).(\gamma \cdot \;\text{p2})-\gamma ^{\mu
}.(\gamma \cdot \;\text{p2}).(\gamma \cdot k)+\gamma ^{\mu }.(\gamma
\cdot \;\text{p2}).(\gamma \cdot \;\text{p2})-(\gamma \cdot k).\gamma
^{\mu }.(\gamma \cdot \;\text{p2})-(\gamma \cdot \;\text{p2}).\gamma
^{\mu }.(\gamma \cdot k)+2 \gamma \cdot \;\text{p2} k^{\mu }-2 \gamma
\cdot \;\text{p2} \;\text{p2}^{\mu }+\gamma ^{\mu } k^2-2 \gamma ^{\mu }
(k\cdot \;\text{p2})+\gamma ^{\mu } \;\text{p2}^2\right) \;\text{e}-4 i
(1-\xi ) \pi ^3
\;\text{C}_0\left(0,\text{p2}^2,\text{p2}^2,0,0,m_e^2\right)
\alpha \left(-\gamma ^{\mu } m_e^2-(\gamma \cdot \;\text{p2}).\gamma
^{\mu } m_e+2 (\gamma \cdot k).(\gamma \cdot k).\gamma ^{\mu }-(\gamma
\cdot k).(\gamma \cdot \;\text{p2}).\gamma ^{\mu }-(\gamma \cdot
\;\text{p2}).(\gamma \cdot k).\gamma ^{\mu }+(\gamma \cdot
\;\text{p2}).(\gamma \cdot \;\text{p2}).\gamma ^{\mu }+2 \gamma \cdot
\;\text{p2} \;\text{p2}^{\mu }-2 \gamma ^{\mu } k^2+2 \gamma ^{\mu }
(k\cdot \;\text{p2})-\gamma ^{\mu } \;\text{p2}^2\right) \;\text{e}-8 i
(1-\xi ) \pi ^3 \;\text{D}_{00}\left(0,\text{p2}^2,k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,m_e^2\right) \alpha \left(-\gamma
^{\mu } m_e^2-\gamma ^{\mu }.(\gamma \cdot k) m_e+\gamma ^{\mu }.(\gamma
\cdot \;\text{p2}) m_e+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }
m_e+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot k)+\gamma
^{\mu } \;\text{p2}^2\right) \;\text{e}-8 i (1-\xi ) \pi ^3 k^{\mu }
\;\text{D}_{11}\left(k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,m_e^2,m_e^2,0,0\right) \alpha \left(-(\gamma
\cdot k) m_e^2-(\gamma \cdot k).(\gamma \cdot k) m_e+(\gamma \cdot
k).(\gamma \cdot \;\text{p2}) m_e+(\gamma \cdot \;\text{p2}).(\gamma
\cdot k) m_e+(\gamma \cdot \;\text{p2}).(\gamma \cdot k).(\gamma \cdot
k)+\gamma \cdot k \;\text{p2}^2\right) \;\text{e}-16 i (1-\xi ) \pi ^3
\;\text{p2}^{\mu } \;\text{D}_{11}\left(0,\text{p2}^2,k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,m_e^2\right) \alpha \left(-(\gamma
\cdot \;\text{p2}) m_e^2-(\gamma \cdot \;\text{p2}).(\gamma \cdot k)
m_e+2 (\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2}) m_e+(\gamma
\cdot \;\text{p2}).(\gamma \cdot \;\text{p2}).(\gamma \cdot k)+\gamma
\cdot \;\text{p2} \;\text{p2}^2\right) \;\text{e}-16 i (1-\xi ) \pi ^3
\;\text{p2}^{\mu } \;\text{D}_{12}\left(\text{p2}^2,0,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,k^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,m_e^2,0,0,m_e^2\right) \alpha \left(-(\gamma
\cdot \;\text{p2}) m_e^2-(\gamma \cdot \;\text{p2}).(\gamma \cdot k)
m_e+2 (\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2}) m_e+(\gamma
\cdot \;\text{p2}).(\gamma \cdot \;\text{p2}).(\gamma \cdot k)+\gamma
\cdot \;\text{p2} \;\text{p2}^2\right) \;\text{e}-16 i (1-\xi ) \pi ^3
\;\text{D}_{12}\left(0,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,k^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,0,0,m_e^2,m_e^2\right)
