One loop effective potential for SO(10) GUT theories in de Sitter space

Zeta-function regularization is applied to the evaluation of the one-loop effective potential for SO(10) grand unified theories in de Sitter cosmologies. When the Higgs scalar field belongs to the 210-dimensional irreducible representation of SO(lO), attention is focused on the mass matrix relevant for the SU( 3 )@SU( 2 )@U( 1) symmetry-breaking direction, to agree with the low-energy phenomenology of the standard model of particle physics. The analysis is restricted to those values of the tree-level potential parameters for which the absolute minimum of the classical potential has been evaluated. As shown in the recent literamre. such minima tum out to be SO(6) @ SO(4) or SU(3) @ SU(2) @ SU(2) @ U(1) invariant. Electroweak phenomenology is more naturally derived, however. from the former minima. Hence the values of the parameters leading to the alternative set of minima have been discarded. Within this framework the flat-space limit and the general form of the one-loop effective potential are studied in detail by using analytic and numerical methods. It turns out that, as far as the absolute-minimum direction is concerned, the flat-space limit of the one-loop calculation with a de Sitter background does not change the results previously obtained in the literature, where the tree-level potential in flat spacelime was studied. Moreover, even when curvature effects are no longer negligible in the one-loop potential. it is found that the early universe can only reach the SO(6) @SO(4) absolute minimum.