FeynCalc manual (development version)

FCFADiracChainJoin

FCFADiracChainJoin[exp] processes the output of FeynArts (after FCFAConvert) with explicit Dirac indices and joins matrices and spinors into closed chains. This is necessary e. g. for models with 4-fermion operators, where FeynArts cannot determine the correct relative signs. When two matrices have a common index but the positions do not match, as in AijBikA_{ij} B_{ik}, it is assumed that we can take the charge conjugate transposed of either matrix to obtain, e.g. (CATC−1)jiBik\left(C A^T C^{-1}\right)_{ji} B_{ik} or (CBTC−1)kiAij\left(C B^TC^{-1}\right)_{ki} A_{ij}.

See also

Overview, DiracChain, DCHN, DiracIndex, DiracIndexDelta, DIDelta, DiracChainCombine, DiracChainExpand, DiracChainFactor, DiracChainJoin, FCCCT.

Examples

Create a closed chain for the 1-loop electron self-energy

-(1/(16 \[Pi]^4)) I el^2 DCHN[Spinor[-Momentum[p, D], me, 1], Dir1]*
   DCHN[Spinor[Momentum[q, D], me, 1], Dir2] DCHN[GAD[Lor1], Dir1, Dir3]*
   DCHN[GAD[Lor2], Dir2, Dir4] DCHN[me - GSD[k], Dir3, Dir4] FAD[{k, me}, k - q] MTD[Lor1, Lor2] 
 
res = FCFADiracChainJoin[%]

−i  el2gLor1  Lor2(γLor1)Dir1  Dir3(γLor2)Dir2  Dir4(me−γ⋅k)Dir3  Dir4(φ(−p,me))Dir1(φ(q,me))Dir216π4(k2−me2).(k−q)2-\frac{i \;\text{el}^2 g^{\text{Lor1}\;\text{Lor2}} \left(\gamma ^{\text{Lor1}}\right){}_{\text{Dir1}\;\text{Dir3}} \left(\gamma ^{\text{Lor2}}\right){}_{\text{Dir2}\;\text{Dir4}} (\text{me}-\gamma \cdot k)_{\text{Dir3}\;\text{Dir4}} (\varphi (-p,\text{me}))_{\text{Dir1}} (\varphi (q,\text{me}))_{\text{Dir2}}}{16 \pi ^4 \left(k^2-\text{me}^2\right).(k-q)^2}

−i  el2gLor1  Lor2(φ(q,me)).γLor2.(−(γ⋅k+me).γLor1).(φ(p,me))16π4(k2−me2).(k−q)2-\frac{i \;\text{el}^2 g^{\text{Lor1}\;\text{Lor2}} (\varphi (q,\text{me})).\gamma ^{\text{Lor2}}.\left(-(\gamma \cdot k+\text{me}).\gamma ^{\text{Lor1}}\right).(\varphi (p,\text{me}))}{16 \pi ^4 \left(k^2-\text{me}^2\right).(k-q)^2}

Sometimes the ordering of the spinors is not the one wants to have. However, we can always transpose the chains to reorder the spinors as we like, which doesn’t change the final result

SpinorChainTranspose[res, Select -> {{Spinor[__], Spinor[__]}}]

−i  el2gLor1  Lor2(φ(−p,me)).γLor1.(me−γ⋅k).γLor2.(φ(−q,me))16π4(k2−me2).(k−q)2-\frac{i \;\text{el}^2 g^{\text{Lor1}\;\text{Lor2}} (\varphi (-p,\text{me})).\gamma ^{\text{Lor1}}.(\text{me}-\gamma \cdot k).\gamma ^{\text{Lor2}}.(\varphi (-q,\text{me}))}{16 \pi ^4 \left(k^2-\text{me}^2\right).(k-q)^2}

Using patterns in the Select option one can create very fine-grained criteria for transposing the chains.