FeynCalc manual (development version)

TID

TID[amp, q] performs tensor decomposition of 1-loop integrals with loop momentum q.

See also

Overview, OneLoopSimplify, TIDL, PaVeLimitTo4.

Examples

FCClearScalarProducts[];
int = FAD[{k, m}, k - Subscript[p, 1], k - Subscript[p, 2]] FVD[k, \[Mu]] // FCI

kμ(k2m2).(kp1)2.(kp2)2\frac{k^{\mu }}{\left(k^2-m^2\right).(k-p_1){}^2.(k-p_2){}^2}

By default, all tensor integrals are reduced to the Passarino-Veltman scalar integrals A0A_0, B0B_0, C0C_0, D0D_0 etc.

TID[int, k]

p12p2μp1μ(p1p2)2((p1p2)2p12p22)k2.((k+p1)2m2)p22(m2+p12)p1μ+p12(m2+p22)p2μ+(m2+p12)(p2μ)(p1p2)(m2+p22)p1μ(p1p2)2((p1p2)2p12p22)(k2m2).(kp2)2.(kp1)2p2μ(p1p2)p22p1μ2((p1p2)2p12p22)k2.((k+p2)2m2)p12p2μ+p22p1μp1μ(p1p2)p2μ(p1p2)2k2.(kp1+p2)2((p1p2)2p12p22)\frac{p_1{}^2 p_2{}^{\mu }-p_1{}^{\mu } \left(p_1\cdot p_2\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right) k^2.\left((k+p_1){}^2-m^2\right)}-\frac{p_2{}^2 \left(m^2+p_1{}^2\right) p_1{}^{\mu }+p_1{}^2 \left(m^2+p_2{}^2\right) p_2{}^{\mu }+\left(m^2+p_1{}^2\right) \left(-p_2{}^{\mu }\right) \left(p_1\cdot p_2\right)-\left(m^2+p_2{}^2\right) p_1{}^{\mu } \left(p_1\cdot p_2\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right) \left(k^2-m^2\right).(k-p_2){}^2.(k-p_1){}^2}-\frac{p_2{}^{\mu } \left(p_1\cdot p_2\right)-p_2{}^2 p_1{}^{\mu }}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right) k^2.\left((k+p_2){}^2-m^2\right)}-\frac{p_1{}^2 p_2{}^{\mu }+p_2{}^2 p_1{}^{\mu }-p_1{}^{\mu } \left(p_1\cdot p_2\right)-p_2{}^{\mu } \left(p_1\cdot p_2\right)}{2 k^2.(k-p_1+p_2){}^2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right)}

Scalar integrals can be converted to the Passarino-Veltman notation via the option ToPaVe

TID[int, k, ToPaVe -> True]

iπ2(p12p2μp1μ(p1p2))  B0(p12,0,m2)2((p1p2)2p12p22)iπ2(p2μ(p1p2)p22p1μ)  B0(p22,0,m2)2((p1p2)2p12p22)iπ2(p12p2μ+p22p1μp1μ(p1p2)p2μ(p1p2))  B0(p122(p1p2)+p22,0,0)2((p1p2)2p12p22)iπ2(p22(m2+p12)p1μ+p12(m2+p22)p2μ+(m2+p12)(p2μ)(p1p2)(m2+p22)p1μ(p1p2))  C0(p12,p22,p122(p1p2)+p22,0,m2,0)2((p1p2)2p12p22)\frac{i \pi ^2 \left(p_1{}^2 p_2{}^{\mu }-p_1{}^{\mu } \left(p_1\cdot p_2\right)\right) \;\text{B}_0\left(p_1{}^2,0,m^2\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right)}-\frac{i \pi ^2 \left(p_2{}^{\mu } \left(p_1\cdot p_2\right)-p_2{}^2 p_1{}^{\mu }\right) \;\text{B}_0\left(p_2{}^2,0,m^2\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right)}-\frac{i \pi ^2 \left(p_1{}^2 p_2{}^{\mu }+p_2{}^2 p_1{}^{\mu }-p_1{}^{\mu } \left(p_1\cdot p_2\right)-p_2{}^{\mu } \left(p_1\cdot p_2\right)\right) \;\text{B}_0\left(p_1{}^2-2 \left(p_1\cdot p_2\right)+p_2{}^2,0,0\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right)}-\frac{i \pi ^2 \left(p_2{}^2 \left(m^2+p_1{}^2\right) p_1{}^{\mu }+p_1{}^2 \left(m^2+p_2{}^2\right) p_2{}^{\mu }+\left(m^2+p_1{}^2\right) \left(-p_2{}^{\mu }\right) \left(p_1\cdot p_2\right)-\left(m^2+p_2{}^2\right) p_1{}^{\mu } \left(p_1\cdot p_2\right)\right) \;\text{C}_0\left(p_1{}^2,p_2{}^2,p_1{}^2-2 \left(p_1\cdot p_2\right)+p_2{}^2,0,m^2,0\right)}{2 \left((p_1\cdot p_2){}^2-p_1{}^2 p_2{}^2\right)}

