Gauge Coupling Unification with Extra Dimensions and Correction due to Higher Dimensional Operators

We study the gauge coupling unification with extra dimensions. We take into account corrections due to the higher dimensional operators. We show the prediction of α3(MZ) is sensitive to such corrections, even if c < Φ > /M = O(0.01). We also discuss the b − τ Yukawa unification. Partially supported by the Academy of Finland (no. 44129). Recently, theories with large extra dimensions have been studied intensively [1]-[5]. If such extra dimensions correspond to a TeV scale that can be a solution of the naturalness problem. One of interesting aspects in the theory with extra dimensions is the power-law behaviour of the running gauge coupling constants shown in Ref. [6], that is, the towers of Kaluza-Klein excitation modes lead to the power-law behaviour. That provides with the possibility that the three gauge coupling constants of the standard model are unified at lower energy scale than 10 GeV and the unified energy scale is just above the energy scale μ0 where the Kaluza-Klein excitation modes appear. However, detailed analyses on the minimal matter content and canonical level of U(1)Y show that the predicted value of α3(MZ) increases as μ0 decreases from 10 16 GeV to a TeV scale, and we obtain incorrect prediction for α3(MZ) [7, 8]. There are several works to obtain a realistic prediction of α3(MZ) leading to the experimental value [9], α3(MZ) = 0.119± 0.002, (1) e.g. by considering the non-canonical level of U(1)Y different from 5/3 or adding extra matter fields [10, 8] . In this paper we consider the case with the minimal matter content and the canonical level of U(1)Y equal to 5/3, and we take into account the correction due to the higher dimensional operator, e.g. c(Φ/M)FF , where c is a coupling constant and M is the cut-off energy scale. We consider the SU(5) grand unified theory (GUT) with the 24 Higgs field Φ as our framework. Within this framework, we have the correction term to the gauge kinetic term [12] , − 1 4 c < Φαβ > M F α μνF βμν . (2) Note that the vacuum expectation value < Φαβ >∝ diag(1/2 √ 15)(2, 2, 2,−3,−3), (3) which corresponds to the SU(3) × SU(2) × U(1)Y preserving direction, contributes to the See also Ref. [11]. In Ref. [13], corrections due to Higgs fields with larger representations are discussed within the framework of the four-dimensional supersymmetric SU(5) GUTs.