GenPaVe
GenPaVe[i, j, ..., {{0, m0}, {Momentum[p1], m1}, {Momentum[p2], m2}, ...]
denotes the invariant (and scalar) Passarino-Veltman integrals, i.e. the coefficient functions of the tensor integral decomposition. In contrast to PaVe
which uses the LoopTools convention, masses and external momenta in GenPaVe
are written in the same order as they appear in the original tensor integral, i.e. FAD[{q,m0},{q-p1,m1},{q-p2,m2},...]
.
See also
Overview , PaVe .
Examples
FVD[ q , \ [ Mu]] FVD[ q , \ [ Nu]] FAD[{ q , m0}, { q + p1, m1}, { q + p2, m2}] / (I * Pi ^ 2 )
TID[ % , q , UsePaVeBasis -> True ]
TID[ %% , q , UsePaVeBasis -> True , GenPaVe -> True ]
− i q μ q ν π 2 ( q 2 − m0 2 ) . ( ( p1 + q ) 2 − m1 2 ) . ( ( p2 + q ) 2 − m2 2 ) -\frac{i q^{\mu } q^{\nu }}{\pi ^2 \left(q^2-\text{m0}^2\right).\left((\text{p1}+q)^2-\text{m1}^2\right).\left((\text{p2}+q)^2-\text{m2}^2\right)} − π 2 ( q 2 − m0 2 ) . ( ( p1 + q ) 2 − m1 2 ) . ( ( p2 + q ) 2 − m2 2 ) i q μ q ν
g μ ν C 00 ( p1 2 , p2 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , m1 2 , m0 2 , m2 2 ) + p1 μ p1 ν C 11 ( p1 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p2 2 , m0 2 , m1 2 , m2 2 ) + p2 μ p2 ν C 11 ( p2 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p1 2 , m0 2 , m2 2 , m1 2 ) + ( p1 ν p2 μ + p1 μ p2 ν ) C 12 ( p1 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p2 2 , m0 2 , m1 2 , m2 2 ) g^{\mu \nu } \;\text{C}_{00}\left(\text{p1}^2,\text{p2}^2,-2 (\text{p1}\cdot \;\text{p2})+\text{p1}^2+\text{p2}^2,\text{m1}^2,\text{m0}^2,\text{m2}^2\right)+\text{p1}^{\mu } \;\text{p1}^{\nu } \;\text{C}_{11}\left(\text{p1}^2,-2 (\text{p1}\cdot \;\text{p2})+\text{p1}^2+\text{p2}^2,\text{p2}^2,\text{m0}^2,\text{m1}^2,\text{m2}^2\right)+\text{p2}^{\mu } \;\text{p2}^{\nu } \;\text{C}_{11}\left(\text{p2}^2,-2 (\text{p1}\cdot \;\text{p2})+\text{p1}^2+\text{p2}^2,\text{p1}^2,\text{m0}^2,\text{m2}^2,\text{m1}^2\right)+\left(\text{p1}^{\nu } \;\text{p2}^{\mu }+\text{p1}^{\mu } \;\text{p2}^{\nu }\right) \;\text{C}_{12}\left(\text{p1}^2,-2 (\text{p1}\cdot \;\text{p2})+\text{p1}^2+\text{p2}^2,\text{p2}^2,\text{m0}^2,\text{m1}^2,\text{m2}^2\right) g μν C 00 ( p1 2 , p2 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , m1 2 , m0 2 , m2 2 ) + p1 μ p1 ν C 11 ( p1 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p2 2 , m0 2 , m1 2 , m2 2 ) + p2 μ p2 ν C 11 ( p2 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p1 2 , m0 2 , m2 2 , m1 2 ) + ( p1 ν p2 μ + p1 μ p2 ν ) C 12 ( p1 2 , − 2 ( p1 ⋅ p2 ) + p1 2 + p2 2 , p2 2 , m0 2 , m1 2 , m2 2 )
g μ ν GenPaVe ( { 0 , 0 } , ( 0 m0 p1 m1 p2 m2 ) ) + p1 μ p1 ν GenPaVe ( { 1 , 1 } , ( 0 m0 p1 m1 p2 m2 ) ) + p2 μ p2 ν GenPaVe ( { 2 , 2 } , ( 0 m0 p1 m1 p2 m2 ) ) + ( p1 ν p2 μ + p1 μ p2 ν ) GenPaVe ( { 1 , 2 } , ( 0 m0 p1 m1 p2 m2 ) ) g^{\mu \nu } \;\text{GenPaVe}\left(\{0,0\},\left(
\begin{array}{cc}
0 & \;\text{m0} \\
\;\text{p1} & \;\text{m1} \\
\;\text{p2} & \;\text{m2} \\
\end{array}
\right)\right)+\text{p1}^{\mu } \;\text{p1}^{\nu } \;\text{GenPaVe}\left(\{1,1\},\left(
\begin{array}{cc}
0 & \;\text{m0} \\
\;\text{p1} & \;\text{m1} \\
\;\text{p2} & \;\text{m2} \\
\end{array}
\right)\right)+\text{p2}^{\mu } \;\text{p2}^{\nu } \;\text{GenPaVe}\left(\{2,2\},\left(
\begin{array}{cc}
0 & \;\text{m0} \\
\;\text{p1} & \;\text{m1} \\
\;\text{p2} & \;\text{m2} \\
\end{array}
\right)\right)+\left(\text{p1}^{\nu } \;\text{p2}^{\mu }+\text{p1}^{\mu } \;\text{p2}^{\nu }\right) \;\text{GenPaVe}\left(\{1,2\},\left(
\begin{array}{cc}
0 & \;\text{m0} \\
\;\text{p1} & \;\text{m1} \\
\;\text{p2} & \;\text{m2} \\
\end{array}
\right)\right) g μν GenPaVe { 0 , 0 } , 0 p1 p2 m0 m1 m2 + p1 μ p1 ν GenPaVe { 1 , 1 } , 0 p1 p2 m0 m1 m2 + p2 μ p2 ν GenPaVe { 2 , 2 } , 0 p1 p2 m0 m1 m2 + ( p1 ν p2 μ + p1 μ p2 ν ) GenPaVe { 1 , 2 } , 0 p1 p2 m0 m1 m2