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Obtain the amplitude
Nicer typesetting
MakeBoxes [ mu, TraditionalForm ] := " \[ Mu]" ;MakeBoxes [ nu, TraditionalForm ] := " \[ Nu]" ;MakeBoxes [ la, TraditionalForm ] := " \[ Lambda]" ;According to Peskin and Schroeder (Ch 19.2), the amplitude for the
first triangle diagram reads
[ 0 ] = ((- 1 ) (- I SMP[ "e" ] )^ 2 DiracTrace[ GAD[ mu] . GA[ 5 ] . [ l - k ] . GAD[ la] . QuarkPropagator[ l ] . [ nu] . QuarkPropagator[ l + p ]] ) // Explicite 2 tr ( γ μ . γ ˉ 5 . i γ ⋅ ( l − k ) ( l − k ) 2 . γ λ . i γ ⋅ l l 2 . γ ν . i γ ⋅ ( l + p ) ( l + p ) 2 ) \text{e}^2 \;\text{tr}\left(\gamma ^{\mu
}.\bar{\gamma }^5.\frac{i \gamma \cdot (l-k)}{(l-k)^2}.\gamma ^{\lambda
}.\frac{i \gamma \cdot l}{l^2}.\gamma ^{\nu }.\frac{i \gamma \cdot
(l+p)}{(l+p)^2}\right) e 2 tr ( γ μ . γ ˉ 5 . ( l − k ) 2 iγ ⋅ ( l − k ) . γ λ . l 2 iγ ⋅ l . γ ν . ( l + p ) 2 iγ ⋅ ( l + p ) )
And the second one follows from the first by interchanging k with p
and la with nu
[ 0 ] = amp1[ 0 ] /. { k -> p , p -> k , la -> nu, nu -> la} e 2 tr ( γ μ . γ ˉ 5 . i γ ⋅ ( l − p ) ( l − p ) 2 . γ ν . i γ ⋅ l l 2 . γ λ . i γ ⋅ ( k + l ) ( k + l ) 2 ) \text{e}^2 \;\text{tr}\left(\gamma ^{\mu
}.\bar{\gamma }^5.\frac{i \gamma \cdot (l-p)}{(l-p)^2}.\gamma ^{\nu
}.\frac{i \gamma \cdot l}{l^2}.\gamma ^{\lambda }.\frac{i \gamma \cdot
(k+l)}{(k+l)^2}\right) e 2 tr ( γ μ . γ ˉ 5 . ( l − p ) 2 iγ ⋅ ( l − p ) . γ ν . l 2 iγ ⋅ l . γ λ . ( k + l ) 2 iγ ⋅ ( k + l ) )
Calculate the amplitude
Contracting both amplitudes with I*(k+p)^mu we can check the
non-conservation of the axial current.
[ 0 ] = Contract[ I * FVD[ k + p , mu] (amp1[ 0 ] + amp2[ 0 ] )] i e 2 tr ( − i ( γ ⋅ ( k + p ) ) . γ ˉ 5 . ( γ ⋅ ( l − p ) ) . γ ν . ( γ ⋅ l ) . γ λ . ( γ ⋅ ( k + l ) ) l 2 ( k + l ) 2 ( l − p ) 2 ) + i e 2 tr ( − i ( γ ⋅ ( k + p ) ) . γ ˉ 5 . ( γ ⋅ ( l − k ) ) . γ λ . ( γ ⋅ l ) . γ ν . ( γ ⋅ ( l + p ) ) l 2 ( l − k ) 2 ( l + p ) 2 ) i \;\text{e}^2 \;\text{tr}\left(-\frac{i
(\gamma \cdot (k+p)).\bar{\gamma }^5.(\gamma \cdot (l-p)).\gamma ^{\nu
}.(\gamma \cdot l).\gamma ^{\lambda }.(\gamma \cdot (k+l))}{l^2 (k+l)^2
(l-p)^2}\right)+i \;\text{e}^2 \;\text{tr}\left(-\frac{i (\gamma \cdot
(k+p)).\bar{\gamma }^5.(\gamma \cdot (l-k)).\gamma ^{\lambda }.(\gamma
\cdot l).\gamma ^{\nu }.(\gamma \cdot (l+p))}{l^2 (l-k)^2
(l+p)^2}\right) i e 2 tr ( − l 2 ( k + l ) 2 ( l − p ) 2 i ( γ ⋅ ( k + p )) . γ ˉ 5 . ( γ ⋅ ( l − p )) . γ ν . ( γ ⋅ l ) . γ λ . ( γ ⋅ ( k + l )) ) + i e 2 tr ( − l 2 ( l − k ) 2 ( l + p ) 2 i ( γ ⋅ ( k + p )) . γ ˉ 5 . ( γ ⋅ ( l − k )) . γ λ . ( γ ⋅ l ) . γ ν . ( γ ⋅ ( l + p )) )
For this calculation it is crucial to use a correct scheme for
gamma^5. As in the book, we use the Breitenlohner-Maison-t’Hooft-Veltman
prescription.
