EpsChisholm
EpsChisholm[exp]
applies the Chisholm identity to a Dirac matrix contracted with a Levi-Civita tensor.
See also
Overview, Chisholm, Eps, DiracGamma.
Examples
LC[\[Mu], \[Nu], \[Rho], \[Sigma]] GA[\[Sigma], 5]
EpsChisholm[%]
γˉσ.γˉ5ϵˉμνρσ
−iγˉμ.γˉν.γˉρ+iγˉρgˉμν−iγˉνgˉμρ+iγˉμgˉνρ
This reproduces the identities given in the Appendix A of arXiv:2111.05153
LC[\[Alpha], \[Nu], \[Beta], \[Rho]] FV[Subscript[p, 1], \[Beta]] SpinorUBar[Subscript[p, 2], SMP["m_s"]] . GA[\[Alpha], 7] . SpinorV[Subscript[p, 1], SMP["m_d"]]
% // EpsChisholm // DiracSimplify // Contract
p1βϵˉανβρuˉ(p2,ms).γˉα.γˉ7.v(p1,md)
imdgˉνρ(φ(p2,ms)).γˉ6.(φ(−p1,md))+ip1ν(φ(p2,ms)).γˉρ.γˉ7.(φ(−p1,md))−ip1ρ(φ(p2,ms)).γˉν.γˉ7.(φ(−p1,md))−imd(φ(p2,ms)).γˉν.γˉρ.γˉ6.(φ(−p1,md))
LC[\[Alpha], \[Nu], \[Beta], \[Rho]] FV[Subscript[p, 2], \[Beta]] SpinorUBar[Subscript[p, 2], SMP["m_s"]] . GA[\[Alpha], 7] . SpinorV[Subscript[p, 1], SMP["m_d"]]
% // EpsChisholm // DiracSimplify // Contract
p2βϵˉανβρuˉ(p2,ms).γˉα.γˉ7.v(p1,md)
−imsgˉνρ(φ(p2,ms)).γˉ7.(φ(−p1,md))−ip2ν(φ(p2,ms)).γˉρ.γˉ7.(φ(−p1,md))+ip2ρ(φ(p2,ms)).γˉν.γˉ7.(φ(−p1,md))+ims(φ(p2,ms)).γˉν.γˉρ.γˉ7.(φ(−p1,md))
LC[\[Alpha], \[Nu], \[Gamma], \[Rho]] FV[Subscript[p, 3], \[Gamma]] SpinorUBar[Subscript[p, 3], SMP["m_s"]] . GA[\[Nu], 7] . SpinorV[Subscript[p, 4], SMP["m_d"]]
% // EpsChisholm // DiracSimplify // Contract
p3γϵˉανγρuˉ(p3,ms).γˉν.γˉ7.v(p4,md)
imsgˉαρ(φ(p3,ms)).γˉ7.(φ(−p4,md))+ip3α(φ(p3,ms)).γˉρ.γˉ7.(φ(−p4,md))−ip3ρ(φ(p3,ms)).γˉα.γˉ7.(φ(−p4,md))−ims(φ(p3,ms)).γˉα.γˉρ.γˉ7.(φ(−p4,md))
LC[\[Beta], \[Gamma], \[Mu], \[Nu]] FV[Subscript[p, 2], \[Gamma]] SpinorUBar[Subscript[p, 3], SMP["m_s"]] . GA[\[Beta], 7] . SpinorV[Subscript[p, 4], SMP["m_d"]]
% // EpsChisholm // DiracSimplify // Contract
p2γϵˉβγμνuˉ(p3,ms).γˉβ.γˉ7.v(p4,md)
igˉμν(φ(p3,ms)).(γˉ⋅p2).γˉ7.(φ(−p4,md))+ip2μ(φ(p3,ms)).γˉν.γˉ7.(φ(−p4,md))−ip2ν(φ(p3,ms)).γˉμ.γˉ7.(φ(−p4,md))−i(φ(p3,ms)).(γˉ⋅p2).γˉμ.γˉν.γˉ7.(φ(−p4,md))