We can force the reduction algorithm to use Passarino-Veltman coefficient functions via the option UsePaVeBasis

TID[int, k, UsePaVeBasis -> True]

iπ2p1μ  C1(p12,p12+p222(p1p2),p22,m2,0,0)iπ2p2μ  C1(p22,p12+p222(p1p2),p12,m2,0,0)-i \pi ^2 p_1{}^{\mu } \;\text{C}_1\left(p_1{}^2,p_1{}^2+p_2{}^2-2 \left(p_1\cdot p_2\right),p_2{}^2,m^2,0,0\right)-i \pi ^2 p_2{}^{\mu } \;\text{C}_1\left(p_2{}^2,p_1{}^2+p_2{}^2-2 \left(p_1\cdot p_2\right),p_1{}^2,m^2,0,0\right)

Very often the integral can be simplified via partial fractioning even before performing the loop reduction. In this case the output will contain a mixture of FAD symbols and Passarino-Veltman functions

TID[SPD[p, q] FAD[q, {q - p, m}] FVD[q, mu], q, UsePaVeBasis -> True]

pmu2(q2m2)+12iπ2(m2p2)pmu((m2p2)  B0(p2,0,m2)2p2A0(m2)2p2)\frac{p^{\text{mu}}}{2 \left(q^2-m^2\right)}+\frac{1}{2} i \pi ^2 \left(m^2-p^2\right) p^{\text{mu}} \left(\frac{\left(m^2-p^2\right) \;\text{B}_0\left(p^2,0,m^2\right)}{2 p^2}-\frac{\text{A}_0\left(m^2\right)}{2 p^2}\right)

This can be avoided by setting both UsePaVeBasis and ToPaVe to True

TID[SPD[p, q] FAD[q, {q - p, m}] FVD[q, mu], q, UsePaVeBasis -> True, ToPaVe -> True]

12iπ2(m2p2)pmu((m2p2)  B0(p2,0,m2)2p2A0(m2)2p2)+12iπ2pmu  A0(m2)\frac{1}{2} i \pi ^2 \left(m^2-p^2\right) p^{\text{mu}} \left(\frac{\left(m^2-p^2\right) \;\text{B}_0\left(p^2,0,m^2\right)}{2 p^2}-\frac{\text{A}_0\left(m^2\right)}{2 p^2}\right)+\frac{1}{2} i \pi ^2 p^{\text{mu}} \;\text{A}_0\left(m^2\right)

Alternatively, we may set ToPaVe to Automatic which will automatically invoke the ToPaVe function if the final result contains even a single Passarino-Veltman function

TID[SPD[p, q] FAD[q, {q - p, m}] FVD[q, mu], q, ToPaVe -> Automatic]

(m2p2)2pmu4p2q2.((qp)2m2)(m23p2)pmu4p2(q2m2)\frac{\left(m^2-p^2\right)^2 p^{\text{mu}}}{4 p^2 q^2.\left((q-p)^2-m^2\right)}-\frac{\left(m^2-3 p^2\right) p^{\text{mu}}}{4 p^2 \left(q^2-m^2\right)}

TID[SPD[p, q] FAD[q, {q - p, m}] FVD[q, mu], q, UsePaVeBasis -> True, ToPaVe -> Automatic]

12iπ2(m2p2)pmu((m2p2)  B0(p2,0,m2)2p2A0(m2)2p2)+12iπ2pmu  A0(m2)\frac{1}{2} i \pi ^2 \left(m^2-p^2\right) p^{\text{mu}} \left(\frac{\left(m^2-p^2\right) \;\text{B}_0\left(p^2,0,m^2\right)}{2 p^2}-\frac{\text{A}_0\left(m^2\right)}{2 p^2}\right)+\frac{1}{2} i \pi ^2 p^{\text{mu}} \;\text{A}_0\left(m^2\right)

The basis of Passarino-Veltman coefficient functions is used automatically if there are zero Gram determinants

