Sphaleron-Bisphaleron bifurcations in a custodial-symmetric two-doublets model

The standard electroweak model is extended by means of a second Brout-Englert-Higgs-doublet. The symmetry breaking potential is chosen is such a way that (i) the Lagrangian possesses a custodial symmetry, (ii) a static, spherically symmetric ansatz of the bosonic fields consistently reduces the Euler-Lagrange equations to a set of differential equations. The potential involves, in particular, products of fields of the two doublets, with a coupling constant $\lambda_3$.Static, finite energy solutions of the classical equations are constructed. The regular, non-trivial solutions having the lowest classical energy can be of two types: sphaleron or bisphaleron, according to the coupling constants. A special emphasis is put to the bifurcation between these two types of solutions which is analyzed in function of the different constants of the model,namely of $\lambda_3$.