Dear Professor Mertig,
My question is related with two cases which are written below (The cases
are in InputForm, this is why they are looking so long, I am sorry, for
this, however in standard form they are simple in fact.). Our aim is to
obtain corresponding matrix elements from amplitude.
In first case, the first matrix element in A.3. is in contradiction with
the fermion currents of the diagram which is taken from A.1.Topology and
insertion and A.2.Corresponding Feynman amplitude.
In second case, the all matrix elements in B.3. is compatible with the
fermion currents of the diagram which is taken from B.1.Topology and
insertion and B.2.Corresponding Feynman amplitude.
The question is: How in the first case does the contradiction appear?
Thank you for your kind considerations.
A.First case: (below; A.1.Topology and insertion, A.2.Corresponding
Feynman amplitude and A.3.Matrix elements)
A.1.Topology and insertion:
dia1 = TopologyList[
Process -> {F[3, {2, c1}], -F[3, {1, c2}]} -> {-F[3, {2, c1}],
F[3, {1, c2}]}, Model -> {"SMQCD"}, GenericModel -> {"Lorentz"},
InsertionLevel -> {Generic, Classes}, ExcludeParticles -> {},
ExcludeFieldPoints -> {}, LastSelections -> {}][
Topology[1][
Propagator[Incoming][Vertex[1][1], Vertex[3][5], Field[1]],
Propagator[Incoming][Vertex[1][2], Vertex[3][6], Field[2]],
Propagator[Outgoing][Vertex[1][3], Vertex[3][7], Field[3]],
Propagator[Outgoing][Vertex[1][4], Vertex[3][8], Field[4]],
Propagator[Loop[1]][Vertex[3][5], Vertex[3][6], Field[5]],
Propagator[Loop[1]][Vertex[3][5], Vertex[3][8], Field[6]],
Propagator[Loop[1]][Vertex[3][6], Vertex[3][7], Field[7]],
Propagator[Loop[1]][Vertex[3][7], Vertex[3][8], Field[8]]] ->
Insertions[Generic][
FeynmanGraph[1, Generic == 7][Field[1] -> F[3, {2, c1}],
Field[2] -> -F[3, {1, c2}], Field[3] -> F[3, {2, c1}],
Field[4] -> -F[3, {1, c2}], Field[5] -> F, Field[6] -> V,
Field[7] -> V, Field[8] -> F] ->
Insertions[Classes][
FeynmanGraph[1, Classes == 1][Field[1] -> F[3, {2, c1}],
Field[2] -> -F[3, {1, c2}], Field[3] -> F[3, {2, c1}],
Field[4] -> -F[3, {1, c2}],
Field[5] -> F[4, {Index[Generation, 5], c1}],
Field[6] -> -V[3], Field[7] -> V[3],
Field[8] -> F[4, {Index[Generation, 6], c1}]]]]]
A.2.Corresponding Feynman amplitude:
FeynAmpList[Process -> {{F[3, {2, c1}], FourMomentum[Incoming, 1], MC},
{-F[3, {1, c2}], FourMomentum[Incoming, 2], MU}} ->
{{-F[3, {2, c1}], FourMomentum[Outgoing, 1], MC},
{F[3, {1, c2}], FourMomentum[Outgoing, 2], MU}}, Model -> {"SMQCD"},
GenericModel -> {"Lorentz"}, AmplitudeLevel -> {Generic, Classes},
ExcludeParticles -> {}, ExcludeFieldPoints -> {}, LastSelections -> {}][
FeynAmp[GraphID[Topology == 1, Generic == 1],
Integral[FourMomentum[Internal, 1]],
((-I/16)*RelativeCF*FermionChain[NonCommutative[
Spinor[-Momentum[FourMomentum[Incoming, 2]], MU, 1]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 3]]],
DiracGamma[6]]*
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 5]], -V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 3]]],
DiracGamma[7]]*
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 5]], -V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[DiracSlash[FourMomentum[Internal, 1]] +
Mass[F[Index[Generic, 5]],
Loop]], NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz,
1]]],
DiracGamma[6]]*G[-1][0][-F[Index[Generic, 5]], F[3, {2, c1}],
-V[Index[Generic,
6]]][NonCommutative[DiracGamma[LorentzIndex[KI1[3]]],
DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 1]]],
DiracGamma[7]]*G[-1][0][-F[Index[Generic, 5]], F[3, {2, c1}],
-V[Index[Generic,
6]]][NonCommutative[DiracGamma[LorentzIndex[KI1[3]]],
DiracGamma[7]]],
NonCommutative[Spinor[Momentum[FourMomentum[Incoming, 1]],
MC, 1]]]*FermionChain[NonCommutative[
Spinor[Momentum[FourMomentum[Outgoing, 2]], MU, 1]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 2]]],
DiracGamma[6]]*
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 8]], V[Index[Generic,
6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 2]]],
DiracGamma[7]]*
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 8]], V[Index[Generic,
6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[DiracSlash[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1]] + Mass[F[Index[Generic, 8]], Loop]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 4]]],
DiracGamma[6]]*
G[-1][0][-F[Index[Generic, 8]], F[3, {2, c1}], V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 4]]],
DiracGamma[7]]*
G[-1][0][-F[Index[Generic, 8]], F[3, {2, c1}], V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[Spinor[-Momentum[FourMomentum[Outgoing, 1]], MC, 1]]]*
FeynAmpDenominator[PropagatorDenominator[FourMomentum[Internal, 1],
Mass[F[Index[Generic, 5]], Loop]], PropagatorDenominator[
FourMomentum[Incoming, 2] + FourMomentum[Internal, 1],
Mass[V[Index[Generic, 7]], Loop]], PropagatorDenominator[
FourMomentum[Incoming, 2] + FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1], Mass[F[Index[Generic, 8]], Loop]],
PropagatorDenominator[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1] - FourMomentum[Outgoing, 2],
Mass[V[Index[Generic, 6]], Loop]]]*MetricTensor[Index[Lorentz, 1],
Index[Lorentz, 2]]*MetricTensor[Index[Lorentz, 3], Index[Lorentz,
4]])/Pi^4,
{Mass[F[Index[Generic, 5]], Loop], Mass[F[Index[Generic, 8]], Loop],
Mass[V[Index[Generic, 6]], Loop], Mass[V[Index[Generic, 7]], Loop],
G[-1][0][-F[Index[Generic, 5]], F[3, {2, c1}], -V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[Index[Generic, 5]], F[3, {2, c1}], -V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][-F[Index[Generic, 8]], F[3, {2, c1}], V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[Index[Generic, 8]], F[3, {2, c1}], V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 5]], -V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 5]], -V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 8]], V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[3, {1, c2}], F[Index[Generic, 8]], V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
RelativeCF} ->
Insertions[Classes][{MQD[Index[Generation, 5]], MQD[Index[Generation,
6]], MW, MW,
0, (I*EL*Conjugate[CKM[2, Index[Generation, 5]]])/(Sqrt[2]*SW), 0,
(I*EL*Conjugate[CKM[2, Index[Generation, 6]]])/(Sqrt[2]*SW), 0,
(I*EL*CKM[1, Index[Generation, 5]])/(Sqrt[2]*SW), 0,
(I*EL*CKM[1, Index[Generation, 6]])/(Sqrt[2]*SW), IndexDelta[c1, c2]*
SumOver[Index[Generation, 5], 3]*SumOver[Index[Generation, 6],
3]}]]]
A.3.Matrix elements (only the elements that include p^0):
{StandardMatrixElement[Spinor[Momentum[p3], MC, 1] . DiracGamma[7] .
Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[6] .
Spinor[-Momentum[p2], MU, 1]], StandardMatrixElement[
Spinor[-Momentum[p2], MU, 1] . DiracGamma[6] . Spinor[Momentum[p1], MC,
1]*
Spinor[-Momentum[p4], MU, 1] . DiracGamma[6] . Spinor[Momentum[p3], MC,
1]],
StandardMatrixElement[Spinor[-Momentum[p2], MU, 1] . DiracGamma[7] .
Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[6] .
Spinor[Momentum[p3], MC, 1]], StandardMatrixElement[
Spinor[-Momentum[p2], MU, 1] . DiracGamma[6] . Spinor[Momentum[p1], MC,
1]*
Spinor[-Momentum[p4], MU, 1] . DiracGamma[7] . Spinor[Momentum[p3], MC,
1]],
StandardMatrixElement[Spinor[-Momentum[p2], MU, 1] . DiracGamma[7] .
Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[7] .
Spinor[Momentum[p3], MC, 1]], StandardMatrixElement[
Spinor[-Momentum[p2], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[6] .
Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[6] .
Spinor[Momentum[p3], MC, 1]],
StandardMatrixElement[Spinor[-Momentum[p2], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[7] .
Spinor[Momentum[p1], MC, 1]*
Spinor[-Momentum[p4], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[6] .
Spinor[Momentum[p3], MC, 1]], StandardMatrixElement[
Spinor[-Momentum[p2], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[6] .
Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[7] .
Spinor[Momentum[p3], MC, 1]],
StandardMatrixElement[Spinor[-Momentum[p2], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[7] .
Spinor[Momentum[p1], MC, 1]*
Spinor[-Momentum[p4], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[7] .
Spinor[Momentum[p3], MC, 1]], StandardMatrixElement[
Spinor[-Momentum[p2], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[LorentzIndex[$MU[2]]] . DiracGamma[6] .
Spinor[Momentum[p1], MC, 1]*
Spinor[-Momentum[p4], MU, 1] . DiracGamma[LorentzIndex[$MU[1]]] .
DiracGamma[LorentzIndex[$MU[2]]] . DiracGamma[6] .
Spinor[Momentum[p3], MC, 1]],
StandardMatrixElement[Spinor[-Momentum[p2], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[LorentzIndex[$MU[2]]] .
DiracGamma[7] . Spinor[Momentum[p1], MC, 1]*Spinor[-Momentum[p4], MU, 1] .
DiracGamma[LorentzIndex[$MU[1]]] . DiracGamma[LorentzIndex[$MU[2]]] .
DiracGamma[7] . Spinor[Momentum[p3], MC, 1]]}
B.Second case (below; B.1.Topology and insertion, B.2.Corresponding
Feynman amplitude and B.3.Matrix elements)
B.1.Topology and insertion:
dia2 = TopologyList[Process -> {F, F} -> {F, F}, Model -> {"SMQCD"},
GenericModel -> {"Lorentz"}, InsertionLevel -> {Generic, Classes},
ExcludeParticles -> {}, ExcludeFieldPoints -> {},
LastSelections -> {}][
Topology[1][
Propagator[Incoming][Vertex[1][1], Vertex[3][5], Field[1]],
Propagator[Incoming][Vertex[1][2], Vertex[3][6], Field[2]],
Propagator[Outgoing][Vertex[1][3], Vertex[3][7], Field[3]],
Propagator[Outgoing][Vertex[1][4], Vertex[3][8], Field[4]],
Propagator[Loop[1]][Vertex[3][5], Vertex[3][6], Field[5]],
Propagator[Loop[1]][Vertex[3][5], Vertex[3][8], Field[6]],
Propagator[Loop[1]][Vertex[3][6], Vertex[3][7], Field[7]],
Propagator[Loop[1]][Vertex[3][7], Vertex[3][8], Field[8]]] ->
Insertions[Generic][
FeynmanGraph[1, Generic == 7][Field[1] -> F, Field[2] -> F,
Field[3] -> F, Field[4] -> F, Field[5] -> F, Field[6] -> V,
Field[7] -> V, Field[8] -> F] -> Insertions[Classes][
FeynmanGraph[1, Classes == 8][
Field[1] -> F[3, {Index[Generation, 1], Index[Colour, 1]}],
Field[2] -> -F[3, {Index[Generation, 2], Index[Colour, 2]}],
Field[3] -> F[3, {Index[Generation, 3], Index[Colour, 3]}],
Field[4] -> -F[3, {Index[Generation, 4], Index[Colour, 4]}],
Field[5] -> F[4, {Index[Generation, 5], Index[Colour, 1]}],