FCClearScalarProducts[]; 
 
SPD[Subscript[p, 1], Subscript[p, 1]] = 0; 
 
SPD[Subscript[p, 2], Subscript[p, 2]] = 0; 
 
SPD[Subscript[p, 1], Subscript[p, 2]] = 0; 
 
TID[FAD[{k, m}, k - Subscript[p, 1], k - Subscript[p, 2]] FVD[k, \[Mu]] // FCI, k] 
 
FCClearScalarProducts[];

iπ2(p1μ+p2μ)  C1(0,0,0,0,0,m2)-i \pi ^2 \left(p_1{}^{\mu }+p_2{}^{\mu }\right) \;\text{C}_1\left(0,0,0,0,0,m^2\right)

In FeynCalc, Passarino-Veltman coefficient functions are defined in the same way as in LoopTools. If one wants to use a different definition, it is useful to activate the option GenPaVe

FCClearScalarProducts[]; 
 
SPD[Subscript[p, 1], Subscript[p, 1]] = 0; 
 
SPD[Subscript[p, 2], Subscript[p, 2]] = 0; 
 
SPD[Subscript[p, 1], Subscript[p, 2]] = 0; 
 
TID[FAD[{k, m}, k - Subscript[p, 1], k - Subscript[p, 2]] FVD[k, \[Mu]] // FCI, k, GenPaVe -> True] 
 
FCClearScalarProducts[];

iπ2p1μ  GenPaVe({1},(0mp10p20))iπ2p2μ  GenPaVe({2},(0mp10p20))-i \pi ^2 p_1{}^{\mu } \;\text{GenPaVe}\left(\{1\},\left( \begin{array}{cc} 0 & m \\ p_1 & 0 \\ p_2 & 0 \\ \end{array} \right)\right)-i \pi ^2 p_2{}^{\mu } \;\text{GenPaVe}\left(\{2\},\left( \begin{array}{cc} 0 & m \\ p_1 & 0 \\ p_2 & 0 \\ \end{array} \right)\right)

To simplify manifestly IR-finite 1-loop results written in terms of Passarino-Veltman functions, we may employ the option PaVeLimitTo4 (must be used together with ToPaVe). The result is valid up to 0th order in Epsilon, i.e. sufficient for 1-loop calculations.

FCClearScalarProducts[]; 
 
int = (D - 1) (D - 2)/(D - 3) FVD[p, mu] FVD[p, nu] FAD[p, p - q]

(D2)(D1)pmupnu(D3)p2.(pq)2\frac{(D-2) (D-1) p^{\text{mu}} p^{\text{nu}}}{(D-3) p^2.(p-q)^2}

TID[int, p, ToPaVe -> True]

iπ2(2D)  B0(q2,0,0)(Dqmuqnuq2gmu  nu)4(3D)\frac{i \pi ^2 (2-D) \;\text{B}_0\left(q^2,0,0\right) \left(D q^{\text{mu}} q^{\text{nu}}-q^2 g^{\text{mu}\;\text{nu}}\right)}{4 (3-D)}

TID[int, p, ToPaVe -> True, PaVeLimitTo4 -> True]

12iπ2  B0(q2,0,0)(4qmuqnuq2gˉmu  nu)+12iπ2(2qmuqnuq2gˉmu  nu)\frac{1}{2} i \pi ^2 \;\text{B}_0\left(\overline{q}^2,0,0\right) \left(4 \overline{q}^{\text{mu}} \overline{q}^{\text{nu}}-\overline{q}^2 \bar{g}^{\text{mu}\;\text{nu}}\right)+\frac{1}{2} i \pi ^2 \left(2 \overline{q}^{\text{mu}} \overline{q}^{\text{nu}}-\overline{q}^2 \bar{g}^{\text{mu}\;\text{nu}}\right)

Sometimes one would like to have external momenta multiplied by symbolic parameters in the propagators. In this case one should first declare the corresponding variables to be of FCVariable type

DataType[a, FCVariable] = True;
DataType[b, FCVariable] = True;
ExpandScalarProduct[SP[P, Q] /. P -> a P1 + b P2] 
 
StandardForm[%] 
  
 

a(P1Q)+b(P2Q)a \left(\overline{\text{P1}}\cdot \overline{Q}\right)+b \left(\overline{\text{P2}}\cdot \overline{Q}\right)

(*a Pair[Momentum[P1], Momentum[Q]] + b Pair[Momentum[P2], Momentum[Q]]*)