Supersymmetric grand unified theories: Two-loop evolution of gauge and Yukawa couplings

Abstract

We make a numerical study of gauge and Yukawa unification in supersymmetric grand unified models and examine the quantitative implications of fermion mass Ansatze at the grand unified scale. Integrating the renormalization group equations with al(Mz) and az(Mz) as inputs, we find as(Mz) --O.lll(0.122) for Msusu = rnt and a3(Mz) N 0.106(0.116) for Msusv = 1 TeV at oneloop (two-loop) order. Including b and T Yukawa couplings in the evolution, we find an upper limit of mt 5 200 GeV from Yukawa unification. For given mt 5 175 GeV, there are two solutions for P: one with tan p > mtlrnb, and one with sin 0 -. 0.78(mt/150 GeV). Taking a popular Ansatz for the mass matrices at the unified scale, we obtain a lower limit on the topquark mass of mt 2 150(115) GeV for a 3 ( M~) = O.ll(0.12) and an upper limit on the supersymmetry parameter tanp 5 50 if a~( M z ) = 0.11. The evolution of the quark mixing matrix elements is also evaluated. PACS number(s): 12.10.Dm, 12.15.Ff I. I N T R O D U C T I O NThere is renewed interest in supersymmetric grand unified theories (GUT's) [I] t o explain gauge couplings, fermion masses, and quark mixings [2-91. Recent measurements of the gauge couplings at the CERN e+e-collider LEP and in other low-energy experiments [lo, 111 are in reasonably good accord with expectations from minimal supersymmetric GUT's with the scale of supersymmetry (SUSY) of order 1 TeV or below [2]. Supersymmetric GUT's are also consistent with the nonobservation to date of proton decay [12]. In addition to the unification of gauge couplings [13], the unification of Yukawa couplings has been considered to predict relations among quark masses [14-161. With equal b-quark and T-lepton Yukawa couplings at the GUT scale, the mb/m, mass ratio is explained by SUSY GUT's [4,15]. With specific Ansatze for the GUT-scale mass matrices (e.g., zero elements, mass hierarchy, relations of quark and lepton elements), other predictions have been obtained from quark masses and mixings that are consistent with measurements [4,6, 7, 17, 181. The consideration of fermion mass relationships has a long history [19-211 and includes single relations and mass matrices ("textures") without evolution [22-241, and single relations and mass matrices with evolution [25].Our approach is to explore supersymmetric GUT's first with the most general assumptions, and then proceed to add additional GUT unification constraints to obtain more predictions at the electroweak scale. The renormalization-group equations (RGE's) used here are for the supersymmetric GUT's [26, 271 with the minimal particle content above the supersymmetry scale and the standard model RGE's [28] below the supersymmetry scale. In Sec. I1 we explore the running of the gauge couplings in the supersymmetric model at the two-loop level and compare the results to those obtained at the one-loop level. Rather than try to predict the scale of supersymmetry (Msusu) which may be sensitive to unknown and model-dependent effects such as particle thresholds at the GUT scale, we choose two value...

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