[ "BMHV" ] ;[ 1 ] = TID[ amp[ 0 ] , l , ToPaVe -> True ] 4 π 2 B 0 ( k 2 , 0 , 0 ) ϵ ˉ λ ν k ‾ p ‾ ( − ( ( 2 − D ) ( k ⋅ p ) 2 ) − ( 2 − D ) k 2 ( k ⋅ p ) − 4 k 2 ( k ⋅ p ) + 2 k 2 ( k ⋅ p − p 2 ) + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) + 4 π 2 B 0 ( p 2 , 0 , 0 ) ϵ ˉ λ ν k ‾ p ‾ ( − ( ( 2 − D ) ( k ⋅ p ) 2 ) − ( 2 − D ) p 2 ( k ⋅ p ) − 4 p 2 ( k ⋅ p ) − 2 ( k 2 − k ⋅ p ) p 2 + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) − 1 ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 4 π 2 C 0 ( k 2 , p 2 , k 2 + 2 ( k ⋅ p ) + p 2 , 0 , 0 , 0 ) ϵ ˉ λ ν k ‾ p ‾ ( − 2 p 2 k 4 − ( 2 − D ) ( k ⋅ p ) 2 k 2 − 2 p 4 k 2 − 2 ( 2 − D ) ( k ⋅ p ) p 2 k 2 − 4 ( k ⋅ p ) p 2 k 2 + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) k 2 − ( 2 − D ) ( k ⋅ p ) 2 p 2 + ( 2 − D ) p 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 + ( 4 π 2 B 0 ( k 2 + 2 ( k ⋅ p ) + p 2 , 0 , 0 ) ϵ ˉ λ ν k ‾ p ‾ ( 4 ( 2 − D ) ( k ⋅ p ) 3 + 4 ( 2 − D ) k 2 ( k ⋅ p ) 2 + 4 ( 2 − D ) p 2 ( k ⋅ p ) 2 + ( 2 − D ) k 4 ( k ⋅ p ) + ( 2 − D ) p 4 ( k ⋅ p ) + 2 ( 2 − D ) k 2 p 2 ( k ⋅ p ) + 16 ( ( k ⋅ p ) 2 − k 2 p 2 ) ( k ⋅ p ) + 4 ( k 2 ( − ( k ⋅ p ) − p 2 ) + 2 ( k ⋅ p ) ( − ( k ⋅ p ) − p 2 ) + p 2 ( − ( k ⋅ p ) − p 2 ) + ( k 2 + 2 ( k ⋅ p ) + p 2 ) ( k 2 + 2 p 2 ) ) ( k ⋅ p ) + 8 k 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) + 8 p 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) + 2 p 2 ( − k 2 ( − k 2 − k ⋅ p ) − 2 ( k ⋅ p ) ( − k 2 − k ⋅ p ) − p 2 ( − k 2 − k ⋅ p ) + ( k 2 − 2 ( k ⋅ p ) ) ( k 2 + 2 ( k ⋅ p ) + p 2 ) ) + 2 k 2 ( k 2 ( − ( k ⋅ p ) − p 2 ) + 2 ( k ⋅ p ) ( − ( k ⋅ p ) − p 2 ) + p 2 ( − ( k ⋅ p ) − p 2 ) + 3 p 2 ( k 2 + 2 ( k ⋅ p ) + p 2 ) ) ) e 2 ) / ( ( 2 − D ) ( k 2 + 2 ( k ⋅ p ) + p 2 ) ( ( k ⋅ p ) 2 − k 2 p 2 ) ) \frac{4 \pi ^2
\;\text{B}_0\left(k^2,0,0\right) \bar{\epsilon }^{\lambda \nu
\overline{k}\overline{p}} \left(-\left((2-D) (k\cdot p)^2\right)-(2-D)
