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Feynman rules for Nucleon-Nucleon Interaction with Tensor Coupling

Physics Asked on August 6, 2021

Hello im trying to Compute the Cross section of an elastic Nucleon-Nucleon scattering and i don’t know what to make of the following Lagrangian:
begin{equation}
mathcal{L}_{rho N N}=-g_{rho} bar{psi} gamma^{mu} boldsymbol{tau} psi cdot boldsymbol{varphi}_{mu}^{(rho)}-frac{f_{rho}}{4 m} bar{psi} sigma^{mu nu} tau psi cdotleft(partial_{mu} boldsymbol{varphi}_{nu}^{(rho)}-partial_{nu} boldsymbol{varphi}_{mu}^{(rho)}right)
end{equation}

Specifcally the second term is the one which complicates things for me. What i know is that the derivative includes a term of the momentum of the vector boson but i don’t know how to continue from there. What im reading suggests the vertex should be of this from
begin{equation}
Gamma^mu=ifrac{f_a}{2m}sigma^{munu}(q’-q)_nu
end{equation}

But i can’t understand how a object with 2 indices namely
$left(partial_{mu} boldsymbol{varphi}_{nu}^{(rho)}-partial_{nu} boldsymbol{varphi}_{mu}^{(rho)}right)$ became $(q’-q)_nu$. Also $sigma_{munu}=frac{i}{2}[gamma_mu,gamma_nu]$

I found out also about Gordon decomposition https://en.wikipedia.org/wiki/Gordon_decomposition which greatly helps once you actually calculated $Gamma_mu$ but i still don’t understand why we don’t get
begin{equation}
Gamma^mu=ifrac{f_a}{4m}sigma^{munu}(k_muvarphi_nu-k_nuvarphi_mu)
end{equation}

Which can’t be right though.

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