\alpha \left(-(\gamma \cdot \;\text{p2}) k^{\mu } m_e^2-\gamma \cdot k
\;\text{p2}^{\mu } m_e^2-(\gamma \cdot \;\text{p2}).(\gamma \cdot k)
k^{\mu } m_e+2 (\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2})
k^{\mu } m_e-(\gamma \cdot k).(\gamma \cdot k) \;\text{p2}^{\mu }
m_e+(\gamma \cdot k).(\gamma \cdot \;\text{p2}) \;\text{p2}^{\mu }
m_e+(\gamma \cdot \;\text{p2}).(\gamma \cdot k) \;\text{p2}^{\mu }
m_e+(\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2}).(\gamma \cdot
k) k^{\mu }+(\gamma \cdot \;\text{p2}).(\gamma \cdot k).(\gamma \cdot k)
\;\text{p2}^{\mu }+\gamma \cdot \;\text{p2} k^{\mu }
\;\text{p2}^2+\gamma \cdot k \;\text{p2}^{\mu } \;\text{p2}^2\right)
\;\text{e}+4 i (1-\xi ) \pi ^3 \;\text{D}_0\left(0,\text{p2}^2,k^2,k^2-2
(k\cdot \;\text{p2})+\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,m_e^2\right) \alpha \left(-\gamma
^{\mu }.(\gamma \cdot k) m_e^3-\gamma ^{\mu }.(\gamma \cdot k).(\gamma
\cdot k) m_e^2+\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot
\;\text{p2}) m_e^2-2 (\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)
m_e^2+(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot \;\text{p2}) m_e^2-2
(\gamma \cdot k).(\gamma \cdot k).\gamma ^{\mu } m_e^2+(\gamma \cdot
\;\text{p2}).\gamma ^{\mu }.(\gamma \cdot k) m_e^2+4 \gamma \cdot k
k^{\mu } m_e^2-2 \gamma \cdot \;\text{p2} k^{\mu } m_e^2+(\gamma \cdot
k).\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot \;\text{p2}) m_e+2
(\gamma \cdot k).(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}) m_e+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot
k).(\gamma \cdot k) m_e+2 (\gamma \cdot \;\text{p2}).(\gamma \cdot
k).\gamma ^{\mu }.(\gamma \cdot k) m_e-2 (\gamma \cdot k).(\gamma \cdot
\;\text{p2}) k^{\mu } m_e-4 (\gamma \cdot \;\text{p2}).(\gamma \cdot k)
k^{\mu } m_e-2 (\gamma \cdot k).(\gamma \cdot \;\text{p2})
\;\text{p2}^{\mu } m_e-\gamma ^{\mu }.(\gamma \cdot \;\text{p2}) k^2
m_e+(\gamma \cdot \;\text{p2}).\gamma ^{\mu } k^2 m_e+2 \gamma ^{\mu
}.(\gamma \cdot \;\text{p2}) (k\cdot \;\text{p2}) m_e-\gamma ^{\mu
}.(\gamma \cdot k) \;\text{p2}^2 m_e+2 (\gamma \cdot k).(\gamma \cdot
k).(\gamma \cdot \;\text{p2}) \;\text{p2}^{\mu }-2 \gamma ^{\mu
}.(\gamma \cdot k).(\gamma \cdot \;\text{p2}) (k\cdot \;\text{p2})-2
(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot \;\text{p2}) (k\cdot
\;\text{p2})+4 \gamma \cdot \;\text{p2} k^{\mu } (k\cdot
\;\text{p2})+(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)
\;\text{p2}^2+(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot \;\text{p2})
\;\text{p2}^2-2 \gamma \cdot k k^{\mu } \;\text{p2}^2-2 \gamma \cdot
\;\text{p2} k^{\mu } \;\text{p2}^2\right) \;\text{e}-4 i (1-\xi ) \pi ^3
\;\text{D}_1\left(k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,m_e^2,m_e^2,0,0\right) \alpha \left(\gamma