Field[6] -> -V[3], Field[7] -> V[3],
Field[8] -> F[4, {Index[Generation, 6], Index[Colour, 3]}]]]]]
B.2.Corresponding Feynman amplitude:
FeynAmpList[Process -> {{F, FourMomentum[Incoming, 1], Mass[F, External]},
{F, FourMomentum[Incoming, 2], Mass[F, External]}} ->
{{F, FourMomentum[Outgoing, 1], Mass[F, External]}, {F,
FourMomentum[Outgoing, 2],
Mass[F, External]}}, Model -> {"SMQCD"}, GenericModel ->
{"Lorentz"},
AmplitudeLevel -> {Generic, Classes}, ExcludeParticles -> {},
ExcludeFieldPoints -> {}, LastSelections -> {}][
FeynAmp[GraphID[Topology == 1, Generic == 1],
Integral[FourMomentum[Internal, 1]],
((-I/16)*RelativeCF*FermionChain[NonCommutative[SpinorType[Index[Generic,
2]][
-FourMomentum[Incoming, 2], Mass[F[Index[Generic, 2]], External]]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 3]]],
DiracGamma[6]]*
G[-1][0][F[Index[Generic, 2]], F[Index[Generic, 5]],
-V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 3]]],
DiracGamma[7]]*
G[-1][0][F[Index[Generic, 2]], F[Index[Generic, 5]],
-V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[DiracSlash[FourMomentum[Internal, 1]] +
Mass[F[Index[Generic, 5]],
Loop]], NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz,
1]]],
DiracGamma[6]]*G[-1][0][-F[Index[Generic, 5]], F[Index[Generic,
1]],
-V[Index[Generic,
6]]][NonCommutative[DiracGamma[LorentzIndex[KI1[3]]],
DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 1]]],
DiracGamma[7]]*G[-1][0][-F[Index[Generic, 5]], F[Index[Generic,
1]],
-V[Index[Generic,
6]]][NonCommutative[DiracGamma[LorentzIndex[KI1[3]]],
DiracGamma[7]]], NonCommutative[SpinorType[Index[Generic, 1]][
FourMomentum[Incoming, 1], Mass[F[Index[Generic, 1]], External]]]]*
FermionChain[NonCommutative[SpinorType[Index[Generic, 4]][
FourMomentum[Outgoing, 2], Mass[F[Index[Generic, 4]], External]]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 2]]],
DiracGamma[6]]*
G[-1][0][F[Index[Generic, 4]], F[Index[Generic, 8]],
V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 2]]],
DiracGamma[7]]*
G[-1][0][F[Index[Generic, 4]], F[Index[Generic, 8]],
V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[DiracSlash[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1]] + Mass[F[Index[Generic, 8]], Loop]],
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 4]]],
DiracGamma[6]]*
G[-1][0][-F[Index[Generic, 8]], F[Index[Generic, 3]],
V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]] +
NonCommutative[DiracGamma[LorentzIndex[Index[Lorentz, 4]]],
DiracGamma[7]]*
G[-1][0][-F[Index[Generic, 8]], F[Index[Generic, 3]],
V[Index[Generic, 7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
NonCommutative[SpinorType[Index[Generic, 3]][-FourMomentum[Outgoing,
1],
Mass[F[Index[Generic, 3]], External]]]]*FeynAmpDenominator[
PropagatorDenominator[FourMomentum[Internal, 1],
Mass[F[Index[Generic, 5]],
Loop]], PropagatorDenominator[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1], Mass[V[Index[Generic, 7]], Loop]],
PropagatorDenominator[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1], Mass[F[Index[Generic, 8]], Loop]],
PropagatorDenominator[FourMomentum[Incoming, 2] +
FourMomentum[Internal, 1] -
FourMomentum[Outgoing, 1] - FourMomentum[Outgoing, 2],
Mass[V[Index[Generic, 6]], Loop]]]*MetricTensor[Index[Lorentz, 1],
Index[Lorentz, 2]]*MetricTensor[Index[Lorentz, 3], Index[Lorentz,
