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Register a new userAuxiliary variables for nonlinear equations with softly broken symmetries
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General methods of solving equations deal with solving N equations in N variables and the solutions are usually a set of discrete values. However, for problems with a softly broken symmetry these methods often first find a point which would be a solution if the symmetry were exact, and is thus an approximate solution. After this, the solver needs to move in the direction of the symmetry to find the actual solution, but that can be very difficult if this direction is not a straight line in the space of variables. The solution can often be found much more quickly by adding the generators of the softly broken symmetry as auxiliary variables. This makes the number of variables more than the equations and hence there will be a family of solutions, any one of which would be acceptable. In this paper we present a procedure for finding solutions in this case, and apply it to several simple examples and an important problem in the physics of false vacuum decay. We also provide a Mathematica package that implements Powells hybrid method with the generalization to allow more variables than equations.
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We develop methods to study the scalar sector of multi-Higgs models with large discrete symmetry groups that are softly broken. While in the exact symmetry limit, the model has very few parameters and can be studied analytically, proliferation of quadratic couplings in the most general softly broken case makes the analysis cumbersome. We identify two sets of soft breaking terms which play different roles: those which preserve the symmetric vacuum expectation value alignment, and the remaining terms which shift it. Focusing on alignment preserving terms, we check which structural features of the symmetric parent model are conserved and which are modified. We find remarkable examples of structural features which are inherited from the parent symmetric model and which persist even when no exact symmetry is left. The general procedure is illustrated with the example of the three-Higgs-doublet model with the softly broken symmetry group $Sigma(36)$.
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