\begin{document}$\kappa$\end{document} framework. Incorporating higher-order quantum corrections from virtual Higgs bosons would significantly alter this functional form, introducing new quartic and cubic power dependencies on the trilinear Higgs self-coupling. To derive this new functional form, we propose a specialized renormalization procedure that tracks all Higgs self-couplings at each calculation step. Additionally, we introduce renormalization constants for coupling modifiers within the \begin{document}$\kappa$\end{document} framework to ensure the cancellation of all ultraviolet divergences. With new functional forms of the cross sections in both the gluon-gluon fusion and vector boson fusion channels, the upper limit of \begin{document}$\kappa_{\lambda_{{\rm{3H}}}}=\lambda_{{\rm{3H}}}/\lambda_{{\rm{3H}}}^{{\rm{SM}}}$\end{document} set by the ATLAS (CMS) collaboration is reduced from 6.6 (6.49) to 5.4 (5.37). However, extracting a meaningful constraint on the quartic Higgs self-coupling \begin{document}$\lambda_{{\rm{4H}}}$\end{document} from Higgs boson pair production data remains challenging. We also present the invariant mass distributions of the Higgs boson pair at different values of the self-couplings, which could assist in setting optimal cuts for experimental analysis."> Improved constraints on Higgs boson self-couplings with quartic and cubic power dependencies of the cross section -
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