ocument}$ and $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
                \usepackage{wasysym} 
                \usepackage{amsfonts} 
                \usepackage{amssymb} 
                \usepackage{amsbsy}
                \usepackage{mathrsfs}
                \usepackage{upgreek}
                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$s_\omega ({\vec {k}}')$$\end{document}$ are both of order $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
                \usepackage{wasysym} 
                \usepackage{amsfonts} 
                \usepackage{amssymb} 
                \usepackage{amsbsy}
                \usepackage{mathrsfs}
                \usepackage{upgreek}
                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$O(|{\vec {k}}'|^2)$$\end{document}$, while the mass satisfies (again, recall that $\documentclass[12pt]{minimal}
                \usepackage{amsmath}
                \usepackage{wasysym} 
                \usepackage{amsfonts} 
                \usepackage{amssymb} 
                \usepackage{amsbsy}
                \usepackage{mathrsfs}
                \usepackage{upgreek}
                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$m_{\text {R},+}=m_{\text {R},-}+6\sqrt{3} t_2\sin \phi $$\end{document}$):$$\documentclass[12pt]{minimal}
                \usepackage{amsmath}
                \usepackage{wasysym} 
                \usepackage{amsfonts} 
                \usepackage{amssymb} 
                \usepackage{amsbsy}
                \usepackage{mathrsfs}
                \usepackage{upgreek}
                \setlength{\oddsidemargin}{-69pt}
                \begin{document}$$\begin{aligned} m_{\omega }({\vec {k}}')=m_{\text {R},\omega }+t_\omega ({\vec {k}}'), \quad \text {with}\quad t_\omega ({\vec {k}}')=O(|{\vec {k}}'|^2). \end{aligned}$$\end{document}$$By definition, the representation above is valid at the initial step, $\documentclass[12pt]{minimal}
                \usep