Spontaneous Twin Symmetry Breaking

We consider a twin Higgs scenario where the $Z_2$ twin symmetry is broken spontaneously, not explicitly. This scenario provides an interesting interpretation of the Higgs metastability in the standard model; the $SU(4)$ breaking scale $f$ is determined by the scale where the Higgs self quartic coupling flips its sign. However, for the misalignment of nonzero vacuum expectation values of the twin Higgs fields, parameter tuning of ${\cal O}(m_h^2/f^2)$ is required like explicit twin symmetry breaking scenarios. For the minimal model with the exact twin symmetry, $f$ is ${\cal O}$($10^{10}$ GeV), so the model is very unnatural. We point out that the tuning can be significantly reduced ($f\gtrsim 2.7$ TeV) if there are twin vector-like leptons with large Yukawa coupling to twin Higgs fields.