Gauge Unification In Six Dimensions

We study the breaking of a supersymmetric SO(10) GUT in 6 dimensions by orbifold compactification. In 4 dimensions we obtain a N=1 supersymmetric theory with the standard model gauge group enlarged by an additional U(1) symmetry. The 4-dimensional gauge symmetry is obtained as intersection of the Pati-Salam and the Georgi-Glashow subgroups of SO(10), which appear as unbroken subgroups in the two 5 dimensional subspaces, respectively. The doublet-triplet splitting arises as in the recently discussed SU(5) GUTs in 5 dimensions. The simplest grand unified theory (GUT) which unifies one generation of quarks and leptons, including the right-handed neutrino, in a single irreducible representation is based on the gauge group SO(10)[1]. Important subgroups, whose phenomenology has been studied in great detail, are the Pati-Salam group SU(4)×SU(2)×SU(2) [2] and the GUT group SU(5) of Georgi and Glashow [3]. The breaking of these GUT groups down to the standard model gauge group is in general rather involved and requires often large Higgs representations. In particular, the mass splitting between the weak doublet and the colour triplet Higgs fields requires either a fine-tuning of parameters or additional, sophisticated mechanisms. An attractive new possibility has been suggested by Kawamura [4]: compactifying a SU(5) GUT in 5 dimensions (5d) on an orbifold, which breaks SU(5) to the standard model group SU(3)×SU(2)×U(1) only on one of the boundary branes, one achieves in 4d the correct gauge symmetry breaking and in addition the wanted doublet-triplet splitting. Various aspects of such 5d SU(5) GUTs, including fermion masses, have recently been studied by several groups [5, 6]. The goal of the present paper is to extend the orbifold breaking to the GUT group SO(10). The breaking of G=SU(5) is achieved by means of a ‘parity’ P , under which the generators of G are either even (S) or odd (A), PSP = S , PAP = −A . (1) Clearly, the set of generators S and A satisfy the relations [S, S] ⊆ S , [S,A] ⊆ A , [A,A] ⊆ S . (2) Hence, the group GS generated by S is a symmetric subgroup of G. Together with the identity P forms the discrete group Z2. In orbifold compactifications of 5d SU(5) GUTs the theory is assumed to be invariant under the parity transformation of gauge fields V (x, y), M = (μ, 5), μ = 0 . . . 3, x = y, and matter fields H±(x, y), PVμ(x,−y + a)P = +Vμ(x, y + a) , PV5(x,−y + a)P = −V5(x, y + a) , (3) PH±(x,−y + a) = ±H±(x, y + a) , (4) where a = 0 or a = πR/2, and R is the radius of the compact dimension. On the orbifold, M = R×S/Z2, the GUT symmetry is then broken on branes which are fixed points of the transformation. Note, that this symmetry breaking preserves the rank of the group. The extension of this procedure to the GUT group SO(10) is not straightforward since the standard model group GSM is not a symmetric subgroup of SO(10)[7, 6]. It is For a detailed discussion of GUT breaking by orbifolding, see e.g. [6].