^{\mu }.(\gamma \cdot k) m_e^3+(\gamma \cdot k).\gamma ^{\mu }
m_e^3+\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot k) m_e^2-\gamma
^{\mu }.(\gamma \cdot k).(\gamma \cdot \;\text{p2}) m_e^2+2 (\gamma
\cdot k).\gamma ^{\mu }.(\gamma \cdot k) m_e^2-(\gamma \cdot k).\gamma
^{\mu }.(\gamma \cdot \;\text{p2}) m_e^2+2 (\gamma \cdot k).(\gamma
\cdot k).\gamma ^{\mu } m_e^2-(\gamma \cdot \;\text{p2}).\gamma ^{\mu
}.(\gamma \cdot k) m_e^2-(\gamma \cdot \;\text{p2}).(\gamma \cdot
k).\gamma ^{\mu } m_e^2-4 \gamma \cdot k k^{\mu } m_e^2+2 \gamma \cdot
\;\text{p2} k^{\mu } m_e^2-(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot
k).(\gamma \cdot \;\text{p2}) m_e-2 (\gamma \cdot k).(\gamma \cdot
k).\gamma ^{\mu }.(\gamma \cdot \;\text{p2}) m_e-(\gamma \cdot
\;\text{p2}).\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot k) m_e-2
(\gamma \cdot \;\text{p2}).(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot
k) m_e-2 (\gamma \cdot k).(\gamma \cdot k) k^{\mu } m_e+4 (\gamma \cdot
k).(\gamma \cdot \;\text{p2}) k^{\mu } m_e+4 (\gamma \cdot
\;\text{p2}).(\gamma \cdot k) k^{\mu } m_e+\gamma ^{\mu }.(\gamma \cdot
k) k^2 m_e+\gamma ^{\mu }.(\gamma \cdot k) \;\text{p2}^2 m_e+(\gamma
\cdot k).\gamma ^{\mu } \;\text{p2}^2 m_e+2 (\gamma \cdot
\;\text{p2}).(\gamma \cdot k).(\gamma \cdot k) k^{\mu }-2 (\gamma \cdot
k).(\gamma \cdot k).(\gamma \cdot \;\text{p2}) \;\text{p2}^{\mu
}-(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot k) k^2+2
\gamma ^{\mu }.(\gamma \cdot k).(\gamma \cdot \;\text{p2}) (k\cdot
\;\text{p2})-4 \gamma \cdot \;\text{p2} k^{\mu } (k\cdot
\;\text{p2})-(\gamma \cdot k).\gamma ^{\mu }.(\gamma \cdot k)
\;\text{p2}^2+4 \gamma \cdot k k^{\mu } \;\text{p2}^2+2 \gamma \cdot
\;\text{p2} k^{\mu } \;\text{p2}^2\right) \;\text{e}+8 i (1-\xi ) \pi ^3
\;\text{D}_1\left(0,\text{p2}^2,k^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,\text{p2}^2,k^2-2 (k\cdot
\;\text{p2})+\text{p2}^2,0,0,m_e^2,m_e^2\right) \alpha \left(-\gamma
^{\mu }.(\gamma \cdot \;\text{p2}) m_e^3-(\gamma \cdot
\;\text{p2}).\gamma ^{\mu } m_e^3-\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}).(\gamma \cdot k) m_e^2+\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}).(\gamma \cdot \;\text{p2}) m_e^2-(\gamma \cdot k).\gamma
^{\mu }.(\gamma \cdot \;\text{p2}) m_e^2-(\gamma \cdot k).(\gamma \cdot
\;\text{p2}).\gamma ^{\mu } m_e^2-(\gamma \cdot \;\text{p2}).\gamma
^{\mu }.(\gamma \cdot k) m_e^2+2 (\gamma \cdot \;\text{p2}).\gamma ^{\mu
}.(\gamma \cdot \;\text{p2}) m_e^2-(\gamma \cdot \;\text{p2}).(\gamma
\cdot k).\gamma ^{\mu } m_e^2+(\gamma \cdot \;\text{p2}).(\gamma \cdot
\;\text{p2}).\gamma ^{\mu } m_e^2+2 \gamma \cdot \;\text{p2} k^{\mu }
m_e^2+2 \gamma \cdot k \;\text{p2}^{\mu } m_e^2-2 \gamma \cdot
\;\text{p2} \;\text{p2}^{\mu } m_e^2+(\gamma \cdot k).\gamma ^{\mu
}.(\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2}) m_e+(\gamma
