Signatures of discrete symmetries in the scalar sector.

I discuss methods to identify the presence of dicrete symmetries in the two-Higgsdoublet model by observing the masses and the cubic and quartic interactions of the scalars. The symmetries considered are a Z2 symmetry under which φ2 → −φ2, and a CP symmetry which enforces real coupling constants in the Higgs potential. Those symmetries are spontaneously broken, and the Z2 symmetry may also be softly broken. I identify the signatures in the interactions of the scalars that these symmetries leave after their breaking. Twenty-one years ago, T. D. Lee [1] pointed out that CP may be spontaneously broken. In his two-Higgs-doublet model, CP is a symmetry of the Lagrangian, which is broken by the relative phase between the vacuum expectation values (VEVs) of the two Higgs doublets. Other models of spontaneous CP violation have been suggested since then [2, 3], and spontaneous CP violation has been used as an ingredient in the building of many models [4, 5]. However, no one has yet attempted to answer the following basic questions: how can we experimentally distinguish between spontaneous and explicit CP violation? If CP violation is spontaneous, does that fact lead to some relationships among the coefficients of the various interaction terms in the Lagrangian, relationships which might be experimentally tested for? (At least in principle, even if the practical measurements might be too difficult.) In the context of the two-Higgs-doublet model, it is usual to assume the existence of a discrete symmetry Z2, under which one of the two doublets changes sign, while the other doublet remains unaffected. That symmetry is softly broken in some models. How can we assert experimentally whether such a symmetry exists or not, and whether it is softly broken or not? After discrete symmetries in the scalar sector are spontaneously or softly broken, do they still leave traces of their presence in the fundamental Lagrangian? I present in this Brief Report a partial answer to these questions. For definiteness, I concentrate on the two-Higgs-doublet model. That model is now very popular, partly because two doublets is the Higgs structure of the minimal supersymmetric ∗On leave of absence from Universidade Técnica de Lisboa, Lisbon, Portugal 1 standard model. If there are more than two doublets the algebra involved becomes extremely heavy. I consider a SU(2)⊗U(1) gauge model with two scalar doublets φ1 and φ2. The most general Higgs potential consistent with renormalizability is V = m1φ † 1φ1 +m2φ † 2φ2 + (m3φ † 1φ2 + h.c.) +a1(φ † 1φ1) 2 + a2(φ † 2φ2) 2 + a3(φ † 1φ1)(φ † 2φ2) + a4(φ † 1φ2)(φ † 2φ1) + [ a5(φ † 1φ2) 2 + a6(φ † 1φ1)(φ † 1φ2) + a7(φ † 2φ2)(φ † 1φ2) + h.c. ] . (1) All the coupling constants, except m3, a5, a6, and a7, are real because of hermiticity. I assume that the VEVs of φ1 and φ2 are aligned, in the sense that they preserve the U(1) of electromagnetism. Those VEVs have a relative phase: the VEV of φ01 is v1, and the VEV of φ 0 2 is v2 exp(iα), v1 and v2 being real and positive. 2 v = √ v 1 + v 2 2 is a measurable quantity, v = 174 GeV. Instead of working with φ1 and φ2, it is convenient to work in the Georgi [6] basis of doublets H1 and H2, with the following defining features: H1 has real and positive VEV v, while H2 has vanishing VEV. The Georgi basis is reached by means of the transformation φ1 = (v1H1 + v2H2)/v , φ2 = e (v2H1 − v1H2)/v . (2) In the Georgi basis, the Higgs potential reads V = μ1H † 1H1 + μ2H † 2H2 + (μ3H † 1H2 + h.c.) +λ1(H † 1H1) 2 + λ2(H † 2H2) 2 + λ3(H † 1H1)(H † 2H2) + λ4(H † 1H2)(H † 2H1) + [ λ5(H † 1H2) 2 + λ6(H † 1H1)(H † 1H2) + λ7(H † 2H2)(H † 1H2) + h.c. ] , (3) in which all the coupling constants, except μ3, λ5, λ6, and λ7, are real by hermiticity. Because only H1 has a non-zero VEV, v, which is real, the stationarity conditions of the vacuum read μ1 = −2λ1v , (4) μ3 = −λ6v . (5) I use these conditions to eliminate μ1 and μ3 as independent variables from V . Because μ3 is complex while μ1 is real, Eqs. 4 and 5 constitute three real equations. They correspond to the three real equations which, in the basis of φ1 and φ2, determine the stability of the vacuum by fixing the partial derivatives of the vacuum potential with respect to v1, v2, and α, to be zero. The Georgi basis is useful because the Goldstone modes are perfectly identified when one uses it. Writing H1 = ( G v + (H + iG)/ √ 2 ) , (6) H2 = ( H (R + iI)/ √ 2 ) , (7) T. D. Lee [1] has shown that this happens if one inequality is satisfied by the coupling constants of the potential. The VEV of φ 1 is made real and positive by a gauge transformation. This represents no loss of generality. 2 G and G are the Goldstone bosons which, in the unitary gauge, become the longitudinal components of the W and of the Z. H, R and I are real neutral fields, which are linear combinations of the three physical scalars Xk,