WWV ( V=γ , Z ) vertex in the Georgi-Machacek model

Abstract

The $CP$-even static form factors $\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{V}^{\ensuremath{'}}$ and $\mathrm{\ensuremath{\Delta}}{Q}_{V}$ ($V=\ensuremath{\gamma}$, $Z$) associated with the $WWV$ vertex are studied in the context of the Georgi-Machacek model (GMM), which predicts nine new scalar bosons accommodated in a singlet, a triplet, and a fiveplet. General expressions for the one-loop contributions to $\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{V}^{\ensuremath{'}}$ and $\mathrm{\ensuremath{\Delta}}{Q}_{V}$ arising from neutral, singly, and doubly charged scalar bosons are obtained in terms of both parametric integrals and Passarino-Veltman scalar functions, which can be numerically evaluated. It is found that the GMM yields 15 (28) distinct contributions to $\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{\ensuremath{\gamma}}^{\ensuremath{'}}$ and $\mathrm{\ensuremath{\Delta}}{Q}_{\ensuremath{\gamma}}$ ($\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{Z}^{\ensuremath{'}}$ and $\mathrm{\ensuremath{\Delta}}{Q}_{Z}$), though several of them are naturally suppressed. A numerical analysis is done in the region of parameter space still consistent with current experimental data and it is found that the largest contributions to $\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{V}^{\ensuremath{'}}$ arise from Feynman diagrams with two nondegenerate scalar bosons in the loop, with values of the order of $a={g}^{2}/(96{\ensuremath{\pi}}^{2})$ reached when there is a large splitting between the masses of these scalar bosons. As for $\mathrm{\ensuremath{\Delta}}{Q}_{V}$, it reaches values as large as ${10}^{\ensuremath{-}2}a$ for the lightest allowed scalar bosons, but it decreases rapidly as one of the masses of the scalar bosons becomes large. Among the new contributions of the GMM to the $\mathrm{\ensuremath{\Delta}}{\ensuremath{\kappa}}_{V}^{\ensuremath{'}}$ and $\mathrm{\ensuremath{\Delta}}{Q}_{V}$ form factors are those induced by the ${H}_{5}^{\ifmmode\pm\else\textpm\fi{}}{W}^{\ensuremath{\mp}}Z$ vertex, which arises at the tree level and is a unique prediction of this model.