CPviolation in weak interactions from orbifold reduction: Possible unification structures

Abstract

We present a mechanism to generate complex phases from real 4 + 1 dimensional couplings in a model of weak interactions through dimensional reduction of a gauge theory. The orbifolding of a 4 + 1 dimensional Sp(4) × U (1) group is the minimal setup which provides both CP violation and an SU (2) × U (1) structure. We show that grand unification requires at least SO(11).CP violation in the standard model, since gauge interactions are naturally CP symmetric, is provided by complex Yukawa couplings which eventually are combined in the CKM matrix in one observable CP violating phase. While this picture has been comforted through B-decay observations [1], the standard model does not tell us more on the origin of CP violation since it is explicitely introduced.On the other hand, a truly unified theory would relate Yukawa couplings to gauge interactions implying that this unified theory would be CP symmetric. In that context a CP breaking mechanism is needed and can be found, as addressed here, in dimensional reduction. One example of these possibilities has been studied in [2]. We present here a realistic realisation of these ideas in the standard model.In the context of five dimensional gauge theory, the reduction from 4 + 1 to 3 + 1 dimensions has to deal with the extra contribution to the energy coming from the extra component of the covariant derivative, that is: D y = ∂ y − ieA y , where the derivative leads to the well known Kaluza-Klein(KK) effective mass n R in 3 + 1 dimensions.For spinors, this contribution is associated to the usual 3 + 1 dimensional pseudoscalar:ψγ 5 ψ, since the Clifford algebra is extended to γ B = (γ µ , iγ 5 ) for 4 + 1 dimensions ( B = 0, 1, · · · , (4 = y)). Thus, whatever the reduction scheme is, the fermionic mass term may receive effective complex masses of the type:This effective complex mass will lead to CP violation( although in a pure minimal-coupling U (1) theory the complex phase can be rotated away by a redefinition of spinors).Several contributions can be considered for X, e.g. the KK mass n R combined with a non-minimal coupling to the photon has been studied by Thirring [3]. Otherwise, in order to distinguish CP violation from the use of exited states, some vacuum expectation value for the extra component of the gauge field, that is the gauge invariant line integral X = A y = dyA y , * ncosme@ulb.ac.be, frere@ulb.ac.be.together with an extention of the gauge group has been considered in [2]. This line integral keeps 3 + 1 dimensional Lorentz invariance and reduces to the usual Wilson loop in the case of a compact extra space.For instance, consider a 4 + 1D SU (2) gauge group with massive doublet Ψ = (ψ 1 , ψ 2 ), and take the expectation value W y = dy W y = w σ 3 to break the group to vectorlike effective interactions in 3 + 1 dimensions:with two massive W ± and one massless W 3 . Then, the Wilson loop contributes to a complex mass matrix:Both phases cannot be redefined and, while making the mass matrix real, a remaining phase appears in the charge current implying a W 3 ...

About Mass, CP and Extra Dimensions

“…In this case, we have shown that the first usable unification group is SO(11) (rather than the usual SU (5) or SO (10)). The source of this extension is precisely in the need to localize both left-and right-handed fermions.…”

“…In a simple example, we generate CP violation from a 4+1 theory with only real couplings. This work is based on reference [9], and its generalization to include grand unified structures [10].…”

A model of fermion masses and mixings triggered by family problem in warped extra dimensions

“…(2.13). The total operations on the fermion fields equal to 21) or 22) where we have omitted the subscripts. In the above, beginning with the 6D action eq.…”

Complete CKM quark mixing via dimensional deconstruction

“…Interestingly, the above CP violation induction via deconstruction can also be visualized in extra dimensional view point. Indeed, because of having the same symmetry transformation property, DD link field can be identified with the Wilson line pointing along a latticized transverse direction (Appendix B), and the latter then can naturally acquire a complex VEV in the generalized Hosotani's mechanism [6,7,8,9] of dynamical symmetry breaking. Apparently, the source of CP violation in this approach comes from the complex effective Yukawa couplings so it can be classified as hard CP violation.…”

“…In this paper we look specifically into two such important mechanisms of extra dimension theories, namely the localization of matter fields in the bulk [3,4,5] and the dynamical breaking of CP symmetry by ED Wilson line [6,7,8,9]. Ultimately, the hybrid of these two mechanisms is just the well-known complex mixing of fermion flavors.…”