\cdot k).(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}) m_e+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}).(\gamma \cdot k) m_e+2 (\gamma \cdot \;\text{p2}).(\gamma
\cdot k).\gamma ^{\mu }.(\gamma \cdot \;\text{p2}) m_e+(\gamma \cdot
\;\text{p2}).(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot k)
m_e-4 (\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p2}) k^{\mu }
m_e+2 (\gamma \cdot k).(\gamma \cdot k) \;\text{p2}^{\mu } m_e-2 (\gamma
\cdot k).(\gamma \cdot \;\text{p2}) \;\text{p2}^{\mu } m_e-2 (\gamma
\cdot \;\text{p2}).(\gamma \cdot k) \;\text{p2}^{\mu } m_e-\gamma ^{\mu
}.(\gamma \cdot \;\text{p2}) k^2 m_e-\gamma ^{\mu }.(\gamma \cdot
\;\text{p2}) \;\text{p2}^2 m_e-(\gamma \cdot \;\text{p2}).\gamma ^{\mu }
\;\text{p2}^2 m_e+2 (\gamma \cdot k).(\gamma \cdot \;\text{p2}).(\gamma
\cdot \;\text{p2}) \;\text{p2}^{\mu }-2 (\gamma \cdot
\;\text{p2}).(\gamma \cdot k).(\gamma \cdot k) \;\text{p2}^{\mu
}+(\gamma \cdot \;\text{p2}).\gamma ^{\mu }.(\gamma \cdot \;\text{p2})
k^2-2 \gamma ^{\mu }.(\gamma \cdot \;\text{p2}).(\gamma \cdot
\;\text{p2}) (k\cdot \;\text{p2})+4 \gamma \cdot \;\text{p2}
\;\text{p2}^{\mu } (k\cdot \;\text{p2})+(\gamma \cdot k).\gamma ^{\mu
}.(\gamma \cdot \;\text{p2}) \;\text{p2}^2-2 \gamma \cdot \;\text{p2}
k^{\mu } \;\text{p2}^2-2 \gamma \cdot k \;\text{p2}^{\mu }
\;\text{p2}^2-2 \gamma \cdot \;\text{p2} \;\text{p2}^{\mu }
\;\text{p2}^2\right) \;\text{e} 8 i ( 1 − ξ ) π 3 B 0 ( 0 , 0 , 0 ) γ μ α e + 4 i ( 2 − D ) π 3 B 0 ( p2 2 , 0 , m e 2 ) γ μ α e − 4 i D ( 1 − ξ ) π 3 γ μ C 00 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 ) α e − 8 i ( 2 − D ) π 3 γ μ C 00 ( k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 γ ⋅ k k μ C 11 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 γ ⋅ p2 p2 μ C 11 ( p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , m e 2 , 0 , m e 2 ) α e − 4 i ( 1 − ξ ) π 3 γ μ ( k 2 − 2 ( k ⋅ p2 ) + p2 2 ) C 11 ( k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , 0 ) α e − 8 i ( 2 − D ) π 3 ( γ ⋅ p2 k μ + γ ⋅ k p2 μ ) C 12 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α e − 2 1 i γ μ ( δ A + 2 δ e + 2 δ ψ ) e + 4 i ( 1 − ξ ) π 3 C 1 ( k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , 0 ) α ( γ μ . ( − ( γ ⋅ ( k − p2 ))) . ( γ ⋅ k ) − γ μ . ( − ( γ ⋅ ( k − p2 ))) . ( γ ⋅ p2 ) − 2 γ ⋅ p2 k μ + 2 γ ⋅ p2 p2 μ − 2 γ μ k 2 + 4 γ μ ( k ⋅ p2 ) − 2 γ μ p2 2 + γ μ . ( − ( γ ⋅ ( k − p2 ))) m e ) e + 4 i ( 1 − ξ ) π 3 C 1 ( p2 2 , p2 2 , 0 , 0 , m e 2 , 0 ) α ( ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ + ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ − ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ − 2 γ μ ( k ⋅ p2 ) + ( γ ⋅ p2 ) . γ μ m e ) e + 4 i π 3 C 0 ( k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + ξ ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) + 2 D γ ⋅ k k μ − 4 γ ⋅ k k μ − 2 ξ γ ⋅ p2 k μ + 2 γ ⋅ p2 k μ + D γ μ . ( γ ⋅ k ) m e − 4 γ μ . ( γ ⋅ k ) m e + 4 k μ m e ) e + 4 i π 3 C 1 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , m e 2 , m e 2 , 0 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) + 4 D γ ⋅ k k μ − 8 γ ⋅ k k μ − 2 ξ γ ⋅ p2 k μ + 2 γ ⋅ p2 k μ + D γ μ . ( γ ⋅ k ) m e − 4 γ μ . ( γ ⋅ k ) m e + D ( γ ⋅ k ) . γ μ m e − 4 ( γ ⋅ k ) . γ μ m e + 8 k μ m e ) e + 4 i π 3 C 1 ( p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , m e 2 , 0 , m e 2 ) α ( − D ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) + 4 ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − 2 ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + 2 D γ ⋅ p2 k μ − 4 γ ⋅ p2 k μ + 2 D γ ⋅ k p2 μ − 4 γ ⋅ k p2 μ − 2 ξ γ ⋅ p2 p2 μ + 2 γ ⋅ p2 p2 μ + D γ μ . ( γ ⋅ p2 ) m e − 4 γ μ . ( γ ⋅ p2 ) m e + D ( γ ⋅ p2 ) . γ μ m e − 4 ( γ ⋅ p2 ) . γ μ m e + 8 p2 μ m e ) e − 4 i ( 1 − ξ ) π 3 C 0 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 ) α ( − γ μ m e 2 + γ μ . ( γ ⋅ k ) m e − γ μ . ( γ ⋅ p2 ) m e + γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) − γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) − γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + 2 γ ⋅ p2 k μ − 2 γ ⋅ p2 p2 μ + γ μ k 2 − 2 γ μ ( k ⋅ p2 ) + γ μ p2 2 ) e − 4 i ( 1 − ξ ) π 3 C 0 ( 0 , p2 2 , p2 2 , 0 , 0 , m e 2 ) α ( − γ μ m e 2 − ( γ ⋅ p2 ) . γ μ m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ − ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ + 2 γ ⋅ p2 p2 μ − 2 γ μ k 2 + 2 γ μ ( k ⋅ p2 ) − γ μ p2 2 ) e − 8 i ( 1 − ξ ) π 3 D 00 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ m e 2 − γ μ . ( γ ⋅ k ) m e + γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) + γ μ p2 2 ) e − 8 i ( 1 − ξ ) π 3 k μ D 11 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 , 0 ) α ( − ( γ ⋅ k ) m e 2 − ( γ ⋅ k ) . ( γ ⋅ k ) m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) + γ ⋅ k p2 2 ) e − 16 i ( 1 − ξ ) π 3 p2 μ D 11 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ ⋅ p2 p2 2 ) e − 16 i ( 1 − ξ ) π 3 p2 μ D 12 ( p2 2 , 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , 0 , 0 , m e 2 ) α ( − ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) + γ ⋅ p2 p2 2 ) e − 16 i ( 1 − ξ ) π 3 D 12 ( 0 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , k 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − ( γ ⋅ p2 ) k μ m e 2 − γ ⋅ k p2 μ m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) k μ m e − ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e + ( γ ⋅ p2 ) . ( γ ⋅ k ) p2 μ m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ + ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ + γ ⋅ p2 k μ p2 2 + γ ⋅ k p2 μ p2 2 ) e + 4 i ( 1 − ξ ) π 3 D 0 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ . ( γ ⋅ k ) m e 3 − γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e 2 + γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e 2 − 