k^2 (k\cdot p)-4 k^2 (k\cdot p)+2 k^2 \left(k\cdot p-p^2\right)+(2-D)
\left((k\cdot p)^2-k^2 p^2\right)\right) \;\text{e}^2}{(2-D)
\left((k\cdot p)^2-k^2 p^2\right)}+\frac{4 \pi ^2
\;\text{B}_0\left(p^2,0,0\right) \bar{\epsilon }^{\lambda \nu
\overline{k}\overline{p}} \left(-\left((2-D) (k\cdot p)^2\right)-(2-D)
p^2 (k\cdot p)-4 p^2 (k\cdot p)-2 \left(k^2-k\cdot p\right) p^2+(2-D)
\left((k\cdot p)^2-k^2 p^2\right)\right) \;\text{e}^2}{(2-D)
\left((k\cdot p)^2-k^2 p^2\right)}-\frac{1}{(2-D) \left((k\cdot p)^2-k^2
p^2\right)}4 \pi ^2 \;\text{C}_0\left(k^2,p^2,k^2+2 (k\cdot
p)+p^2,0,0,0\right) \bar{\epsilon }^{\lambda \nu
\overline{k}\overline{p}} \left(-2 p^2 k^4-(2-D) (k\cdot p)^2 k^2-2 p^4
k^2-2 (2-D) (k\cdot p) p^2 k^2-4 (k\cdot p) p^2 k^2+(2-D) \left((k\cdot
p)^2-k^2 p^2\right) k^2-(2-D) (k\cdot p)^2 p^2+(2-D) p^2 \left((k\cdot
p)^2-k^2 p^2\right)\right) \;\text{e}^2+\left(4 \pi ^2
\;\text{B}_0\left(k^2+2 (k\cdot p)+p^2,0,0\right) \bar{\epsilon
}^{\lambda \nu \overline{k}\overline{p}} \left(4 (2-D) (k\cdot p)^3+4
(2-D) k^2 (k\cdot p)^2+4 (2-D) p^2 (k\cdot p)^2+(2-D) k^4 (k\cdot
p)+(2-D) p^4 (k\cdot p)+2 (2-D) k^2 p^2 (k\cdot p)+16 \left((k\cdot
p)^2-k^2 p^2\right) (k\cdot p)+4 \left(k^2 \left(-(k\cdot
p)-p^2\right)+2 (k\cdot p) \left(-(k\cdot p)-p^2\right)+p^2
\left(-(k\cdot p)-p^2\right)+\left(k^2+2 (k\cdot p)+p^2\right)
\left(k^2+2 p^2\right)\right) (k\cdot p)+8 k^2 \left((k\cdot p)^2-k^2
p^2\right)+8 p^2 \left((k\cdot p)^2-k^2 p^2\right)+2 p^2 \left(-k^2
\left(-k^2-k\cdot p\right)-2 (k\cdot p) \left(-k^2-k\cdot p\right)-p^2
\left(-k^2-k\cdot p\right)+\left(k^2-2 (k\cdot p)\right) \left(k^2+2
(k\cdot p)+p^2\right)\right)+2 k^2 \left(k^2 \left(-(k\cdot
p)-p^2\right)+2 (k\cdot p) \left(-(k\cdot p)-p^2\right)+p^2
\left(-(k\cdot p)-p^2\right)+3 p^2 \left(k^2+2 (k\cdot
p)+p^2\right)\right)\right) \;\text{e}^2\right)/\left((2-D) \left(k^2+2
(k\cdot p)+p^2\right) \left((k\cdot p)^2-k^2
