FeynCalc manual (development version)

FCLoopIsolate

FCLoopIsolate[expr, {q1, q2, ...}] wraps loop integrals into heads specified by the user. This is useful when you want to know which loop integrals appear in the given expression.

See also

Overview, FCLoopExtract.

Examples

FCI[GSD[q - p1] . (GSD[q - p2] + M) . GSD[p3] SPD[q, p2] FAD[q, q - p1, {q - p2, m}]] 
 
FCLoopIsolate[%, {q}, Head -> loopInt] 
 
Cases2[%, loopInt]

(p2q)(γ(qp1)).(M+γ(qp2)).(γ  p3)q2.(qp1)2.((qp2)2m2)\frac{(\text{p2}\cdot q) (\gamma \cdot (q-\text{p1})).(M+\gamma \cdot (q-\text{p2})).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}

((γ  p1).(γ  p2).(γ  p3)M(γ  p1).(γ  p3))  loopInt(p2qq2.(qp1)2.((qp2)2m2))+M  loopInt((p2q)(γq).(γ  p3)q2.(qp1)2.((qp2)2m2))loopInt((p2q)(γ  p1).(γq).(γ  p3)q2.(qp1)2.((qp2)2m2))loopInt((p2q)(γq).(γ  p2).(γ  p3)q2.(qp1)2.((qp2)2m2))+loopInt((p2q)(γq).(γq).(γ  p3)q2.(qp1)2.((qp2)2m2))((\gamma \cdot \;\text{p1}).(\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p3})-M (\gamma \cdot \;\text{p1}).(\gamma \cdot \;\text{p3})) \;\text{loopInt}\left(\frac{\text{p2}\cdot q}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)+M \;\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)-\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot \;\text{p1}).(\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)-\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)+\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)

{loopInt(p2qq2.(qp1)2.((qp2)2m2)),loopInt((p2q)(γq).(γ  p3)q2.(qp1)2.((qp2)2m2)),loopInt((p2q)(γ  p1).(γq).(γ  p3)q2.(qp1)2.((qp2)2m2)),loopInt((p2q)(γq).(γ  p2).(γ  p3)q2.(qp1)2.((qp2)2m2)),loopInt((p2q)(γq).(γq).(γ  p3)q2.(qp1)2.((qp2)2m2))}\left\{\text{loopInt}\left(\frac{\text{p2}\cdot q}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right),\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right),\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot \;\text{p1}).(\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right),\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot \;\text{p2}).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right),\text{loopInt}\left(\frac{(\text{p2}\cdot q) (\gamma \cdot q).(\gamma \cdot q).(\gamma \cdot \;\text{p3})}{q^2.(q-\text{p1})^2.\left((q-\text{p2})^2-m^2\right)}\right)\right\}

TID[FVD[q, \[Mu]] FVD[q, \[Nu]] FAD[{q, m}, {q + p, m}, {q + r, m}], q, UsePaVeBasis -> True] 
 
FCLoopIsolate[%, {q}, Head -> l] 
 
Cases2[%, l]

iπ2gμν  C00(p2,r2,2(pr)+p2+r2,m2,m2,m2)+iπ2pμpν  C11(p2,2(pr)+p2+r2,r2,m2,m2,m2)+iπ2rμrν  C11(r2,2(pr)+p2+r2,p2,m2,m2,m2)+iπ2(pνrμ+pμrν)  C12(p2,2(pr)+p2+r2,r2,m2,m2,m2)i \pi ^2 g^{\mu \nu } \;\text{C}_{00}\left(p^2,r^2,-2 (p\cdot r)+p^2+r^2,m^2,m^2,m^2\right)+i \pi ^2 p^{\mu } p^{\nu } \;\text{C}_{11}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)+i \pi ^2 r^{\mu } r^{\nu } \;\text{C}_{11}\left(r^2,-2 (p\cdot r)+p^2+r^2,p^2,m^2,m^2,m^2\right)+i \pi ^2 \left(p^{\nu } r^{\mu }+p^{\mu } r^{\nu }\right) \;\text{C}_{12}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)

iπ2gμνl(C00(p2,r2,2(pr)+p2+r2,m2,m2,m2))+iπ2pμpνl(C11(p2,2(pr)+p2+r2,r2,m2,m2,m2))+iπ2rμrνl(C11(r2,2(pr)+p2+r2,p2,m2,m2,m2))+iπ2(pνrμ+pμrν)l(C12(p2,2(pr)+p2+r2,r2,m2,m2,m2))i \pi ^2 g^{\mu \nu } l\left(\text{C}_{00}\left(p^2,r^2,-2 (p\cdot r)+p^2+r^2,m^2,m^2,m^2\right)\right)+i \pi ^2 p^{\mu } p^{\nu } l\left(\text{C}_{11}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)\right)+i \pi ^2 r^{\mu } r^{\nu } l\left(\text{C}_{11}\left(r^2,-2 (p\cdot r)+p^2+r^2,p^2,m^2,m^2,m^2\right)\right)+i \pi ^2 \left(p^{\nu } r^{\mu }+p^{\mu } r^{\nu }\right) l\left(\text{C}_{12}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)\right)

{l(C00(p2,r2,2(pr)+p2+r2,m2,m2,m2)),l(C11(p2,2(pr)+p2+r2,r2,m2,m2,m2)),l(C11(r2,2(pr)+p2+r2,p2,m2,m2,m2)),l(C12(p2,2(pr)+p2+r2,r2,m2,m2,m2))}\left\{l\left(\text{C}_{00}\left(p^2,r^2,-2 (p\cdot r)+p^2+r^2,m^2,m^2,m^2\right)\right),l\left(\text{C}_{11}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)\right),l\left(\text{C}_{11}\left(r^2,-2 (p\cdot r)+p^2+r^2,p^2,m^2,m^2,m^2\right)\right),l\left(\text{C}_{12}\left(p^2,-2 (p\cdot r)+p^2+r^2,r^2,m^2,m^2,m^2\right)\right)\right\}

SPD[q, q]^2 FAD[{q, m}] + SPD[q, q] 
 
FCLoopIsolate[%, {q}, DropScaleless -> True]

q4q2m2+q2\frac{q^4}{q^2-m^2}+q^2

FCGV(LoopInt)(q4q2m2)\text{FCGV}(\text{LoopInt})\left(\frac{q^4}{q^2-m^2}\right)

a FAD[{q1, m}, {q2, m}] + b FAD[{q1, m, 2}] 
 
FCLoopIsolate[%, {q1, q2}] 
 
FCLoopIsolate[%%, {q1, q2}, MultiLoop -> True]

a(q12m2).(q22m2)+b(q12m2)2\frac{a}{\left(\text{q1}^2-m^2\right).\left(\text{q2}^2-m^2\right)}+\frac{b}{\left(\text{q1}^2-m^2\right)^2}

a  FCGV(LoopInt)(1(q12m2).(q22m2))+b  FCGV(LoopInt)(1(q12m2)2)a \;\text{FCGV}(\text{LoopInt})\left(\frac{1}{\left(\text{q1}^2-m^2\right).\left(\text{q2}^2-m^2\right)}\right)+b \;\text{FCGV}(\text{LoopInt})\left(\frac{1}{\left(\text{q1}^2-m^2\right)^2}\right)

a  FCGV(LoopInt)(1(q12m2).(q22m2))+b(q12m2)2a \;\text{FCGV}(\text{LoopInt})\left(\frac{1}{\left(\text{q1}^2-m^2\right).\left(\text{q2}^2-m^2\right)}\right)+\frac{b}{\left(\text{q1}^2-m^2\right)^2}