Imposing additional symmetries to the Higgs doublets in the Two-Higgs doublets model, like the $Z_2$ symmetry, aims to reduce the number of parameters in the potential. But it is said that the mass term $m^2_{12}$ term breaks it "softly" what is the meaning of softly in this particular case ?
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$\begingroup$ Look at this answer of mine physics.stackexchange.com/a/421945/47373 for what we mean by "soft". $\endgroup$– apt45Aug 12, 2018 at 18:37
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Actually, imposing $Z_2$ on the two Higgs doublet model (2HDM) is usually a strategy to prevent flavor changing neutral currents, which are very restricted by experiment. Alternatively, it is also a way of impose CP invariance in the model.
With this in mind, $Z_2$ symmetry removes the lagrangian parameters $\lambda_6$, $\lambda_7$ and $m_{12}$ (as they are usually denominated). This is an exact symmetry on the theory. When considering $m_{12}\neq 0$, we are still retaining CP conservation on the theory, but we "softly" break $Z_2$ as to keep the desired physical properties of the model but not oversimplifying it.
