FermionSpinSum[exp] converts products of closed spinor
chains in exp into Dirac traces. Both Dirac and Majorana
particles are supported. It is understood, that exp
represents a squared amplitude.
Overview, Spinor, ComplexConjugate, DiracTrace.
FeynCalc uses the customary relativistic normalization of the spinors.
SpinorUBar[Momentum[p], m] . SpinorU[Momentum[p], m]
FermionSpinSum[%]
DiracSimplify[%]\bar{u}\left(\overline{p},m\right).u\left(\overline{p},m\right)
\text{tr}\left(\bar{\gamma }\cdot \overline{p}+m\right)
4 m
SpinorVBar[Momentum[p], m] . SpinorV[Momentum[p], m]
FermionSpinSum[%]
DiracSimplify[%]\bar{v}\left(\overline{p},m\right).v\left(\overline{p},m\right)
\text{tr}\left(\bar{\gamma }\cdot \overline{p}-m\right)
-4 m
amp = SpinorUBar[k1, m] . GS[p] . GA[5] . SpinorU[p1, m]
ampSq = amp ComplexConjugate[amp]\bar{u}(\text{k1},m).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.u(\text{p1},m)
\bar{u}(\text{k1},m).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.u(\text{p1},m) \left(-\left(\varphi (\overline{\text{p1}},m)\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{p}\right).\left(\varphi (\overline{\text{k1}},m)\right)\right)
FermionSpinSum[ampSq]
DiracSimplify[%]-\text{tr}\left(\left(\bar{\gamma }\cdot \overline{\text{k1}}+m\right).\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{\text{p1}}+m\right).\bar{\gamma }^5.\left(\bar{\gamma }\cdot \overline{p}\right)\right)
-4 \overline{p}^2 \left(\overline{\text{k1}}\cdot \overline{\text{p1}}\right)+8 \left(\overline{\text{k1}}\cdot \overline{p}\right) \left(\overline{p}\cdot \overline{\text{p1}}\right)-4 m^2 \overline{p}^2