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                \begin{document}$${\vec {\ell }}_i\cdot {\vec {G}}_j=2\pi \delta _{i,j}$$\end{document}$):$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} 0 = i {\widehat{K}}^{\text {R}}_{i,j}(\mathbf{0}) + \frac{\partial }{\partial p_{i}} {{\widehat{S}}}^{\text {R}}_{j}(\mathbf{0}). \end{aligned}$$\end{document}$$Now, recall the definition of $\documentclass[12pt]{minimal}
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                \begin{document}$${\widehat{S}}^{\beta ,L;\text {R}}_{j}(\mathbf{p})$$\end{document}$ from Eq. ([3.8](#Equ32){ref-type=""}): $\documentclass[12pt]{minimal}
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                \begin{document}$${\widehat{S}}^{\beta ,L;\text {R}}_{j}(\mathbf{0})=- i\sum _{{\vec {x}}} e^{-i{\vec {p}}\cdot {\vec {x}}} \langle [n_{{\vec {x}}}\, , j