p^2\right)\right) ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 4 π 2 B 0 ( k 2 , 0 , 0 ) ϵ ˉ λ ν k p ( − ( ( 2 − D ) ( k ⋅ p ) 2 ) − ( 2 − D ) k 2 ( k ⋅ p ) − 4 k 2 ( k ⋅ p ) + 2 k 2 ( k ⋅ p − p 2 ) + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 4 π 2 B 0 ( p 2 , 0 , 0 ) ϵ ˉ λ ν k p ( − ( ( 2 − D ) ( k ⋅ p ) 2 ) − ( 2 − D ) p 2 ( k ⋅ p ) − 4 p 2 ( k ⋅ p ) − 2 ( k 2 − k ⋅ p ) p 2 + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 − ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 1 4 π 2 C 0 ( k 2 , p 2 , k 2 + 2 ( k ⋅ p ) + p 2 , 0 , 0 , 0 ) ϵ ˉ λ ν k p ( − 2 p 2 k 4 − ( 2 − D ) ( k ⋅ p ) 2 k 2 − 2 p 4 k 2 − 2 ( 2 − D ) ( k ⋅ p ) p 2 k 2 − 4 ( k ⋅ p ) p 2 k 2 + ( 2 − D ) ( ( k ⋅ p ) 2 − k 2 p 2 ) k 2 − ( 2 − D ) ( k ⋅ p ) 2 p 2 + ( 2 − D ) p 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) ) e 2 + ( 4 π 2 B 0 ( k 2 + 2 ( k ⋅ p ) + p 2 , 0 , 0 ) ϵ ˉ λ ν k p ( 4 ( 2 − D ) ( k ⋅ p ) 3 + 4 ( 2 − D ) k 2 ( k ⋅ p ) 2 + 4 ( 2 − D ) p 2 ( k ⋅ p ) 2 + ( 2 − D ) k 4 ( k ⋅ p ) + ( 2 − D ) p 4 ( k ⋅ p ) + 2 ( 2 − D ) k 2 p 2 ( k ⋅ p ) + 16 ( ( k ⋅ p ) 2 − k 2 p 2 ) ( k ⋅ p ) + 4 ( k 2 ( − ( k ⋅ p ) − p 2 ) + 2 ( k ⋅ p ) ( − ( k ⋅ p ) − p 2 ) + p 2 ( − ( k ⋅ p ) − p 2 ) + ( k 2 + 2 ( k ⋅ p ) + p 2 ) ( k 2 + 2 p 2 ) ) ( k ⋅ p ) + 8 k 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) + 8 p 2 ( ( k ⋅ p ) 2 − k 2 p 2 ) + 2 p 2 ( − k 2 ( − k 2 − k ⋅ p ) − 2 ( k ⋅ p ) ( − k 2 − k ⋅ p ) − p 2 ( − k 2 − k ⋅ p ) + ( k 2 − 2 ( k ⋅ p ) ) ( k 2 + 2 ( k ⋅ p ) + p 2 ) ) + 2 k 2 ( k 2 ( − ( k ⋅ p ) − p 2 ) + 2 ( k ⋅ p ) ( − ( k ⋅ p ) − p 2 ) + p 2 ( − ( k ⋅ p ) − p 2 ) + 3 p 2 ( k 2 + 2 ( k ⋅ p ) + p 2 ) ) ) e 2 ) / ( ( 2 − D ) ( k 2 + 2 ( k ⋅ p ) + p 2 ) ( ( k ⋅ p ) 2 − k 2 p 2 ) )
[] ;[ k , D | D - 4 ] = Momentum[ k ] ;[ p , D | D - 4 ] = Momentum[ p ] ;The explicit values for the PaVe functions B0 and C0 can be obtained
e.g. from H. Patel’s Package-X. Here we just insert the known results.
