Erratum to: Spontaneous symmetry breaking in the S3-symmetric scalar sector

D. Emmanuel-Costa, O.M. Ogreid, P. Osland and M.N. Rebelo Centro de F́ısica Teórica de Part́ıculas – CFTP, Instituto Superior Técnico – IST, Universidade de Lisboa, Av. Rovisco Pais, P-1049-001 Lisboa, Portugal Bergen University College, Bergen, Norway Department of Physics and Technology, University of Bergen, Postboks 7803, N-5020 Bergen, Norway Theory Department, CERN, CH 1211 Geneva 23, Switzerland


JHEP08(2016)169
Vacuum λ 4 SCPV Vacuum λ 4 SCPV Vacuum λ 4 SCPV C-I-a X no C-III-f,g 0 no C-IV-c X yes C-III-a X yes C-III-h X yes C-IV-d 0 no C-III-b 0 no C-III-i X no C-IV-e 0 no C-III-c 0 no C-IV-a 0 no C-IV-f X yes C-III-d,e X no C-IV-b 0 no C-V 0 no Table 6. Spontaneous CP violation.
In our discussion of Spontaneous CP violation two cases must be corrected. In " Table 6: Spontaneous CP Violation." cases C-III-c and C-IV-e corresponding to λ 4 = 0 are indicated as having SPCV (spontaneous CP violation), however this is not the case. The correct table is given above.
As a result one may conclude that S 3 symmetric models with λ 4 = 0 cannot violate CP spontaneously. Still, there are cases with λ 4 = 0 where spontaneous CP violation may occur in three Higgs doublet models with S 3 symmetry.
The explanation for the absence of spontaneous CP violation in these two cases lies in the fact that models with λ 4 = 0 have an additional SO(2) symmetry. This symmetry can be used to build a matrix U verifying eq. (8.3) for cases C-III-c and C-IV-e.
In order to prove that case C-III-c does not violate CP spontaneously we start from the corresponding set of vevs (ŵ 1 e iσ ,ŵ 2 , 0) and perform a Higgs basis transformation on the Higgs doublets h 1 and h 2 by an SO(2) rotation into: such that the vevs of the new S 3 doublet fields now have the same modulus and are now of the form (ae iδ 1 , ae iδ 2 , 0). This requires Obviously the Lagrangian remains invariant. Next we perform an overall phase rotation of the three Higgs doublets with the phase factor exp[−i(δ 1 + δ 2 )/2], leading now to the following vevs: (ae iδ , ae −iδ , 0). Making use of the symmetry for the interchange h ′ 1 ↔ h ′ 2 we can verify eq. (8.3) in the following way: In terms of the initial vevs, this equation translates into Notice that (ae iδ , ae −iδ , 0) is a special case of the PS vacuum, given in eq. (5.10).
In order to prove that C-IV-e does not violate CP spontaneously we start with the corresponding set of vevs: (ŵ 1 e iσ 1 ,ŵ 2 e iσ 2 ,ŵ S ) wherê in this phase convention. In general one should write sin(2σ 1 − 2σ S ) and sin(2σ 2 − 2σ S ) in the latter relation, where σ S would be the phase of the third vev. We now perform an SO(2) rotation, similar to the one specified above, with which once again will lead to equal moduli for the S 3 doublet fields. In this case, the vevs will acquire the form (be iγ 1 , be iγ 2 ,ŵ S ). Unlike in case C-III-c, an overall phase rotation would also affect the vev of h S . However, it turns out that condition (8.10) enforces As a result, this SO(2) rotation takes us to the PS vacuum, which, as we discussed previously, does not violate CP spontaneously.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.