FeynCalc manual (development version)

GordonSimplify

GordonSimplify[exp] rewrites spinor chains describing a vector or an axial-vector current using Gordon identities.

See also

Overview, DiracGamma, Spinor, SpinorChainTrick.

Examples

SpinorUBar[p1, m1] . GA[\[Mu]] . SpinorU[p2, m2] 
 
GordonSimplify[%]

uˉ(p1,m1).γˉμ.u(p2,m2)\bar{u}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.u(\text{p2},\text{m2})

(p1+p2)μ(φ(p1,m1)).(φ(p2,m2))m1+m2+i(φ(p1,m1)).σμp1p2.(φ(p2,m2))m1+m2\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (\overline{\text{p1}},\text{m1})\right).\left(\varphi (\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}+\frac{i \left(\varphi (\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\left(\varphi (\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}

SpinorUBar[p1, m1] . GA[\[Mu], 5] . SpinorV[p2, m2] 
 
GordonSimplify[%]

uˉ(p1,m1).γˉμ.γˉ5.v(p2,m2)\bar{u}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.\bar{\gamma }^5.v(\text{p2},\text{m2})

(p1+p2)μ(φ(p1,m1)).γˉ5.(φ(p2,m2))m1+m2+i(φ(p1,m1)).σμp1p2.γˉ5.(φ(p2,m2))m1+m2\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^5.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}+\frac{i \left(\varphi (\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\bar{\gamma }^5.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}

Relations involving projectors can be used to trade the right projector for a left one

SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2] 
 
GordonSimplify[%]

vˉ(p1,m1).γˉμ.γˉ6.v(p2,m2)\bar{v}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.\bar{\gamma }^6.v(\text{p2},\text{m2})

i(φ(p1,m1)).σμp1p2.γˉ6.(φ(p2,m2))m1m2(φ(p1,m1)).γˉμ.γˉ7.(φ(p2,m2))m1(p1+p2)μ(φ(p1,m1)).γˉ6.(φ(p2,m2))m1-\frac{i \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\bar{\gamma }^6.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\text{m2} \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^{\mu }.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^6.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}

Use the Select option to achieve the opposite

ex = SpinorVBar[p1, m1] . GA[\[Mu], 7] . SpinorV[p2, m2] 
 
GordonSimplify[ex]

vˉ(p1,m1).γˉμ.γˉ7.v(p2,m2)\bar{v}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.\bar{\gamma }^7.v(\text{p2},\text{m2})

(φ(p1,m1)).γˉμ.γˉ7.(φ(p2,m2))\left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^{\mu }.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)

GordonSimplify[ex, Select -> {{Spinor[__], DiracGamma[__], GA[7], Spinor[__]}}]

i(φ(p1,m1)).σμp1p2.γˉ7.(φ(p2,m2))m1m2(φ(p1,m1)).γˉμ.γˉ6.(φ(p2,m2))m1(p1+p2)μ(φ(p1,m1)).γˉ7.(φ(p2,m2))m1-\frac{i \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\text{m2} \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^{\mu }.\bar{\gamma }^6.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}

We can choose between having expressions proportional to 1/m11/m_1 (mass of the first spinor) or 1/m21/m_2 (mass of the second spinor)

GordonSimplify[SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2], Inverse -> First]

i(φ(p1,m1)).σμp1p2.γˉ6.(φ(p2,m2))m1m2(φ(p1,m1)).γˉμ.γˉ7.(φ(p2,m2))m1(p1+p2)μ(φ(p1,m1)).γˉ6.(φ(p2,m2))m1-\frac{i \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\bar{\gamma }^6.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\text{m2} \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^{\mu }.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}-\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^6.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m1}}

GordonSimplify[SpinorVBar[p1, m1] . GA[\[Mu], 6] . SpinorV[p2, m2], Inverse -> Last]

i(φ(p1,m1)).σμp1p2.γˉ7.(φ(p2,m2))m2m1(φ(p1,m1)).γˉμ.γˉ7.(φ(p2,m2))m2(p1+p2)μ(φ(p1,m1)).γˉ7.(φ(p2,m2))m2-\frac{i \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m2}}-\frac{\text{m1} \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^{\mu }.\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m2}}-\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (-\overline{\text{p1}},\text{m1})\right).\bar{\gamma }^7.\left(\varphi (-\overline{\text{p2}},\text{m2})\right)}{\text{m2}}

In DD-dimensions chiral Gordon identities are scheme dependent!

ex = SpinorVBarD[p1, m1] . GAD[\[Mu], 5] . SpinorVD[p2, m2]

vˉ(p1,m1).γμ.γˉ5.v(p2,m2)\bar{v}(\text{p1},\text{m1}).\gamma ^{\mu }.\bar{\gamma }^5.v(\text{p2},\text{m2})

FCGetDiracGammaScheme[] 
 
GordonSimplify[ex]

NDR\text{NDR}

(p1+p2)μ(φ(p1,m1)).γˉ5.(φ(p2,m2))m1m2i(φ(p1,m1)).σμ  p1p2.γˉ5.(φ(p2,m2))m1m2-\frac{(\text{p1}+\text{p2})^{\mu } (\varphi (-\text{p1},\text{m1})).\bar{\gamma }^5.(\varphi (-\text{p2},\text{m2}))}{\text{m1}-\text{m2}}-\frac{i (\varphi (-\text{p1},\text{m1})).\sigma ^{\mu \;\text{p1}-\text{p2}}.\bar{\gamma }^5.(\varphi (-\text{p2},\text{m2}))}{\text{m1}-\text{m2}}

FCSetDiracGammaScheme["BMHV"] 
 
GordonSimplify[ex]

BMHV\text{BMHV}

i(φ(p1,m1)).σμ  p1p2.γˉ5.(φ(p2,m2))m1m2(p1+p2)μ(φ(p1,m1)).γˉ5.(φ(p2,m2))m1m2+2(φ(p1,m1)).γμ.(γ^p2^).γˉ5.(φ(p2,m2))m1m2-\frac{i (\varphi (-\text{p1},\text{m1})).\sigma ^{\mu \;\text{p1}-\text{p2}}.\bar{\gamma }^5.(\varphi (-\text{p2},\text{m2}))}{\text{m1}-\text{m2}}-\frac{(\text{p1}+\text{p2})^{\mu } (\varphi (-\text{p1},\text{m1})).\bar{\gamma }^5.(\varphi (-\text{p2},\text{m2}))}{\text{m1}-\text{m2}}+\frac{2 (\varphi (-\text{p1},\text{m1})).\gamma ^{\mu }.\left(\hat{\gamma }\cdot \hat{\text{p2}}\right).\bar{\gamma }^5.(\varphi (-\text{p2},\text{m2}))}{\text{m1}-\text{m2}}

FCSetDiracGammaScheme["NDR"]

NDR\text{NDR}