The C0 function is finite here, so because of the prefactor (D-4) it
gives no contribution in the D->4 limit.
[ 2 ] = Collect2[ amp[ 1 ], { B0, C0}] / /. { [ FCI@SP[ p_ , p_ ], 0 , 0 ] :> 1 / (16 Epsilon \ [ Pi ] ^ 4 ) - (- 2 + EulerGamma )/ (16 \ [ Pi ] ^ 4 ) + Log [ - ((4 \ [ Pi ] ScaleMu^ 2 )/ Pair[ Momentum[ p ], Momentum[ p ]] )] / (16 \ [ Pi ] ^ 4 ), [ FCI[ SP[ p , p ] + 2 SP[ p , k ] + SP[ k , k ]], 0 , 0 ] :> [ FCI[ SP[ k + p , k + p ]], 0 , 0 ], D - 4 ) ExpandScalarProduct[ C0[ SP[ k ], SP[ p ], SP[ k + p ], 0 , 0 , 0 ]] -> 0 } − 4 π 2 ( D − 4 ) e 2 p ‾ 2 ( k ‾ ⋅ p ‾ + k ‾ 2 ) ( log ( − 4 π μ 2 p ‾ 2 ) 16 π 4 + 1 16 π 4 ε − γ − 2 16 π 4 ) ϵ ˉ λ ν k ‾ p ‾ ( D − 2 ) ( ( k ‾ ⋅ p ‾ ) 2 − k ‾ 2 p ‾ 2 ) − 4 π 2 ( D − 4 ) e 2 k ‾ 2 ( k ‾ ⋅ p ‾ + p ‾ 2 ) ( log ( − 4 π μ 2 k ‾ 2 ) 16 π 4 + 1 16 π 4 ε − γ − 2 16 π 4 ) ϵ ˉ λ ν k ‾ p ‾ ( D − 2 ) ( ( k ‾ ⋅ p ‾ ) 2 − k ‾ 2 p ‾ 2 ) − 4 π 2 ( D − 4 ) e 2 ( k ‾ ⋅ p ‾ ) ( 2 ( k ‾ ⋅ p ‾ ) + k ‾ 2 + p ‾ 2 ) ϵ ˉ λ ν k ‾ p ‾ ( log ( − 4 π μ 2 ( k ‾ + p ‾ ) 2 ) 16 π 4 + 1 16 π 4 ε − γ − 2 16 π 4 ) ( D − 2 ) ( k ‾ 2 p ‾ 2 − ( k ‾ ⋅ p ‾ ) 2 ) -\frac{4 \pi ^2 (D-4) \;\text{e}^2
\overline{p}^2 \left(\overline{k}\cdot
\overline{p}+\overline{k}^2\right) \left(\frac{\log \left(-\frac{4
\pi \mu ^2}{\overline{p}^2}\right)}{16 \pi ^4}+\frac{1}{16 \pi ^4
\varepsilon }-\frac{\gamma -2}{16 \pi ^4}\right) \bar{\epsilon
}^{\lambda \nu \overline{k}\overline{p}}}{(D-2) \left((\overline{k}\cdot
\overline{p})^2-\overline{k}^2 \overline{p}^2\right)}-\frac{4 \pi ^2
(D-4) \;\text{e}^2 \overline{k}^2 \left(\overline{k}\cdot
\overline{p}+\overline{p}^2\right) \left(\frac{\log \left(-\frac{4
\pi \mu ^2}{\overline{k}^2}\right)}{16 \pi ^4}+\frac{1}{16 \pi ^4
\varepsilon }-\frac{\gamma -2}{16 \pi ^4}\right) \bar{\epsilon
}^{\lambda \nu \overline{k}\overline{p}}}{(D-2) \left((\overline{k}\cdot
\overline{p})^2-\overline{k}^2 \overline{p}^2\right)}-\frac{4 \pi ^2
(D-4) \;\text{e}^2 \left(\overline{k}\cdot \overline{p}\right) \left(2
\left(\overline{k}\cdot