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ m e 2 + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 + 4 γ ⋅ k k μ m e 2 − 2 γ ⋅ p2 k μ m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) k μ m e − 4 ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e − γ μ . ( γ ⋅ p2 ) k 2 m e + ( γ ⋅ p2 ) . γ μ k 2 m e + 2 γ μ . ( γ ⋅ p2 ) ( k ⋅ p2 ) m e − γ μ . ( γ ⋅ k ) p2 2 m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ − 2 γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) − 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) ( k ⋅ p2 ) + 4 γ ⋅ p2 k μ ( k ⋅ p2 ) + ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) p2 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) p2 2 − 2 γ ⋅ k k μ p2 2 − 2 γ ⋅ p2 k μ p2 2 ) e − 4 i ( 1 − ξ ) π 3 D 1 ( k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , m e 2 , m e 2 , 0 , 0 ) α ( γ μ . ( γ ⋅ k ) m e 3 + ( γ ⋅ k ) . γ μ m e 3 + γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e 2 − γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e 2 + 2 ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 + 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ m e 2 − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ m e 2 − 4 γ ⋅ k k μ m e 2 + 2 γ ⋅ p2 k μ m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) . ( γ ⋅ k ) m e − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) m e − 2 ( γ ⋅ k ) . ( γ ⋅ k ) k μ m e + 4 ( γ ⋅ k ) . ( γ ⋅ p2 ) k μ m e + 4 ( γ ⋅ p2 ) . ( γ ⋅ k ) k μ m e + γ μ . ( γ ⋅ k ) k 2 m e + γ μ . ( γ ⋅ k ) p2 2 m e + ( γ ⋅ k ) . γ μ p2 2 m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) k μ − 2 ( γ ⋅ k ) . ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) k 2 + 2 γ μ . ( γ ⋅ k ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) − 4 γ ⋅ p2 k μ ( k ⋅ p2 ) − ( γ ⋅ k ) . γ μ . ( γ ⋅ k ) p2 2 + 4 γ ⋅ k k μ p2 2 + 2 γ ⋅ p2 k μ p2 2 ) e + 8 i ( 1 − ξ ) π 3 D 1 ( 0 , p2 2 , k 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , p2 2 , k 2 − 2 ( k ⋅ p2 ) + p2 2 , 0 , 0 , m e 2 , m e 2 ) α ( − γ μ . ( γ ⋅ p2 ) m e 3 − ( γ ⋅ p2 ) . γ μ m e 3 − γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) m e 2 + γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ m e 2 − ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e 2 + 2 ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) m e 2 − ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ m e 2 + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ m e 2 + 2 γ ⋅ p2 k μ m e 2 + 2 γ ⋅ k p2 μ m e 2 − 2 γ ⋅ p2 p2 μ m e 2 + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) m e + ( γ ⋅ k ) . ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) . ( γ ⋅ k ) m e + 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) m e + ( γ ⋅ p2 ) . ( γ ⋅ p2 ) . γ μ . ( γ ⋅ k ) m e − 4 ( γ ⋅ p2 ) . ( γ ⋅ p2 ) k μ m e + 2 ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ m e − 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) p2 μ m e − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) p2 μ m e − γ μ . ( γ ⋅ p2 ) k 2 m e − γ μ . ( γ ⋅ p2 ) p2 2 m e − ( γ ⋅ p2 ) . γ μ p2 2 m e + 2 ( γ ⋅ k ) . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) p2 μ − 2 ( γ ⋅ p2 ) . ( γ ⋅ k ) . ( γ ⋅ k ) p2 μ + ( γ ⋅ p2 ) . γ μ . ( γ ⋅ p2 ) k 2 − 2 γ μ . ( γ ⋅ p2 ) . ( γ ⋅ p2 ) ( k ⋅ p2 ) + 4 γ ⋅ p2 p2 μ ( k ⋅ p2 ) + ( γ ⋅ k ) . γ μ . ( γ ⋅ p2 ) p2 2 − 2 γ ⋅ p2 k μ p2 2 − 2 γ ⋅ k p2 μ p2 2 − 2 γ ⋅ p2 p2 μ p2 2 ) e
Discard all the finite pieces of the 1-loop amplitude
[ 0 ] = PaVeUVPart[ amp3[ 2 ], Prefactor -> 1 / (2 Pi )^ D ] // DiracSimplify // [ #, D -> 4 - 2 Epsilon] & // Series [ #, { Epsilon, 0 , 0 }] & // Normal // // SelectNotFree2[ #, { SMP[ "Delta" ], SMP[ "d_A" ], [ "d_e" ], SMP[ "d_psi" ]}] & // Simplify − i e γ μ ( α Δ ξ + 2 π δ A + 4 π δ ψ + 4 π δ e ) 4 π -\frac{i \;\text{e} \gamma ^{\mu }
\left(\alpha \Delta \xi +2 \pi \delta _A+4 \pi \delta _{\psi }+4
\pi \delta _e\right)}{4 \pi } − 4 π i e γ μ ( α Δ ξ + 2 π δ A + 4 π δ ψ + 4 π δ e )
[ 0 ] = Simplify [ amp3Div[ 0 ] / (- FCI [ I SMP[ "e" ] GAD[ mu]] ) == 0 ] α Δ ξ + 2 π δ A + 4 π δ ψ + 4 π δ e = 0 \alpha \Delta \xi +2 \pi \delta _A+4
\pi \delta _{\psi }+4 \pi \delta _e=0 α Δ ξ + 2 π δ A + 4 π δ ψ + 4 π δ e = 0
[ 0 ] = Simplify [ ward[ 0 ] /. Epsilon -> 1 / SMP[ "Delta" ] /. { SMP[ "d_psi" ] -> SMP[ "d_psi^MSbar" ]} /. solMSbar1] [ 0 ] = Simplify [ ward[ 0 ] /. { SMP[ "d_psi" ] -> SMP[ "d_psi^MSbar" ]} /. ] δ A + 2 δ e = 0 \delta _A+2 \delta _e=0 δ A + 2 δ e = 0
δ A + 2 δ e = 0 \delta _A+2 \delta _e=0 δ A + 2 δ e = 0
= { SMP[ "d_A" ] + 2 * SMP[ "d_e" ] == 0 , [ "d_A" ] + 2 * SMP[ "d_e" ] == 0 } ;[{ wardMS[ 0 ], wardMSbar[ 0 ]}, knownResults, Text -> { " \t Verify Ward's identity:" , "CORRECT." , "WRONG!" }, Interrupt -> { Hold [ Quit [ 1 ]], Automatic }] ;\ tVerify Ward’s identity: CORRECT. \text{$\backslash $tVerify Ward's
identity:} \;\text{CORRECT.} \tVerify Ward’s identity: CORRECT.
Check the final results
```mathematica knownResult = { SMP[“d_psi^MS”] ->
-(GaugeXiSMP[“alpha_fs”])/(4 EpsilonPi), SMP[“d_m^MS”] ->
(-3 SMP[“alpha_fs”])/(4Epsilon Pi), SMP[“d_A^MS”] ->
-SMP[“alpha_fs”]/(3Epsilon Pi), SMP[“d_psi^MSbar”] ->
-(GaugeXiSMP[“alpha_fs”] SMP[“Delta”])/(4Pi),
SMP[“d_m^MSbar”] ->
(-3 SMP[“alpha_fs”]SMP[“Delta”])/(4 Pi), SMP[“d_A^MSbar”]
-> -(SMP[“alpha_fs”]SMP[“Delta”])/(3 Pi) };
FCCompareResults[Join[solMS1, solMS2, solMSbar1, solMSbar2],
knownResult, Text -> {“the final result:”, “CORRECT.”, “WRONG!”},
Interrupt -> {Hold[Quit[1]], Automatic}]; Print[“Time used:”,
Round[N[TimeUsed[], 4], 0.001], ” s.”];
```mathematica
\ tCheck the final result: CORRECT. \text{$\backslash $tCheck the final
result:} \;\text{CORRECT.} \tCheck the final result: CORRECT.
\ tCPU Time used: 26.364 s. \text{$\backslash $tCPU Time used:
}26.364\text{ s.} \tCPU Time used: 26.364 s.
