Trigonometric Parity for the Composite Higgs

We identify trigonometric parity as the key ingredient behind models of neutral naturalness for the Higgs potential. We show that any symmetric coset space readily includes such a trigonometric parity, which is simply a combination of a $\pi /2$ rotation along a broken direction and a Higgs parity transformation. We explain how to extend the top sector such that this $Z_2$ remains intact while the rest of the shift symmetry is explicitly broken, resulting in the cancelation of the quadratic divergences in the Higgs potential. Assuming additional structure (for example partial compositeness with maximal symmetry) can render the Higgs potential completely finite and with minimal tuning. We apply our principles to construct the minimal model realizing trigonometric parity based on $SO(6)/SO(5) \simeq SU(4)/Sp(4)$, yielding the simplest model of neutral naturalness. An added advantage of this model is that a simple fermionic UV completion can be easily identified. We analyze the tuning of the Higgs potential and find that the top partners can be quite heavy while vector mesons need to be relatively light to obtain minimal tuning. Finally we briefly comment on some novel phenomenology, including a possible six top final state at the LHC appearing in this model.