\overline{p}\right)+\overline{k}^2+\overline{p}^2\right) \bar{\epsilon
}^{\lambda \nu \overline{k}\overline{p}} \left(\frac{\log \left(-\frac{4
\pi \mu ^2}{(\overline{k}+\overline{p})^2}\right)}{16 \pi
^4}+\frac{1}{16 \pi ^4 \varepsilon }-\frac{\gamma -2}{16 \pi
^4}\right)}{(D-2) \left(\overline{k}^2 \overline{p}^2-(\overline{k}\cdot
\overline{p})^2\right)} − ( D − 2 ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 4 π 2 ( D − 4 ) e 2 p 2 ( k ⋅ p + k 2 ) 16 π 4 l o g ( − p 2 4 π μ 2 ) + 16 π 4 ε 1 − 16 π 4 γ − 2 ϵ ˉ λ ν k p − ( D − 2 ) ( ( k ⋅ p ) 2 − k 2 p 2 ) 4 π 2 ( D − 4 ) e 2 k 2 ( k ⋅ p + p 2 ) 16 π 4 l o g ( − k 2 4 π μ 2 ) + 16 π 4 ε 1 − 16 π 4 γ − 2 ϵ ˉ λ ν k p − ( D − 2 ) ( k 2 p 2 − ( k ⋅ p ) 2 ) 4 π 2 ( D − 4 ) e 2 ( k ⋅ p ) ( 2 ( k ⋅ p ) + k 2 + p 2 ) ϵ ˉ λ ν k p 16 π 4 l o g ( − ( k + p ) 2 4 π μ 2 ) + 16 π 4 ε 1 − 16 π 4 γ − 2
Now we insert the explicit values, convert the external momenta to 4
dimensions and expand in Epsilon
[ 3 ] = amp[ 2 ] // FCReplaceD[ #, D -> 4 - 2 Epsilon] & // Series [ #, { Epsilon, 0 , 0 }] & // Normal − e 2 ϵ ˉ λ ν k ‾ p ‾ 2 π 2 -\frac{\text{e}^2 \bar{\epsilon }^{\lambda
\nu \overline{k}\overline{p}}}{2 \pi ^2} − 2 π 2 e 2 ϵ ˉ λ ν k p
The result should be twice Eq. 19.59 in Peskin and Schroeder
Check the final results
= 2 (SMP[ "e" ] ^ 2 / (4 Pi ^ 2 ) LC[ al, la, be, nu] FV[ k , al] FV[ p , be] ) // Contract;[ amp[ 3 ], knownResult, Text -> { " \t Compare to Peskin and Schroeder, An Introduction to QFT, Eq 19.59:" , "CORRECT." , "WRONG!" }, Interrupt -> { Hold [ Quit [ 1 ]], Automatic }] ;Print [ " \t CPU Time used: " , Round [ N [ TimeUsed [], 4 ], 0.001 ], " s." ] ;\ tCompare to Peskin and Schroeder, An Introduction to QFT, Eq 19.59: CORRECT. \text{$\backslash $tCompare to Peskin and
Schroeder, An Introduction to QFT, Eq 19.59:}
\;\text{CORRECT.} \tCompare to Peskin and Schroeder, An Introduction to QFT, Eq 19.59: CORRECT.
\ tCPU Time used: 17.634 s. \text{$\backslash $tCPU Time used:
}17.634\text{ s.} \tCPU Time used: 17.634 s.