4]])/Pi^4,
{Mass[F[Index[Generic, 1]], External], Mass[F[Index[Generic, 2]],
External],
Mass[F[Index[Generic, 3]], External], Mass[F[Index[Generic, 4]],
External],
Mass[F[Index[Generic, 5]], Loop], Mass[F[Index[Generic, 8]], Loop],
Mass[V[Index[Generic, 6]], Loop], Mass[V[Index[Generic, 7]], Loop],
G[-1][0][F[Index[Generic, 2]], F[Index[Generic, 5]], -V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][F[Index[Generic, 2]], F[Index[Generic, 5]], -V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][F[Index[Generic, 4]], F[Index[Generic, 8]], V[Index[Generic,
6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][F[Index[Generic, 4]], F[Index[Generic, 8]], V[Index[Generic,
6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][-F[Index[Generic, 5]], F[Index[Generic, 1]],
-V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[Index[Generic, 5]], F[Index[Generic, 1]],
-V[Index[Generic, 6]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
G[-1][0][-F[Index[Generic, 8]], F[Index[Generic, 3]], V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[6]]],
G[-1][0][-F[Index[Generic, 8]], F[Index[Generic, 3]], V[Index[Generic,
7]]][
NonCommutative[DiracGamma[LorentzIndex[KI1[3]]], DiracGamma[7]]],
RelativeCF} ->
Insertions[Classes][{MQU[Index[Generation, 1]], MQU[Index[Generation,
2]],
MQU[Index[Generation, 3]], MQU[Index[Generation, 4]],
MQD[Index[Generation, 5]],
MQD[Index[Generation, 6]], MW, MW, 0,
(I*EL*CKM[Index[Generation, 2], Index[Generation, 5]])/(Sqrt[2]*SW),
0,
(I*EL*CKM[Index[Generation, 4], Index[Generation, 6]])/(Sqrt[2]*SW),
0,
(I*EL*Conjugate[CKM[Index[Generation, 1], Index[Generation,
5]]])/(Sqrt[2]*SW),
0, (I*EL*Conjugate[CKM[Index[Generation, 3], Index[Generation, 6]]])/
(Sqrt[2]*SW), IndexDelta[Index[Colour, 1], Index[Colour, 2]]*
IndexDelta[Index[Colour, 3], Index[Colour,
4]]*SumOver[Index[Generation, 5], 3]*
SumOver[Index[Generation, 6], 3]*SumOver[Index[Colour, 1], 3,
External]*
SumOver[Index[Colour, 2], 3, External]*SumOver[Index[Colour, 3], 3,
External]*
SumOver[Index[Colour, 4], 3, External]*SumOver[Index[Generation, 1],
3,
External]*SumOver[Index[Generation, 2], 3, External]*
SumOver[Index[Generation, 3], 3, External]*SumOver[Index[Generation,
4], 3,
External]}]]]
B.3.Matrix elements (only the elements that include p^0):
{StandardMatrixElement[Spinor[Momentum[Index[Generic, 2]], 0,
1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[6] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 1]], 0,
1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]4]] .
DiracGamma[7] . Spinor[Momentum[Index[Generic, 3]], 0,
1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[6] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]], StandardMatrixElement[
Spinor[Momentum[Index[Generic, 2]], 0, 1][-Momentum[p2],
"Mass[F[Index[Generic, 2]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 1]], 0, 1][Momentum[p1],
"Mass[F[Index[Generic, 1]], External]"]*
Spinor[Momentum[Index[Generic, 4]], 0, 1][-Momentum[p4],
"Mass[F[Index[Generic, 4]], External]"] .
DiracGamma[LorentzIndex[\[Mu]2]] .
DiracGamma[LorentzIndex[\[Mu]4]] . DiracGamma[7] .
Spinor[Momentum[Index[Generic, 3]], 0, 1][Momentum[p3],
"Mass[F[Index[Generic, 3]], External]"]]}
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